using bagged posteriors for robust inference
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Using Bagged Posteriors for Robust Inference Jonathan Huggins - PowerPoint PPT Presentation

Dec 07, 2023 •150 likes •1.72k views

Using Bagged Posteriors for Robust Inference Jonathan Huggins Harvard University Joint work with Je ff Miller 1 Bagged posterior corrects for model misspecification [ H & Miller 2019] 2 Bagged posterior corrects for model

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  4. <latexit sha1_base64="hH24RH9xGP3+VSMItbuKC+PC0aA=">AB/3icbVDLTgJBEJz1ifhCPXqZSEy8SHbRI4kXjxCIg8DGzI79MLIzOxmZtaEbDh49qrf4M149VP8BP/CAfYgYCWdVKq6090VxJxp47rfztr6xubWdm4nv7u3f3BYODpu6ihRFBo04pFqB0QDZxIahkO7VgBEQGHVjC6nfqtJ1CaRfLejGPwBRlIFjJKjJXqD71C0S25M+BV4mWkiDLUeoWfbj+iQBpKCdadzw3Nn5KlGUwyTfTEhI7IADqWSiJA+ns0Ak+t0ofh5GyJQ2eqX8nUiK0HovAdgpihnrZm4r/eZ3EhBU/ZTJODEg6XxQmHJsIT7/GfaAGj62hFDF7K2YDoki1NhsFrY8DoeXgmlanuRtNt5yEqukWS5V6Vy/bpYrWQp5dApOkMXyEM3qIruUA01EWAXtArenOenXfnw/mct6452cwJWoDz9QvBKpYd</latexit> <latexit sha1_base64="20pbYpT3sLDj1aIwDWwHqkeyJo8=">ACBnicbVDLSsNAFJ3UV62vqks3g0VwY0mqYJcFNy4r2Ie0oUymk2bsPMLMRCghe9du9RvciVt/w0/wL5y2WdjWAxcO59zLvfcEMaPauO63U1hb39jcKm6Xdnb39g/Kh0dtLROFSQtLJlU3QJowKkjLUMNIN1YE8YCRTjC+mfqdJ6I0leLeTGLiczQSNKQYGSt1HwZpIKXJBuWKW3VngKvEy0kF5GgOyj/9ocQJ8JghrTueW5s/BQpQzEjWamfaBIjPEYj0rNUIE60n87uzeCZVYwlMqWMHCm/p1IEd6wgPbyZGJ9LI3Ff/zeokJ635KRZwYIvB8UZgwaCScPg+HVBFs2MQShBW1t0IcIYWwsREtbHmMogtONa5lJZuNt5zEKmnXqt5ltXZ3VWnU85SK4AScgnPgWvQALegCVoAwZewCt4c56d+fD+Zy3Fpx85hgswPn6BbJrmW4=</latexit> <latexit sha1_base64="yNrYmQo/EdIzEea4bdhMv6/bSc=">ACD3icbVDLSgNBEJz1GeMjqx69LAbBi2E3CnoMevEYwTwgCWF20puMmX0w0yOGJR/h2at+gzfx6if4Cf6Fk2QPJrGgoajqpryE8EVu63tbK6tr6xmdvKb+/s7hXs/YO6irVkUGOxiGXTpwoEj6CGHAU0Ewk09AU0/OHNxG8glQ8ju5xlEAnpP2IB5xRNFLXLlS7bYQnVEGKUsO4axfdkjuFs0y8jBRJhmrX/mn3YqZDiJAJqlTLcxPspFQiZwLG+bZWkFA2pH1oGRrREFQnT4+dk6M0nOCWJqJ0Jmqfy9SGio1Cn2zGVIcqEVvIv7ntTQGV52UR4lGiNgsKNDCwdiZtOD0uASGYmQIZKbXx02oJIyNF3NpTwMBmchV6w8zptuvMUmlkm9XPLOS+W7i2LlOmspR47IMTklHrkFXJLqRGNHkhbySN+vZerc+rM/Z6oqV3RySOVhfvx9jnOw=</latexit> <latexit sha1_base64="gJPiAjmVJN6pCfYEi9mg/afJSB4=">ACEHicbVDLSgNBEJz1GeNr1aOXxSDEg2E3CuY8OIxgolKEsLspDcZnZldZnqDYclPePaq3+BNvPoHfoJ/4STuwVdBQ1HVTUVJoIb9P13Z25+YXFpubBSXF1b39h0t7ZbJk41gyaLRayvQmpAcAVN5CjgKtFAZSjgMrw9nfqXI9CGx+oCxwl0JR0oHnFG0Uo91+0g3KGJMglUTcrXBz235Ff8Gby/JMhJieRo9NyPTj9mqQSFTFBj2oGfYDejGjkTMCl2UgMJZbd0AG1LFZVgutns84m3b5W+F8XajkJvpn6/yKg0ZixDuykpDs1vbyr+57VTjGrdjKskRVDsKyhKhYexN63B63MNDMXYEso0t796bEg1ZWjL+pFyMxweSm5YdVK03QS/m/hLWtVKcFSpnh+X6rW8pQLZJXukTAJyQurkjDRIkzAyIg/kTw5986z8+K8fq3OfnNDvkB5+0TD5icyA=</latexit> <latexit sha1_base64="gJPiAjmVJN6pCfYEi9mg/afJSB4=">ACEHicbVDLSgNBEJz1GeNr1aOXxSDEg2E3CuY8OIxgolKEsLspDcZnZldZnqDYclPePaq3+BNvPoHfoJ/4STuwVdBQ1HVTUVJoIb9P13Z25+YXFpubBSXF1b39h0t7ZbJk41gyaLRayvQmpAcAVN5CjgKtFAZSjgMrw9nfqXI9CGx+oCxwl0JR0oHnFG0Uo91+0g3KGJMglUTcrXBz235Ff8Gby/JMhJieRo9NyPTj9mqQSFTFBj2oGfYDejGjkTMCl2UgMJZbd0AG1LFZVgutns84m3b5W+F8XajkJvpn6/yKg0ZixDuykpDs1vbyr+57VTjGrdjKskRVDsKyhKhYexN63B63MNDMXYEso0t796bEg1ZWjL+pFyMxweSm5YdVK03QS/m/hLWtVKcFSpnh+X6rW8pQLZJXukTAJyQurkjDRIkzAyIg/kTw5986z8+K8fq3OfnNDvkB5+0TD5icyA=</latexit> Bootstrapping • Data Y = ( Y 1 , …, Y n ) , where Y i ~ P true mean ( Y ) P true • Interested in parameter that best explains distribution [e.g. mean of independent normal observations] Y • Want sampling uncertainty [e.g. distribution of mean ( Y ) under P true ] Y boot distribution of… mean ( Y ) 5 [Efron 1979]

  5. <latexit sha1_base64="hH24RH9xGP3+VSMItbuKC+PC0aA=">AB/3icbVDLTgJBEJz1ifhCPXqZSEy8SHbRI4kXjxCIg8DGzI79MLIzOxmZtaEbDh49qrf4M149VP8BP/CAfYgYCWdVKq6090VxJxp47rfztr6xubWdm4nv7u3f3BYODpu6ihRFBo04pFqB0QDZxIahkO7VgBEQGHVjC6nfqtJ1CaRfLejGPwBRlIFjJKjJXqD71C0S25M+BV4mWkiDLUeoWfbj+iQBpKCdadzw3Nn5KlGUwyTfTEhI7IADqWSiJA+ns0Ak+t0ofh5GyJQ2eqX8nUiK0HovAdgpihnrZm4r/eZ3EhBU/ZTJODEg6XxQmHJsIT7/GfaAGj62hFDF7K2YDoki1NhsFrY8DoeXgmlanuRtNt5yEqukWS5V6Vy/bpYrWQp5dApOkMXyEM3qIruUA01EWAXtArenOenXfnw/mct6452cwJWoDz9QvBKpYd</latexit> <latexit sha1_base64="20pbYpT3sLDj1aIwDWwHqkeyJo8=">ACBnicbVDLSsNAFJ3UV62vqks3g0VwY0mqYJcFNy4r2Ie0oUymk2bsPMLMRCghe9du9RvciVt/w0/wL5y2WdjWAxcO59zLvfcEMaPauO63U1hb39jcKm6Xdnb39g/Kh0dtLROFSQtLJlU3QJowKkjLUMNIN1YE8YCRTjC+mfqdJ6I0leLeTGLiczQSNKQYGSt1HwZpIKXJBuWKW3VngKvEy0kF5GgOyj/9ocQJ8JghrTueW5s/BQpQzEjWamfaBIjPEYj0rNUIE60n87uzeCZVYwlMqWMHCm/p1IEd6wgPbyZGJ9LI3Ff/zeokJ635KRZwYIvB8UZgwaCScPg+HVBFs2MQShBW1t0IcIYWwsREtbHmMogtONa5lJZuNt5zEKmnXqt5ltXZ3VWnU85SK4AScgnPgWvQALegCVoAwZewCt4c56d+fD+Zy3Fpx85hgswPn6BbJrmW4=</latexit> <latexit sha1_base64="n170n2F0XF87NI7Ry/Z7rxiNpA=">ACAXicbVDLTgJBEOzF+IL9ehlIjHxItlFEz0SvXjEKI8ENmR2mIWRmdnNzKwJ2XDy7FW/wZvx6pf4Cf6FA+xBwEo6qVR1p7sriDnTxnW/ndzK6tr6Rn6zsLW9s7tX3D9o6ChRhNZJxCPVCrCmnElaN8xw2oVxSLgtBkMbyZ+84kqzSL5YEYx9QXuSxYygo2V7mtd2S2W3LI7BVomXkZKkKHWLf50ehFJBJWGcKx123Nj46dYGUY4HRc6iaYxJkPcp21LJRZU+n01DE6sUoPhZGyJQ2aqn8nUiy0HonAdgpsBnrRm4j/e3EhFd+ymScGCrJbFGYcGQiNPkb9ZixPCRJZgoZm9FZIAVJsamM7flcTA4E0yTyrhgs/EWk1gmjUrZOy9X7i5K1espTwcwTGcgeXUIVbqEdCPThBV7hzXl23p0P53PWmnOymUOYg/P1C0ntlv8=</latexit> <latexit sha1_base64="gJPiAjmVJN6pCfYEi9mg/afJSB4=">ACEHicbVDLSgNBEJz1GeNr1aOXxSDEg2E3CuY8OIxgolKEsLspDcZnZldZnqDYclPePaq3+BNvPoHfoJ/4STuwVdBQ1HVTUVJoIb9P13Z25+YXFpubBSXF1b39h0t7ZbJk41gyaLRayvQmpAcAVN5CjgKtFAZSjgMrw9nfqXI9CGx+oCxwl0JR0oHnFG0Uo91+0g3KGJMglUTcrXBz235Ff8Gby/JMhJieRo9NyPTj9mqQSFTFBj2oGfYDejGjkTMCl2UgMJZbd0AG1LFZVgutns84m3b5W+F8XajkJvpn6/yKg0ZixDuykpDs1vbyr+57VTjGrdjKskRVDsKyhKhYexN63B63MNDMXYEso0t796bEg1ZWjL+pFyMxweSm5YdVK03QS/m/hLWtVKcFSpnh+X6rW8pQLZJXukTAJyQurkjDRIkzAyIg/kTw5986z8+K8fq3OfnNDvkB5+0TD5icyA=</latexit> <latexit sha1_base64="gJPiAjmVJN6pCfYEi9mg/afJSB4=">ACEHicbVDLSgNBEJz1GeNr1aOXxSDEg2E3CuY8OIxgolKEsLspDcZnZldZnqDYclPePaq3+BNvPoHfoJ/4STuwVdBQ1HVTUVJoIb9P13Z25+YXFpubBSXF1b39h0t7ZbJk41gyaLRayvQmpAcAVN5CjgKtFAZSjgMrw9nfqXI9CGx+oCxwl0JR0oHnFG0Uo91+0g3KGJMglUTcrXBz235Ff8Gby/JMhJieRo9NyPTj9mqQSFTFBj2oGfYDejGjkTMCl2UgMJZbd0AG1LFZVgutns84m3b5W+F8XajkJvpn6/yKg0ZixDuykpDs1vbyr+57VTjGrdjKskRVDsKyhKhYexN63B63MNDMXYEso0t796bEg1ZWjL+pFyMxweSm5YdVK03QS/m/hLWtVKcFSpnh+X6rW8pQLZJXukTAJyQurkjDRIkzAyIg/kTw5986z8+K8fq3OfnNDvkB5+0TD5icyA=</latexit> <latexit sha1_base64="yNrYmQo/EdIzEea4bdhMv6/bSc=">ACD3icbVDLSgNBEJz1GeMjqx69LAbBi2E3CnoMevEYwTwgCWF20puMmX0w0yOGJR/h2at+gzfx6if4Cf6Fk2QPJrGgoajqpryE8EVu63tbK6tr6xmdvKb+/s7hXs/YO6irVkUGOxiGXTpwoEj6CGHAU0Ewk09AU0/OHNxG8glQ8ju5xlEAnpP2IB5xRNFLXLlS7bYQnVEGKUsO4axfdkjuFs0y8jBRJhmrX/mn3YqZDiJAJqlTLcxPspFQiZwLG+bZWkFA2pH1oGRrREFQnT4+dk6M0nOCWJqJ0Jmqfy9SGio1Cn2zGVIcqEVvIv7ntTQGV52UR4lGiNgsKNDCwdiZtOD0uASGYmQIZKbXx02oJIyNF3NpTwMBmchV6w8zptuvMUmlkm9XPLOS+W7i2LlOmspR47IMTklHrkFXJLqRGNHkhbySN+vZerc+rM/Z6oqV3RySOVhfvx9jnOw=</latexit> Bootstrapping • Data Y = ( Y 1 , …, Y n ) , where Y i ~ P true mean ( Y ) P true • Interested in parameter that best P n explains distribution [e.g. mean of independent normal observations] Y • Want sampling uncertainty [e.g. distribution of mean ( Y ) under P true ] Y boot • Bootstrap: replace P true with P n distribution of… mean ( Y ) 5 [Efron 1979]

  6. <latexit sha1_base64="hH24RH9xGP3+VSMItbuKC+PC0aA=">AB/3icbVDLTgJBEJz1ifhCPXqZSEy8SHbRI4kXjxCIg8DGzI79MLIzOxmZtaEbDh49qrf4M149VP8BP/CAfYgYCWdVKq6090VxJxp47rfztr6xubWdm4nv7u3f3BYODpu6ihRFBo04pFqB0QDZxIahkO7VgBEQGHVjC6nfqtJ1CaRfLejGPwBRlIFjJKjJXqD71C0S25M+BV4mWkiDLUeoWfbj+iQBpKCdadzw3Nn5KlGUwyTfTEhI7IADqWSiJA+ns0Ak+t0ofh5GyJQ2eqX8nUiK0HovAdgpihnrZm4r/eZ3EhBU/ZTJODEg6XxQmHJsIT7/GfaAGj62hFDF7K2YDoki1NhsFrY8DoeXgmlanuRtNt5yEqukWS5V6Vy/bpYrWQp5dApOkMXyEM3qIruUA01EWAXtArenOenXfnw/mct6452cwJWoDz9QvBKpYd</latexit> <latexit sha1_base64="20pbYpT3sLDj1aIwDWwHqkeyJo8=">ACBnicbVDLSsNAFJ3UV62vqks3g0VwY0mqYJcFNy4r2Ie0oUymk2bsPMLMRCghe9du9RvciVt/w0/wL5y2WdjWAxcO59zLvfcEMaPauO63U1hb39jcKm6Xdnb39g/Kh0dtLROFSQtLJlU3QJowKkjLUMNIN1YE8YCRTjC+mfqdJ6I0leLeTGLiczQSNKQYGSt1HwZpIKXJBuWKW3VngKvEy0kF5GgOyj/9ocQJ8JghrTueW5s/BQpQzEjWamfaBIjPEYj0rNUIE60n87uzeCZVYwlMqWMHCm/p1IEd6wgPbyZGJ9LI3Ff/zeokJ635KRZwYIvB8UZgwaCScPg+HVBFs2MQShBW1t0IcIYWwsREtbHmMogtONa5lJZuNt5zEKmnXqt5ltXZ3VWnU85SK4AScgnPgWvQALegCVoAwZewCt4c56d+fD+Zy3Fpx85hgswPn6BbJrmW4=</latexit> <latexit sha1_base64="n170n2F0XF87NI7Ry/Z7rxiNpA=">ACAXicbVDLTgJBEOzF+IL9ehlIjHxItlFEz0SvXjEKI8ENmR2mIWRmdnNzKwJ2XDy7FW/wZvx6pf4Cf6FA+xBwEo6qVR1p7sriDnTxnW/ndzK6tr6Rn6zsLW9s7tX3D9o6ChRhNZJxCPVCrCmnElaN8xw2oVxSLgtBkMbyZ+84kqzSL5YEYx9QXuSxYygo2V7mtd2S2W3LI7BVomXkZKkKHWLf50ehFJBJWGcKx123Nj46dYGUY4HRc6iaYxJkPcp21LJRZU+n01DE6sUoPhZGyJQ2aqn8nUiy0HonAdgpsBnrRm4j/e3EhFd+ymScGCrJbFGYcGQiNPkb9ZixPCRJZgoZm9FZIAVJsamM7flcTA4E0yTyrhgs/EWk1gmjUrZOy9X7i5K1espTwcwTGcgeXUIVbqEdCPThBV7hzXl23p0P53PWmnOymUOYg/P1C0ntlv8=</latexit> <latexit sha1_base64="gJPiAjmVJN6pCfYEi9mg/afJSB4=">ACEHicbVDLSgNBEJz1GeNr1aOXxSDEg2E3CuY8OIxgolKEsLspDcZnZldZnqDYclPePaq3+BNvPoHfoJ/4STuwVdBQ1HVTUVJoIb9P13Z25+YXFpubBSXF1b39h0t7ZbJk41gyaLRayvQmpAcAVN5CjgKtFAZSjgMrw9nfqXI9CGx+oCxwl0JR0oHnFG0Uo91+0g3KGJMglUTcrXBz235Ff8Gby/JMhJieRo9NyPTj9mqQSFTFBj2oGfYDejGjkTMCl2UgMJZbd0AG1LFZVgutns84m3b5W+F8XajkJvpn6/yKg0ZixDuykpDs1vbyr+57VTjGrdjKskRVDsKyhKhYexN63B63MNDMXYEso0t796bEg1ZWjL+pFyMxweSm5YdVK03QS/m/hLWtVKcFSpnh+X6rW8pQLZJXukTAJyQurkjDRIkzAyIg/kTw5986z8+K8fq3OfnNDvkB5+0TD5icyA=</latexit> <latexit sha1_base64="gJPiAjmVJN6pCfYEi9mg/afJSB4=">ACEHicbVDLSgNBEJz1GeNr1aOXxSDEg2E3CuY8OIxgolKEsLspDcZnZldZnqDYclPePaq3+BNvPoHfoJ/4STuwVdBQ1HVTUVJoIb9P13Z25+YXFpubBSXF1b39h0t7ZbJk41gyaLRayvQmpAcAVN5CjgKtFAZSjgMrw9nfqXI9CGx+oCxwl0JR0oHnFG0Uo91+0g3KGJMglUTcrXBz235Ff8Gby/JMhJieRo9NyPTj9mqQSFTFBj2oGfYDejGjkTMCl2UgMJZbd0AG1LFZVgutns84m3b5W+F8XajkJvpn6/yKg0ZixDuykpDs1vbyr+57VTjGrdjKskRVDsKyhKhYexN63B63MNDMXYEso0t796bEg1ZWjL+pFyMxweSm5YdVK03QS/m/hLWtVKcFSpnh+X6rW8pQLZJXukTAJyQurkjDRIkzAyIg/kTw5986z8+K8fq3OfnNDvkB5+0TD5icyA=</latexit> <latexit sha1_base64="yNrYmQo/EdIzEea4bdhMv6/bSc=">ACD3icbVDLSgNBEJz1GeMjqx69LAbBi2E3CnoMevEYwTwgCWF20puMmX0w0yOGJR/h2at+gzfx6if4Cf6Fk2QPJrGgoajqpryE8EVu63tbK6tr6xmdvKb+/s7hXs/YO6irVkUGOxiGXTpwoEj6CGHAU0Ewk09AU0/OHNxG8glQ8ju5xlEAnpP2IB5xRNFLXLlS7bYQnVEGKUsO4axfdkjuFs0y8jBRJhmrX/mn3YqZDiJAJqlTLcxPspFQiZwLG+bZWkFA2pH1oGRrREFQnT4+dk6M0nOCWJqJ0Jmqfy9SGio1Cn2zGVIcqEVvIv7ntTQGV52UR4lGiNgsKNDCwdiZtOD0uASGYmQIZKbXx02oJIyNF3NpTwMBmchV6w8zptuvMUmlkm9XPLOS+W7i2LlOmspR47IMTklHrkFXJLqRGNHkhbySN+vZerc+rM/Z6oqV3RySOVhfvx9jnOw=</latexit> Bootstrapping • Data Y = ( Y 1 , …, Y n ) , where Y i ~ P true mean ( Y ) P true • Interested in parameter that best P n explains distribution [e.g. mean of independent normal observations] Y • Want sampling uncertainty [e.g. distribution of mean ( Y ) under P true ] Y boot • Bootstrap: replace P true with P n distribution of… mean ( Y ) 5 [Efron 1979]

  7. <latexit sha1_base64="Iesqmfu0rBSKVSkuUwpJZrxV6B0=">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</latexit> <latexit sha1_base64="gJPiAjmVJN6pCfYEi9mg/afJSB4=">ACEHicbVDLSgNBEJz1GeNr1aOXxSDEg2E3CuY8OIxgolKEsLspDcZnZldZnqDYclPePaq3+BNvPoHfoJ/4STuwVdBQ1HVTUVJoIb9P13Z25+YXFpubBSXF1b39h0t7ZbJk41gyaLRayvQmpAcAVN5CjgKtFAZSjgMrw9nfqXI9CGx+oCxwl0JR0oHnFG0Uo91+0g3KGJMglUTcrXBz235Ff8Gby/JMhJieRo9NyPTj9mqQSFTFBj2oGfYDejGjkTMCl2UgMJZbd0AG1LFZVgutns84m3b5W+F8XajkJvpn6/yKg0ZixDuykpDs1vbyr+57VTjGrdjKskRVDsKyhKhYexN63B63MNDMXYEso0t796bEg1ZWjL+pFyMxweSm5YdVK03QS/m/hLWtVKcFSpnh+X6rW8pQLZJXukTAJyQurkjDRIkzAyIg/kTw5986z8+K8fq3OfnNDvkB5+0TD5icyA=</latexit> <latexit sha1_base64="yNrYmQo/EdIzEea4bdhMv6/bSc=">ACD3icbVDLSgNBEJz1GeMjqx69LAbBi2E3CnoMevEYwTwgCWF20puMmX0w0yOGJR/h2at+gzfx6if4Cf6Fk2QPJrGgoajqpryE8EVu63tbK6tr6xmdvKb+/s7hXs/YO6irVkUGOxiGXTpwoEj6CGHAU0Ewk09AU0/OHNxG8glQ8ju5xlEAnpP2IB5xRNFLXLlS7bYQnVEGKUsO4axfdkjuFs0y8jBRJhmrX/mn3YqZDiJAJqlTLcxPspFQiZwLG+bZWkFA2pH1oGRrREFQnT4+dk6M0nOCWJqJ0Jmqfy9SGio1Cn2zGVIcqEVvIv7ntTQGV52UR4lGiNgsKNDCwdiZtOD0uASGYmQIZKbXx02oJIyNF3NpTwMBmchV6w8zptuvMUmlkm9XPLOS+W7i2LlOmspR47IMTklHrkFXJLqRGNHkhbySN+vZerc+rM/Z6oqV3RySOVhfvx9jnOw=</latexit> <latexit sha1_base64="hH24RH9xGP3+VSMItbuKC+PC0aA=">AB/3icbVDLTgJBEJz1ifhCPXqZSEy8SHbRI4kXjxCIg8DGzI79MLIzOxmZtaEbDh49qrf4M149VP8BP/CAfYgYCWdVKq6090VxJxp47rfztr6xubWdm4nv7u3f3BYODpu6ihRFBo04pFqB0QDZxIahkO7VgBEQGHVjC6nfqtJ1CaRfLejGPwBRlIFjJKjJXqD71C0S25M+BV4mWkiDLUeoWfbj+iQBpKCdadzw3Nn5KlGUwyTfTEhI7IADqWSiJA+ns0Ak+t0ofh5GyJQ2eqX8nUiK0HovAdgpihnrZm4r/eZ3EhBU/ZTJODEg6XxQmHJsIT7/GfaAGj62hFDF7K2YDoki1NhsFrY8DoeXgmlanuRtNt5yEqukWS5V6Vy/bpYrWQp5dApOkMXyEM3qIruUA01EWAXtArenOenXfnw/mct6452cwJWoDz9QvBKpYd</latexit> <latexit sha1_base64="n170n2F0XF87NI7Ry/Z7rxiNpA=">ACAXicbVDLTgJBEOzF+IL9ehlIjHxItlFEz0SvXjEKI8ENmR2mIWRmdnNzKwJ2XDy7FW/wZvx6pf4Cf6FA+xBwEo6qVR1p7sriDnTxnW/ndzK6tr6Rn6zsLW9s7tX3D9o6ChRhNZJxCPVCrCmnElaN8xw2oVxSLgtBkMbyZ+84kqzSL5YEYx9QXuSxYygo2V7mtd2S2W3LI7BVomXkZKkKHWLf50ehFJBJWGcKx123Nj46dYGUY4HRc6iaYxJkPcp21LJRZU+n01DE6sUoPhZGyJQ2aqn8nUiy0HonAdgpsBnrRm4j/e3EhFd+ymScGCrJbFGYcGQiNPkb9ZixPCRJZgoZm9FZIAVJsamM7flcTA4E0yTyrhgs/EWk1gmjUrZOy9X7i5K1espTwcwTGcgeXUIVbqEdCPThBV7hzXl23p0P53PWmnOymUOYg/P1C0ntlv8=</latexit> <latexit sha1_base64="gJPiAjmVJN6pCfYEi9mg/afJSB4=">ACEHicbVDLSgNBEJz1GeNr1aOXxSDEg2E3CuY8OIxgolKEsLspDcZnZldZnqDYclPePaq3+BNvPoHfoJ/4STuwVdBQ1HVTUVJoIb9P13Z25+YXFpubBSXF1b39h0t7ZbJk41gyaLRayvQmpAcAVN5CjgKtFAZSjgMrw9nfqXI9CGx+oCxwl0JR0oHnFG0Uo91+0g3KGJMglUTcrXBz235Ff8Gby/JMhJieRo9NyPTj9mqQSFTFBj2oGfYDejGjkTMCl2UgMJZbd0AG1LFZVgutns84m3b5W+F8XajkJvpn6/yKg0ZixDuykpDs1vbyr+57VTjGrdjKskRVDsKyhKhYexN63B63MNDMXYEso0t796bEg1ZWjL+pFyMxweSm5YdVK03QS/m/hLWtVKcFSpnh+X6rW8pQLZJXukTAJyQurkjDRIkzAyIg/kTw5986z8+K8fq3OfnNDvkB5+0TD5icyA=</latexit> <latexit sha1_base64="20pbYpT3sLDj1aIwDWwHqkeyJo8=">ACBnicbVDLSsNAFJ3UV62vqks3g0VwY0mqYJcFNy4r2Ie0oUymk2bsPMLMRCghe9du9RvciVt/w0/wL5y2WdjWAxcO59zLvfcEMaPauO63U1hb39jcKm6Xdnb39g/Kh0dtLROFSQtLJlU3QJowKkjLUMNIN1YE8YCRTjC+mfqdJ6I0leLeTGLiczQSNKQYGSt1HwZpIKXJBuWKW3VngKvEy0kF5GgOyj/9ocQJ8JghrTueW5s/BQpQzEjWamfaBIjPEYj0rNUIE60n87uzeCZVYwlMqWMHCm/p1IEd6wgPbyZGJ9LI3Ff/zeokJ635KRZwYIvB8UZgwaCScPg+HVBFs2MQShBW1t0IcIYWwsREtbHmMogtONa5lJZuNt5zEKmnXqt5ltXZ3VWnU85SK4AScgnPgWvQALegCVoAwZewCt4c56d+fD+Zy3Fpx85hgswPn6BbJrmW4=</latexit> Bootstrapping • Data Y = ( Y 1 , …, Y n ) , where Y i ~ P true mean ( Y ) P true • Interested in parameter that best P n explains distribution [e.g. mean of independent normal observations] Y • Want sampling uncertainty [e.g. distribution of mean ( Y ) under P true ] Y boot • Bootstrap: replace P true with P n mean ( Y (1) mean ( Y (2) boot ) · · · mean ( Y ( B ) boot ) boot ) • Sample B bootstrap datasets to get distribution of… empirical distribution [e.g. mean ( Y boot )] mean ( Y ) 5 [Efron 1979]

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  10. <latexit sha1_base64="Fv0JrZMx0DwM91zg0r9FPy/3dps=">ACKnicbVBNSyNBEO1xXWzq8bdo5fGILgLG2aiYI4BL3tUMDEhCaGnU5O09sfQXRMw/wHf4hnr+5v2Fvw6sV/sZ2YhfXjQcHjvSq6sWpFA7DcBasfFj9uLa+8an0+cvm1nZ52vLmcxyaHIjW3HzIEUGpoUEI7tcBULOEivjqZ+xcTsE4YfY7TFPqKjbRIBGfopUH5Rw/hGl2SK2C6OgM8tgYL7TfzodiQloWtDOoFwJq+EC9C2JlqRCljgdlJ96Q8MzBRq5ZM51ozDFfs4sCi6hKPUyBynjV2wEXU81U+D6+eKngu57ZUgTY31pAv1/4mcKemKvadiuHYvfbm4nteN8Ok3s+FTjMEzZ8XJZmkaOg8IDoUFjKqSeMW+FvpXzMLOPoY3yx5XI8/qmE47Wi5LOJXifxlrRq1eiwWjs7qjTqy5Q2yC7ZIwckIsekQX6RU9IknNyQO3JPfge3wZ9gFjw8t64Ey5lv5AWCx7/0H6e7</latexit> <latexit sha1_base64="gJPiAjmVJN6pCfYEi9mg/afJSB4=">ACEHicbVDLSgNBEJz1GeNr1aOXxSDEg2E3CuY8OIxgolKEsLspDcZnZldZnqDYclPePaq3+BNvPoHfoJ/4STuwVdBQ1HVTUVJoIb9P13Z25+YXFpubBSXF1b39h0t7ZbJk41gyaLRayvQmpAcAVN5CjgKtFAZSjgMrw9nfqXI9CGx+oCxwl0JR0oHnFG0Uo91+0g3KGJMglUTcrXBz235Ff8Gby/JMhJieRo9NyPTj9mqQSFTFBj2oGfYDejGjkTMCl2UgMJZbd0AG1LFZVgutns84m3b5W+F8XajkJvpn6/yKg0ZixDuykpDs1vbyr+57VTjGrdjKskRVDsKyhKhYexN63B63MNDMXYEso0t796bEg1ZWjL+pFyMxweSm5YdVK03QS/m/hLWtVKcFSpnh+X6rW8pQLZJXukTAJyQurkjDRIkzAyIg/kTw5986z8+K8fq3OfnNDvkB5+0TD5icyA=</latexit> <latexit sha1_base64="yNrYmQo/EdIzEea4bdhMv6/bSc=">ACD3icbVDLSgNBEJz1GeMjqx69LAbBi2E3CnoMevEYwTwgCWF20puMmX0w0yOGJR/h2at+gzfx6if4Cf6Fk2QPJrGgoajqpryE8EVu63tbK6tr6xmdvKb+/s7hXs/YO6irVkUGOxiGXTpwoEj6CGHAU0Ewk09AU0/OHNxG8glQ8ju5xlEAnpP2IB5xRNFLXLlS7bYQnVEGKUsO4axfdkjuFs0y8jBRJhmrX/mn3YqZDiJAJqlTLcxPspFQiZwLG+bZWkFA2pH1oGRrREFQnT4+dk6M0nOCWJqJ0Jmqfy9SGio1Cn2zGVIcqEVvIv7ntTQGV52UR4lGiNgsKNDCwdiZtOD0uASGYmQIZKbXx02oJIyNF3NpTwMBmchV6w8zptuvMUmlkm9XPLOS+W7i2LlOmspR47IMTklHrkFXJLqRGNHkhbySN+vZerc+rM/Z6oqV3RySOVhfvx9jnOw=</latexit> <latexit sha1_base64="Iesqmfu0rBSKVSkuUwpJZrxV6B0=">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</latexit> <latexit sha1_base64="hH24RH9xGP3+VSMItbuKC+PC0aA=">AB/3icbVDLTgJBEJz1ifhCPXqZSEy8SHbRI4kXjxCIg8DGzI79MLIzOxmZtaEbDh49qrf4M149VP8BP/CAfYgYCWdVKq6090VxJxp47rfztr6xubWdm4nv7u3f3BYODpu6ihRFBo04pFqB0QDZxIahkO7VgBEQGHVjC6nfqtJ1CaRfLejGPwBRlIFjJKjJXqD71C0S25M+BV4mWkiDLUeoWfbj+iQBpKCdadzw3Nn5KlGUwyTfTEhI7IADqWSiJA+ns0Ak+t0ofh5GyJQ2eqX8nUiK0HovAdgpihnrZm4r/eZ3EhBU/ZTJODEg6XxQmHJsIT7/GfaAGj62hFDF7K2YDoki1NhsFrY8DoeXgmlanuRtNt5yEqukWS5V6Vy/bpYrWQp5dApOkMXyEM3qIruUA01EWAXtArenOenXfnw/mct6452cwJWoDz9QvBKpYd</latexit> <latexit sha1_base64="n170n2F0XF87NI7Ry/Z7rxiNpA=">ACAXicbVDLTgJBEOzF+IL9ehlIjHxItlFEz0SvXjEKI8ENmR2mIWRmdnNzKwJ2XDy7FW/wZvx6pf4Cf6FA+xBwEo6qVR1p7sriDnTxnW/ndzK6tr6Rn6zsLW9s7tX3D9o6ChRhNZJxCPVCrCmnElaN8xw2oVxSLgtBkMbyZ+84kqzSL5YEYx9QXuSxYygo2V7mtd2S2W3LI7BVomXkZKkKHWLf50ehFJBJWGcKx123Nj46dYGUY4HRc6iaYxJkPcp21LJRZU+n01DE6sUoPhZGyJQ2aqn8nUiy0HonAdgpsBnrRm4j/e3EhFd+ymScGCrJbFGYcGQiNPkb9ZixPCRJZgoZm9FZIAVJsamM7flcTA4E0yTyrhgs/EWk1gmjUrZOy9X7i5K1espTwcwTGcgeXUIVbqEdCPThBV7hzXl23p0P53PWmnOymUOYg/P1C0ntlv8=</latexit> <latexit sha1_base64="20pbYpT3sLDj1aIwDWwHqkeyJo8=">ACBnicbVDLSsNAFJ3UV62vqks3g0VwY0mqYJcFNy4r2Ie0oUymk2bsPMLMRCghe9du9RvciVt/w0/wL5y2WdjWAxcO59zLvfcEMaPauO63U1hb39jcKm6Xdnb39g/Kh0dtLROFSQtLJlU3QJowKkjLUMNIN1YE8YCRTjC+mfqdJ6I0leLeTGLiczQSNKQYGSt1HwZpIKXJBuWKW3VngKvEy0kF5GgOyj/9ocQJ8JghrTueW5s/BQpQzEjWamfaBIjPEYj0rNUIE60n87uzeCZVYwlMqWMHCm/p1IEd6wgPbyZGJ9LI3Ff/zeokJ635KRZwYIvB8UZgwaCScPg+HVBFs2MQShBW1t0IcIYWwsREtbHmMogtONa5lJZuNt5zEKmnXqt5ltXZ3VWnU85SK4AScgnPgWvQALegCVoAwZewCt4c56d+fD+Zy3Fpx85hgswPn6BbJrmW4=</latexit> <latexit sha1_base64="gJPiAjmVJN6pCfYEi9mg/afJSB4=">ACEHicbVDLSgNBEJz1GeNr1aOXxSDEg2E3CuY8OIxgolKEsLspDcZnZldZnqDYclPePaq3+BNvPoHfoJ/4STuwVdBQ1HVTUVJoIb9P13Z25+YXFpubBSXF1b39h0t7ZbJk41gyaLRayvQmpAcAVN5CjgKtFAZSjgMrw9nfqXI9CGx+oCxwl0JR0oHnFG0Uo91+0g3KGJMglUTcrXBz235Ff8Gby/JMhJieRo9NyPTj9mqQSFTFBj2oGfYDejGjkTMCl2UgMJZbd0AG1LFZVgutns84m3b5W+F8XajkJvpn6/yKg0ZixDuykpDs1vbyr+57VTjGrdjKskRVDsKyhKhYexN63B63MNDMXYEso0t796bEg1ZWjL+pFyMxweSm5YdVK03QS/m/hLWtVKcFSpnh+X6rW8pQLZJXukTAJyQurkjDRIkzAyIg/kTw5986z8+K8fq3OfnNDvkB5+0TD5icyA=</latexit> Bootstrapping • Data Y = ( Y 1 , …, Y n ) , where Y i ~ P true mean ( Y ) P true • Interested in parameter that best P n explains distribution [e.g. mean of independent normal observations] Y • Want sampling uncertainty [e.g. distribution of mean ( Y ) under P true ] Y boot • Bootstrap: replace P true with P n mean ( Y (1) mean ( Y (2) boot ) · · · mean ( Y ( B ) boot ) boot ) • Sample B bootstrap datasets to get distribution of… empirical distribution [e.g. mean ( Y boot )] mean ( Y boot ) given Y • Benefits: no assumptions about P true , mean ( Y ) easy to use, can parallelize across B • Challenges: B large (1,000-100,000), finite-sample properties 5 [Efron 1979]

  11. <latexit sha1_base64="20pbYpT3sLDj1aIwDWwHqkeyJo8=">ACBnicbVDLSsNAFJ3UV62vqks3g0VwY0mqYJcFNy4r2Ie0oUymk2bsPMLMRCghe9du9RvciVt/w0/wL5y2WdjWAxcO59zLvfcEMaPauO63U1hb39jcKm6Xdnb39g/Kh0dtLROFSQtLJlU3QJowKkjLUMNIN1YE8YCRTjC+mfqdJ6I0leLeTGLiczQSNKQYGSt1HwZpIKXJBuWKW3VngKvEy0kF5GgOyj/9ocQJ8JghrTueW5s/BQpQzEjWamfaBIjPEYj0rNUIE60n87uzeCZVYwlMqWMHCm/p1IEd6wgPbyZGJ9LI3Ff/zeokJ635KRZwYIvB8UZgwaCScPg+HVBFs2MQShBW1t0IcIYWwsREtbHmMogtONa5lJZuNt5zEKmnXqt5ltXZ3VWnU85SK4AScgnPgWvQALegCVoAwZewCt4c56d+fD+Zy3Fpx85hgswPn6BbJrmW4=</latexit> <latexit sha1_base64="yNrYmQo/EdIzEea4bdhMv6/bSc=">ACD3icbVDLSgNBEJz1GeMjqx69LAbBi2E3CnoMevEYwTwgCWF20puMmX0w0yOGJR/h2at+gzfx6if4Cf6Fk2QPJrGgoajqpryE8EVu63tbK6tr6xmdvKb+/s7hXs/YO6irVkUGOxiGXTpwoEj6CGHAU0Ewk09AU0/OHNxG8glQ8ju5xlEAnpP2IB5xRNFLXLlS7bYQnVEGKUsO4axfdkjuFs0y8jBRJhmrX/mn3YqZDiJAJqlTLcxPspFQiZwLG+bZWkFA2pH1oGRrREFQnT4+dk6M0nOCWJqJ0Jmqfy9SGio1Cn2zGVIcqEVvIv7ntTQGV52UR4lGiNgsKNDCwdiZtOD0uASGYmQIZKbXx02oJIyNF3NpTwMBmchV6w8zptuvMUmlkm9XPLOS+W7i2LlOmspR47IMTklHrkFXJLqRGNHkhbySN+vZerc+rM/Z6oqV3RySOVhfvx9jnOw=</latexit> <latexit sha1_base64="n170n2F0XF87NI7Ry/Z7rxiNpA=">ACAXicbVDLTgJBEOzF+IL9ehlIjHxItlFEz0SvXjEKI8ENmR2mIWRmdnNzKwJ2XDy7FW/wZvx6pf4Cf6FA+xBwEo6qVR1p7sriDnTxnW/ndzK6tr6Rn6zsLW9s7tX3D9o6ChRhNZJxCPVCrCmnElaN8xw2oVxSLgtBkMbyZ+84kqzSL5YEYx9QXuSxYygo2V7mtd2S2W3LI7BVomXkZKkKHWLf50ehFJBJWGcKx123Nj46dYGUY4HRc6iaYxJkPcp21LJRZU+n01DE6sUoPhZGyJQ2aqn8nUiy0HonAdgpsBnrRm4j/e3EhFd+ymScGCrJbFGYcGQiNPkb9ZixPCRJZgoZm9FZIAVJsamM7flcTA4E0yTyrhgs/EWk1gmjUrZOy9X7i5K1espTwcwTGcgeXUIVbqEdCPThBV7hzXl23p0P53PWmnOymUOYg/P1C0ntlv8=</latexit> <latexit sha1_base64="gJPiAjmVJN6pCfYEi9mg/afJSB4=">ACEHicbVDLSgNBEJz1GeNr1aOXxSDEg2E3CuY8OIxgolKEsLspDcZnZldZnqDYclPePaq3+BNvPoHfoJ/4STuwVdBQ1HVTUVJoIb9P13Z25+YXFpubBSXF1b39h0t7ZbJk41gyaLRayvQmpAcAVN5CjgKtFAZSjgMrw9nfqXI9CGx+oCxwl0JR0oHnFG0Uo91+0g3KGJMglUTcrXBz235Ff8Gby/JMhJieRo9NyPTj9mqQSFTFBj2oGfYDejGjkTMCl2UgMJZbd0AG1LFZVgutns84m3b5W+F8XajkJvpn6/yKg0ZixDuykpDs1vbyr+57VTjGrdjKskRVDsKyhKhYexN63B63MNDMXYEso0t796bEg1ZWjL+pFyMxweSm5YdVK03QS/m/hLWtVKcFSpnh+X6rW8pQLZJXukTAJyQurkjDRIkzAyIg/kTw5986z8+K8fq3OfnNDvkB5+0TD5icyA=</latexit> <latexit sha1_base64="hH24RH9xGP3+VSMItbuKC+PC0aA=">AB/3icbVDLTgJBEJz1ifhCPXqZSEy8SHbRI4kXjxCIg8DGzI79MLIzOxmZtaEbDh49qrf4M149VP8BP/CAfYgYCWdVKq6090VxJxp47rfztr6xubWdm4nv7u3f3BYODpu6ihRFBo04pFqB0QDZxIahkO7VgBEQGHVjC6nfqtJ1CaRfLejGPwBRlIFjJKjJXqD71C0S25M+BV4mWkiDLUeoWfbj+iQBpKCdadzw3Nn5KlGUwyTfTEhI7IADqWSiJA+ns0Ak+t0ofh5GyJQ2eqX8nUiK0HovAdgpihnrZm4r/eZ3EhBU/ZTJODEg6XxQmHJsIT7/GfaAGj62hFDF7K2YDoki1NhsFrY8DoeXgmlanuRtNt5yEqukWS5V6Vy/bpYrWQp5dApOkMXyEM3qIruUA01EWAXtArenOenXfnw/mct6452cwJWoDz9QvBKpYd</latexit> Bootstrapping Bayes mean ( Y ) P true P n Y Y boot 6 [Douady et al. 2003, Bühlmann 2014, H & Miller 2019]

  12. <latexit sha1_base64="hH24RH9xGP3+VSMItbuKC+PC0aA=">AB/3icbVDLTgJBEJz1ifhCPXqZSEy8SHbRI4kXjxCIg8DGzI79MLIzOxmZtaEbDh49qrf4M149VP8BP/CAfYgYCWdVKq6090VxJxp47rfztr6xubWdm4nv7u3f3BYODpu6ihRFBo04pFqB0QDZxIahkO7VgBEQGHVjC6nfqtJ1CaRfLejGPwBRlIFjJKjJXqD71C0S25M+BV4mWkiDLUeoWfbj+iQBpKCdadzw3Nn5KlGUwyTfTEhI7IADqWSiJA+ns0Ak+t0ofh5GyJQ2eqX8nUiK0HovAdgpihnrZm4r/eZ3EhBU/ZTJODEg6XxQmHJsIT7/GfaAGj62hFDF7K2YDoki1NhsFrY8DoeXgmlanuRtNt5yEqukWS5V6Vy/bpYrWQp5dApOkMXyEM3qIruUA01EWAXtArenOenXfnw/mct6452cwJWoDz9QvBKpYd</latexit> <latexit sha1_base64="20pbYpT3sLDj1aIwDWwHqkeyJo8=">ACBnicbVDLSsNAFJ3UV62vqks3g0VwY0mqYJcFNy4r2Ie0oUymk2bsPMLMRCghe9du9RvciVt/w0/wL5y2WdjWAxcO59zLvfcEMaPauO63U1hb39jcKm6Xdnb39g/Kh0dtLROFSQtLJlU3QJowKkjLUMNIN1YE8YCRTjC+mfqdJ6I0leLeTGLiczQSNKQYGSt1HwZpIKXJBuWKW3VngKvEy0kF5GgOyj/9ocQJ8JghrTueW5s/BQpQzEjWamfaBIjPEYj0rNUIE60n87uzeCZVYwlMqWMHCm/p1IEd6wgPbyZGJ9LI3Ff/zeokJ635KRZwYIvB8UZgwaCScPg+HVBFs2MQShBW1t0IcIYWwsREtbHmMogtONa5lJZuNt5zEKmnXqt5ltXZ3VWnU85SK4AScgnPgWvQALegCVoAwZewCt4c56d+fD+Zy3Fpx85hgswPn6BbJrmW4=</latexit> <latexit sha1_base64="gJPiAjmVJN6pCfYEi9mg/afJSB4=">ACEHicbVDLSgNBEJz1GeNr1aOXxSDEg2E3CuY8OIxgolKEsLspDcZnZldZnqDYclPePaq3+BNvPoHfoJ/4STuwVdBQ1HVTUVJoIb9P13Z25+YXFpubBSXF1b39h0t7ZbJk41gyaLRayvQmpAcAVN5CjgKtFAZSjgMrw9nfqXI9CGx+oCxwl0JR0oHnFG0Uo91+0g3KGJMglUTcrXBz235Ff8Gby/JMhJieRo9NyPTj9mqQSFTFBj2oGfYDejGjkTMCl2UgMJZbd0AG1LFZVgutns84m3b5W+F8XajkJvpn6/yKg0ZixDuykpDs1vbyr+57VTjGrdjKskRVDsKyhKhYexN63B63MNDMXYEso0t796bEg1ZWjL+pFyMxweSm5YdVK03QS/m/hLWtVKcFSpnh+X6rW8pQLZJXukTAJyQurkjDRIkzAyIg/kTw5986z8+K8fq3OfnNDvkB5+0TD5icyA=</latexit> <latexit sha1_base64="n170n2F0XF87NI7Ry/Z7rxiNpA=">ACAXicbVDLTgJBEOzF+IL9ehlIjHxItlFEz0SvXjEKI8ENmR2mIWRmdnNzKwJ2XDy7FW/wZvx6pf4Cf6FA+xBwEo6qVR1p7sriDnTxnW/ndzK6tr6Rn6zsLW9s7tX3D9o6ChRhNZJxCPVCrCmnElaN8xw2oVxSLgtBkMbyZ+84kqzSL5YEYx9QXuSxYygo2V7mtd2S2W3LI7BVomXkZKkKHWLf50ehFJBJWGcKx123Nj46dYGUY4HRc6iaYxJkPcp21LJRZU+n01DE6sUoPhZGyJQ2aqn8nUiy0HonAdgpsBnrRm4j/e3EhFd+ymScGCrJbFGYcGQiNPkb9ZixPCRJZgoZm9FZIAVJsamM7flcTA4E0yTyrhgs/EWk1gmjUrZOy9X7i5K1espTwcwTGcgeXUIVbqEdCPThBV7hzXl23p0P53PWmnOymUOYg/P1C0ntlv8=</latexit> <latexit sha1_base64="yNrYmQo/EdIzEea4bdhMv6/bSc=">ACD3icbVDLSgNBEJz1GeMjqx69LAbBi2E3CnoMevEYwTwgCWF20puMmX0w0yOGJR/h2at+gzfx6if4Cf6Fk2QPJrGgoajqpryE8EVu63tbK6tr6xmdvKb+/s7hXs/YO6irVkUGOxiGXTpwoEj6CGHAU0Ewk09AU0/OHNxG8glQ8ju5xlEAnpP2IB5xRNFLXLlS7bYQnVEGKUsO4axfdkjuFs0y8jBRJhmrX/mn3YqZDiJAJqlTLcxPspFQiZwLG+bZWkFA2pH1oGRrREFQnT4+dk6M0nOCWJqJ0Jmqfy9SGio1Cn2zGVIcqEVvIv7ntTQGV52UR4lGiNgsKNDCwdiZtOD0uASGYmQIZKbXx02oJIyNF3NpTwMBmchV6w8zptuvMUmlkm9XPLOS+W7i2LlOmspR47IMTklHrkFXJLqRGNHkhbySN+vZerc+rM/Z6oqV3RySOVhfvx9jnOw=</latexit> Bootstrapping Bayes mean ( Y ) P true • Recall: posterior given data Y is denoted π ( 𝜄 | Y ) P n Y Y boot uncertainty about true mean standard 
 posterior 6 [Douady et al. 2003, Bühlmann 2014, H & Miller 2019]

  13. <latexit sha1_base64="ea0D8ayWdb9tL2ni7JEGIQU5DU=">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</latexit> 
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 <latexit sha1_base64="yNrYmQo/EdIzEea4bdhMv6/bSc=">ACD3icbVDLSgNBEJz1GeMjqx69LAbBi2E3CnoMevEYwTwgCWF20puMmX0w0yOGJR/h2at+gzfx6if4Cf6Fk2QPJrGgoajqpryE8EVu63tbK6tr6xmdvKb+/s7hXs/YO6irVkUGOxiGXTpwoEj6CGHAU0Ewk09AU0/OHNxG8glQ8ju5xlEAnpP2IB5xRNFLXLlS7bYQnVEGKUsO4axfdkjuFs0y8jBRJhmrX/mn3YqZDiJAJqlTLcxPspFQiZwLG+bZWkFA2pH1oGRrREFQnT4+dk6M0nOCWJqJ0Jmqfy9SGio1Cn2zGVIcqEVvIv7ntTQGV52UR4lGiNgsKNDCwdiZtOD0uASGYmQIZKbXx02oJIyNF3NpTwMBmchV6w8zptuvMUmlkm9XPLOS+W7i2LlOmspR47IMTklHrkFXJLqRGNHkhbySN+vZerc+rM/Z6oqV3RySOVhfvx9jnOw=</latexit> Bootstrapping Bayes mean ( Y ) P true • Recall: posterior given data Y is denoted π ( 𝜄 | Y ) P n • BayesBag method: Sample B bootstrap datasets and average Y over posteriors 
 Y boot B π BB ( θ | Y ) = 1 π ( θ | Y ( b ) X boot ) B b =1 uncertainty about true mean standard 
 posterior bagged posterior 6 [Douady et al. 2003, Bühlmann 2014, H & Miller 2019]

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 Bootstrapping Bayes mean ( Y ) P true • Recall: posterior given data Y is denoted π ( 𝜄 | Y ) P n • BayesBag method: Sample B bootstrap datasets and average Y over posteriors 
 Y boot B π BB ( θ | Y ) = 1 π ( θ | Y ( b ) X boot ) B b =1 uncertainty about true mean • Same benefits as bootstrap: no correct model assumption, easy- standard 
 to-use, can parallelize across B posterior bagged posterior 6 [Douady et al. 2003, Bühlmann 2014, H & Miller 2019]

  15. <latexit sha1_base64="ea0D8ayWdb9tL2ni7JEGIQU5DU=">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</latexit> 
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 Bootstrapping Bayes mean ( Y ) P true • Recall: posterior given data Y is denoted π ( 𝜄 | Y ) P n • BayesBag method: Sample B bootstrap datasets and average Y over posteriors 
 Y boot B π BB ( θ | Y ) = 1 π ( θ | Y ( b ) X boot ) B b =1 uncertainty about true mean • Same benefits as bootstrap: no correct model assumption, easy- standard 
 to-use, can parallelize across B posterior • Su ffi ces to take B = 50 or 100 bagged posterior 6 [Douady et al. 2003, Bühlmann 2014, H & Miller 2019]

  16. <latexit sha1_base64="ea0D8ayWdb9tL2ni7JEGIQU5DU=">ACUHicbVBNbxMxFHwbCpTw0QBHLlYjpORAtJsiwYFKVXrhGCTSFmXTldfxdt3a65X9Fimy/Lf4IdyQegWJf8ANnDSVmpaRLI9m5unZk9dSWIzjH1Hr3tb9Bw+3H7UfP3n6bKfz/MWR1Y1hfMK01OYkp5ZLUfEJCpT8pDacqlzy4/zicOkf+XGCl19xkXNZ4qeVaIQjGKQs4rUXmRiPfS7HkSEmqxJx86ZN9khaGMpd4N/KpbVTm8v3En47CwGY2GFqjP3W9vO/7WacbD+IVyF2SrEkX1hnd/pXLNG8QqZpNZOk7jGmaMGBZPct9PG8pqyC3rGp4FWVHE7c6ufe/I6KHNSaBNOhWSl3pxwVFm7UHlIKoqlve0txf950waL9zMnqrpBXrGrRUjCWqyrJHMheEM5SIQyowIbyWspKEwDGVvbDkvyzdKWDb07dBNcruJu+RoOEj2BsNPb7sH9YtbcMr2IUeJPAODuAjGECDL7BJfyEX9H36E/0txVdRa9veAkbaLX/AVB4s54=</latexit> 
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 Bootstrapping Bayes mean ( Y ) P true • Recall: posterior given data Y is denoted π ( 𝜄 | Y ) P n • BayesBag method: Sample B bootstrap datasets and average Y over posteriors 
 Y boot B π BB ( θ | Y ) = 1 π ( θ | Y ( b ) X boot ) B b =1 uncertainty about true mean • Same benefits as bootstrap: no correct model assumption, easy- standard 
 to-use, can parallelize across B posterior • Su ffi ces to take B = 50 or 100 bagged posterior • Finite-sample benefits of Bayes 6 [Douady et al. 2003, Bühlmann 2014, H & Miller 2019]

  17. Better parameter inference with BayesBag • Assumed model: Gaussian linear regression with conjugate priors • Data-generating distribution P true : can be correct or misspecified • θ opt = optimal parameter that is “closest” to P true • Performance metric is di ff erence in log posterior density at θ opt : log π BB ( θ opt | Y ) − log π ( θ opt | Y ) 7 [ H & Miller 2019]

  18. Better parameter inference with BayesBag • Assumed model: Gaussian linear regression with conjugate priors • Data-generating distribution P true : can be correct or misspecified • θ opt = optimal parameter that is “closest” to P true • Performance metric is di ff erence in log posterior density at θ opt : log π BB ( θ opt | Y ) − log π ( θ opt | Y ) 7 [ H & Miller 2019]

  19. Better parameter inference with BayesBag • Assumed model: Gaussian linear regression with conjugate priors • Data-generating distribution P true : can be correct or misspecified • θ opt = optimal parameter that is “closest” to P true • Performance metric is di ff erence in log posterior density at θ opt : log π BB ( θ opt | Y ) − log π ( θ opt | Y ) 7 [ H & Miller 2019]

  20. Better parameter inference with BayesBag • Assumed model: Gaussian linear regression with conjugate priors • Data-generating distribution P true : can be correct or misspecified • θ opt = optimal parameter that is “closest” to P true • Performance metric is di ff erence in log posterior density at θ opt : log π BB ( θ opt | Y ) − log π ( θ opt | Y ) 7 [ H & Miller 2019]

  21. Better parameter inference with BayesBag • Assumed model: Gaussian linear regression with conjugate priors • Data-generating distribution P true : can be correct or misspecified • θ opt = optimal parameter that is “closest” to P true • Performance metric is di ff erence in log posterior density at θ opt : log π BB ( θ opt | Y ) − log π ( θ opt | Y ) 7 [ H & Miller 2019]

  22. Better parameter inference with BayesBag • Assumed model: Gaussian linear regression with conjugate priors • Data-generating distribution P true : can be correct or misspecified • θ opt = optimal parameter that is “closest” to P true • Performance metric is di ff erence in log posterior density at θ opt : log π BB ( θ opt | Y ) − log π ( θ opt | Y ) better data-generating distribution model correct model incorrect 7 [ H & Miller 2019]

  23. Agenda • BayesBag for parameter inference (and prediction) ➡ BayesBag theory and methodology • BayesBag for model selection 8

  24. <latexit sha1_base64="mCRhlLKG1JGk8ncnxH3wIYkImLY=">ADp3icnVJdaxNBFJ1k/ajR1raCPvgyGpQGMSpVH0QiXgi1jBpJFMCLOzd3fH7s4sM7OhcRjwb/oL/BtOtrGYtip4H5az95659yPsMi4Np3O91o9uHb9xs2NW43bdza37m7v7I60LBWDIZOZVOQasi4gKHhJoNxoYDmYQbH4cnbZfx4DkpzKT6ZRQHTnCaCx5xR412zndo3Mqdqb/kxKRg6s4eHDpOEz0Hgzy389A0mpYhAhYoysIRGkTaUnS0mHR7hZmSwYCEPCGWPpbpkNpTSuVXGdm1miy7DKZImBU6NjS4RUOc0/woYTgtgBiJcSG1Acal8KoKvovp6nAoGzjn87F9iPXmldjC4IPU/lZJ5kVJhZG4L90sKlvEf5Z43hHOgwouebTc7U5l+DLorkATrezI72yLRJKVOQjDMqr1pNvxvVnfDGcZuAYpNReL01g4qGgOeiprW7F4SfeE+HY14+lMLjy/v7C0lzrR56Zk5Nqi/Gls6rYpPSxK+mlouiNCDYWaG4zLCReHl4OLKrzRbeECZ4l4rZin1i/LjWK/yJU2f51yznmus+aslgEj8qfsmq7/3XkajmuDrpR2cz+syGPXa3f32/scXzX5/NcsN9BA9Rnuoi16iPnqHjtAQsdqP+mb9fv1B0Ao+BKNgfEat1Zv7qE1C+hPNcY3BQ=</latexit> BayesBag incorporates model- and sampling-based uncertainty � � Var( ϑ BB | Y ) = E Var( ϑ BB | Y boot ) + Var E ( ϑ BB | Y boot ) | {z } | {z } expected posterior variance of variance posterior mean 9 [ H & Miller 2019]

  25. <latexit sha1_base64="mCRhlLKG1JGk8ncnxH3wIYkImLY=">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</latexit> <latexit sha1_base64="ugYbrCBJi1MVkecYuNzHImxZ5iY=">ACGXicbVDLSsNAFJ3UV62vqitxEyxCXViSVnzsim5cVrAPaUKZTKfN2EkmzNwUSgh+iGu3+g3uxK0rP8G/ME2LWOuBC4dz7uXe5yAMwWG8alFhaXleyq7m19Y3Nrfz2TkOJUBJaJ4IL2XKwopz5tA4MOG0FkmLP4bTpDK7GfnNIpWLCv4VRQG0P93WYwRDInXye9YQSyuyXHAp4OJdJ3KEgPjIijv5glEyUujzxJySApqi1sl/WV1BQo/6QDhWqm0aAdgRlsAIp3HOChUNMBngPm0n1MceVXaUvhDrh4nS1XtCJuWDnq/JyLsKTXynKTw+Cqv95Y/M9rh9A7tyPmByFQn0wW9UKug9DHehdJikBPkoIJpIlt+rExRITSFKb2XLvusceU6Qc59JsLsY4/UlinjTKJbNSqtycFKqX05SyaB8doCIy0RmqomtUQ3VE0AN6Qs/oRXvUXrU37X3SmtGmM7toBtrHN5wRoQY=</latexit> BayesBag incorporates model- and sampling-based uncertainty Var { ˆ θ ( Y boot ) } Bootstrap variance: � � Var( ϑ BB | Y ) = E Var( ϑ BB | Y boot ) + Var E ( ϑ BB | Y boot ) | {z } | {z } expected posterior variance of variance posterior mean 9 [ H & Miller 2019]

  26. <latexit sha1_base64="ugYbrCBJi1MVkecYuNzHImxZ5iY=">ACGXicbVDLSsNAFJ3UV62vqitxEyxCXViSVnzsim5cVrAPaUKZTKfN2EkmzNwUSgh+iGu3+g3uxK0rP8G/ME2LWOuBC4dz7uXe5yAMwWG8alFhaXleyq7m19Y3Nrfz2TkOJUBJaJ4IL2XKwopz5tA4MOG0FkmLP4bTpDK7GfnNIpWLCv4VRQG0P93WYwRDInXye9YQSyuyXHAp4OJdJ3KEgPjIijv5glEyUujzxJySApqi1sl/WV1BQo/6QDhWqm0aAdgRlsAIp3HOChUNMBngPm0n1MceVXaUvhDrh4nS1XtCJuWDnq/JyLsKTXynKTw+Cqv95Y/M9rh9A7tyPmByFQn0wW9UKug9DHehdJikBPkoIJpIlt+rExRITSFKb2XLvusceU6Qc59JsLsY4/UlinjTKJbNSqtycFKqX05SyaB8doCIy0RmqomtUQ3VE0AN6Qs/oRXvUXrU37X3SmtGmM7toBtrHN5wRoQY=</latexit> <latexit sha1_base64="mCRhlLKG1JGk8ncnxH3wIYkImLY=">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</latexit> BayesBag incorporates model- and sampling-based uncertainty Var { ˆ θ ( Y boot ) } Bootstrap variance: point estimate � � Var( ϑ BB | Y ) = E Var( ϑ BB | Y boot ) + Var E ( ϑ BB | Y boot ) | {z } | {z } expected posterior variance of variance posterior mean 9 [ H & Miller 2019]

  27. <latexit sha1_base64="ugYbrCBJi1MVkecYuNzHImxZ5iY=">ACGXicbVDLSsNAFJ3UV62vqitxEyxCXViSVnzsim5cVrAPaUKZTKfN2EkmzNwUSgh+iGu3+g3uxK0rP8G/ME2LWOuBC4dz7uXe5yAMwWG8alFhaXleyq7m19Y3Nrfz2TkOJUBJaJ4IL2XKwopz5tA4MOG0FkmLP4bTpDK7GfnNIpWLCv4VRQG0P93WYwRDInXye9YQSyuyXHAp4OJdJ3KEgPjIijv5glEyUujzxJySApqi1sl/WV1BQo/6QDhWqm0aAdgRlsAIp3HOChUNMBngPm0n1MceVXaUvhDrh4nS1XtCJuWDnq/JyLsKTXynKTw+Cqv95Y/M9rh9A7tyPmByFQn0wW9UKug9DHehdJikBPkoIJpIlt+rExRITSFKb2XLvusceU6Qc59JsLsY4/UlinjTKJbNSqtycFKqX05SyaB8doCIy0RmqomtUQ3VE0AN6Qs/oRXvUXrU37X3SmtGmM7toBtrHN5wRoQY=</latexit> <latexit sha1_base64="mCRhlLKG1JGk8ncnxH3wIYkImLY=">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</latexit> BayesBag incorporates model- and sampling-based uncertainty Var { ˆ θ ( Y boot ) } Bootstrap variance: sampling uncertainty point estimate � � Var( ϑ BB | Y ) = E Var( ϑ BB | Y boot ) + Var E ( ϑ BB | Y boot ) | {z } | {z } expected posterior variance of variance posterior mean 9 [ H & Miller 2019]

  28. <latexit sha1_base64="ugYbrCBJi1MVkecYuNzHImxZ5iY=">ACGXicbVDLSsNAFJ3UV62vqitxEyxCXViSVnzsim5cVrAPaUKZTKfN2EkmzNwUSgh+iGu3+g3uxK0rP8G/ME2LWOuBC4dz7uXe5yAMwWG8alFhaXleyq7m19Y3Nrfz2TkOJUBJaJ4IL2XKwopz5tA4MOG0FkmLP4bTpDK7GfnNIpWLCv4VRQG0P93WYwRDInXye9YQSyuyXHAp4OJdJ3KEgPjIijv5glEyUujzxJySApqi1sl/WV1BQo/6QDhWqm0aAdgRlsAIp3HOChUNMBngPm0n1MceVXaUvhDrh4nS1XtCJuWDnq/JyLsKTXynKTw+Cqv95Y/M9rh9A7tyPmByFQn0wW9UKug9DHehdJikBPkoIJpIlt+rExRITSFKb2XLvusceU6Qc59JsLsY4/UlinjTKJbNSqtycFKqX05SyaB8doCIy0RmqomtUQ3VE0AN6Qs/oRXvUXrU37X3SmtGmM7toBtrHN5wRoQY=</latexit> <latexit sha1_base64="C2LlLSP6VsdCkr9/JlGFoG10P1c=">ACJHicbVDLTgIxFO3gC/GFunTiCa4kAxgfOyIblxiIg/DENIpF6bS6UzaDgkh/IAf4tqtfoM748KNe/CAhMj4kmanJxzb87tcUPOlLbtDyuxsLi0vJcTa2tb2xupbd3qiqIJIUKDXg6y5RwJmAimaQz2UQHyXQ83tXY39Wh+kYoG41YMQmj7pCtZhlGgjtdIHTp9I7YEm2GmbOyELOvEQpf1QeC7o1Y6Y+fsCfA8yckg2KUW+kvpx3QyAehKSdKNfJ2qJtDk8Qoh1HKiRSEhPZIFxqGCuKDag4nvxnhQ6O0cSeQ5gmNJ+rvjSHxlRr4rpn0ifbUX28s/uc1It05bw6ZCMNgk6DOhHOsDjanCbSaCaDwhVDJzK6YekYRqU+BMyr3nHftM0cIoNenmYozTnybmSbWQyxdzxZuTOkybimJ9tA+yqI8OkMldI3KqIoekBP6Bm9WI/Wq/VmvU9HE1a8s4tmYH1+A8bspKE=</latexit> <latexit sha1_base64="mCRhlLKG1JGk8ncnxH3wIYkImLY=">ADp3icnVJdaxNBFJ1k/ajR1raCPvgyGpQGMSpVH0QiXgi1jBpJFMCLOzd3fH7s4sM7OhcRjwb/oL/BtOtrGYtip4H5az95659yPsMi4Np3O91o9uHb9xs2NW43bdza37m7v7I60LBWDIZOZVOQasi4gKHhJoNxoYDmYQbH4cnbZfx4DkpzKT6ZRQHTnCaCx5xR412zndo3Mqdqb/kxKRg6s4eHDpOEz0Hgzy389A0mpYhAhYoysIRGkTaUnS0mHR7hZmSwYCEPCGWPpbpkNpTSuVXGdm1miy7DKZImBU6NjS4RUOc0/woYTgtgBiJcSG1Acal8KoKvovp6nAoGzjn87F9iPXmldjC4IPU/lZJ5kVJhZG4L90sKlvEf5Z43hHOgwouebTc7U5l+DLorkATrezI72yLRJKVOQjDMqr1pNvxvVnfDGcZuAYpNReL01g4qGgOeiprW7F4SfeE+HY14+lMLjy/v7C0lzrR56Zk5Nqi/Gls6rYpPSxK+mlouiNCDYWaG4zLCReHl4OLKrzRbeECZ4l4rZin1i/LjWK/yJU2f51yznmus+aslgEj8qfsmq7/3XkajmuDrpR2cz+syGPXa3f32/scXzX5/NcsN9BA9Rnuoi16iPnqHjtAQsdqP+mb9fv1B0Ao+BKNgfEat1Zv7qE1C+hPNcY3BQ=</latexit> BayesBag incorporates model- and sampling-based uncertainty Var { ˆ θ ( Y boot ) } Bootstrap variance: sampling uncertainty point estimate Sample from posterior: ϑ ∼ π ( θ | Y ) � � Var( ϑ BB | Y ) = E Var( ϑ BB | Y boot ) + Var E ( ϑ BB | Y boot ) | {z } | {z } expected posterior variance of variance posterior mean 9 [ H & Miller 2019]

  29. <latexit sha1_base64="C2LlLSP6VsdCkr9/JlGFoG10P1c=">ACJHicbVDLTgIxFO3gC/GFunTiCa4kAxgfOyIblxiIg/DENIpF6bS6UzaDgkh/IAf4tqtfoM748KNe/CAhMj4kmanJxzb87tcUPOlLbtDyuxsLi0vJcTa2tb2xupbd3qiqIJIUKDXg6y5RwJmAimaQz2UQHyXQ83tXY39Wh+kYoG41YMQmj7pCtZhlGgjtdIHTp9I7YEm2GmbOyELOvEQpf1QeC7o1Y6Y+fsCfA8yckg2KUW+kvpx3QyAehKSdKNfJ2qJtDk8Qoh1HKiRSEhPZIFxqGCuKDag4nvxnhQ6O0cSeQ5gmNJ+rvjSHxlRr4rpn0ifbUX28s/uc1It05bw6ZCMNgk6DOhHOsDjanCbSaCaDwhVDJzK6YekYRqU+BMyr3nHftM0cIoNenmYozTnybmSbWQyxdzxZuTOkybimJ9tA+yqI8OkMldI3KqIoekBP6Bm9WI/Wq/VmvU9HE1a8s4tmYH1+A8bspKE=</latexit> <latexit sha1_base64="VlY6LCwIOMGS1v8DN+h8EKiNslg=">ACGHicbVDLTgIxFO3gC/GFujFx0hMcCGZAeNjR3TjEhN5GCkUy5MpdOZtB0SQvBDXLvVb3Bn3LrzE/wLO0CMiCdpc3LOfbTHDTlT2rY/rcTC4tLySnI1tba+sbmV3t6pqCSFMo04IGsuUQBZwLKmkOtVAC8V0OVbd3FfvVPkjFAnGrByE0fdIVrMo0UZqpfcafSKz8aU90AQ3uqwPAt8dtdIZO2ePgeJMyUZNEWplf5qtAMa+SA05USpumOHujk0gxnlMEo1IgUhoT3ShbqhgvigmsPxD0b40Cht3AmkOULjsfq7Y0h8pQa+ayp9oj314vF/7x6pDvnzSETYaRB0MmiTsSxDnAcB24zCVTzgSGESmbeiqlHJKHahDaz5d7zjn2maH6UGmdzEeP0J4l5UsnEKucHOSKV5OU0qifXSAshBZ6iIrlEJlRFD+gJPaMX69F6td6s90lpwpr27KIZWB/fkgmf1w=</latexit> <latexit sha1_base64="mCRhlLKG1JGk8ncnxH3wIYkImLY=">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</latexit> <latexit sha1_base64="ugYbrCBJi1MVkecYuNzHImxZ5iY=">ACGXicbVDLSsNAFJ3UV62vqitxEyxCXViSVnzsim5cVrAPaUKZTKfN2EkmzNwUSgh+iGu3+g3uxK0rP8G/ME2LWOuBC4dz7uXe5yAMwWG8alFhaXleyq7m19Y3Nrfz2TkOJUBJaJ4IL2XKwopz5tA4MOG0FkmLP4bTpDK7GfnNIpWLCv4VRQG0P93WYwRDInXye9YQSyuyXHAp4OJdJ3KEgPjIijv5glEyUujzxJySApqi1sl/WV1BQo/6QDhWqm0aAdgRlsAIp3HOChUNMBngPm0n1MceVXaUvhDrh4nS1XtCJuWDnq/JyLsKTXynKTw+Cqv95Y/M9rh9A7tyPmByFQn0wW9UKug9DHehdJikBPkoIJpIlt+rExRITSFKb2XLvusceU6Qc59JsLsY4/UlinjTKJbNSqtycFKqX05SyaB8doCIy0RmqomtUQ3VE0AN6Qs/oRXvUXrU37X3SmtGmM7toBtrHN5wRoQY=</latexit> BayesBag incorporates model- and sampling-based uncertainty Var { ˆ θ ( Y boot ) } Bootstrap variance: sampling uncertainty point estimate Sample from posterior: ϑ ∼ π ( θ | Y ) Var( ϑ | Y ) Posterior variance: � � Var( ϑ BB | Y ) = E Var( ϑ BB | Y boot ) + Var E ( ϑ BB | Y boot ) | {z } | {z } expected posterior variance of variance posterior mean 9 [ H & Miller 2019]

  30. <latexit sha1_base64="mCRhlLKG1JGk8ncnxH3wIYkImLY=">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</latexit> <latexit sha1_base64="ugYbrCBJi1MVkecYuNzHImxZ5iY=">ACGXicbVDLSsNAFJ3UV62vqitxEyxCXViSVnzsim5cVrAPaUKZTKfN2EkmzNwUSgh+iGu3+g3uxK0rP8G/ME2LWOuBC4dz7uXe5yAMwWG8alFhaXleyq7m19Y3Nrfz2TkOJUBJaJ4IL2XKwopz5tA4MOG0FkmLP4bTpDK7GfnNIpWLCv4VRQG0P93WYwRDInXye9YQSyuyXHAp4OJdJ3KEgPjIijv5glEyUujzxJySApqi1sl/WV1BQo/6QDhWqm0aAdgRlsAIp3HOChUNMBngPm0n1MceVXaUvhDrh4nS1XtCJuWDnq/JyLsKTXynKTw+Cqv95Y/M9rh9A7tyPmByFQn0wW9UKug9DHehdJikBPkoIJpIlt+rExRITSFKb2XLvusceU6Qc59JsLsY4/UlinjTKJbNSqtycFKqX05SyaB8doCIy0RmqomtUQ3VE0AN6Qs/oRXvUXrU37X3SmtGmM7toBtrHN5wRoQY=</latexit> <latexit sha1_base64="C2LlLSP6VsdCkr9/JlGFoG10P1c=">ACJHicbVDLTgIxFO3gC/GFunTiCa4kAxgfOyIblxiIg/DENIpF6bS6UzaDgkh/IAf4tqtfoM748KNe/CAhMj4kmanJxzb87tcUPOlLbtDyuxsLi0vJcTa2tb2xupbd3qiqIJIUKDXg6y5RwJmAimaQz2UQHyXQ83tXY39Wh+kYoG41YMQmj7pCtZhlGgjtdIHTp9I7YEm2GmbOyELOvEQpf1QeC7o1Y6Y+fsCfA8yckg2KUW+kvpx3QyAehKSdKNfJ2qJtDk8Qoh1HKiRSEhPZIFxqGCuKDag4nvxnhQ6O0cSeQ5gmNJ+rvjSHxlRr4rpn0ifbUX28s/uc1It05bw6ZCMNgk6DOhHOsDjanCbSaCaDwhVDJzK6YekYRqU+BMyr3nHftM0cIoNenmYozTnybmSbWQyxdzxZuTOkybimJ9tA+yqI8OkMldI3KqIoekBP6Bm9WI/Wq/VmvU9HE1a8s4tmYH1+A8bspKE=</latexit> <latexit sha1_base64="VlY6LCwIOMGS1v8DN+h8EKiNslg=">ACGHicbVDLTgIxFO3gC/GFujFx0hMcCGZAeNjR3TjEhN5GCkUy5MpdOZtB0SQvBDXLvVb3Bn3LrzE/wLO0CMiCdpc3LOfbTHDTlT2rY/rcTC4tLySnI1tba+sbmV3t6pqCSFMo04IGsuUQBZwLKmkOtVAC8V0OVbd3FfvVPkjFAnGrByE0fdIVrMo0UZqpfcafSKz8aU90AQ3uqwPAt8dtdIZO2ePgeJMyUZNEWplf5qtAMa+SA05USpumOHujk0gxnlMEo1IgUhoT3ShbqhgvigmsPxD0b40Cht3AmkOULjsfq7Y0h8pQa+ayp9oj314vF/7x6pDvnzSETYaRB0MmiTsSxDnAcB24zCVTzgSGESmbeiqlHJKHahDaz5d7zjn2maH6UGmdzEeP0J4l5UsnEKucHOSKV5OU0qifXSAshBZ6iIrlEJlRFD+gJPaMX69F6td6s90lpwpr27KIZWB/fkgmf1w=</latexit> BayesBag incorporates model- and sampling-based uncertainty Var { ˆ θ ( Y boot ) } Bootstrap variance: sampling uncertainty point estimate Sample from posterior: ϑ ∼ π ( θ | Y ) Var( ϑ | Y ) Posterior variance: model-based uncertainty � � Var( ϑ BB | Y ) = E Var( ϑ BB | Y boot ) + Var E ( ϑ BB | Y boot ) | {z } | {z } expected posterior variance of variance posterior mean 9 [ H & Miller 2019]

  31. <latexit sha1_base64="mCRhlLKG1JGk8ncnxH3wIYkImLY=">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</latexit> <latexit sha1_base64="C2LlLSP6VsdCkr9/JlGFoG10P1c=">ACJHicbVDLTgIxFO3gC/GFunTiCa4kAxgfOyIblxiIg/DENIpF6bS6UzaDgkh/IAf4tqtfoM748KNe/CAhMj4kmanJxzb87tcUPOlLbtDyuxsLi0vJcTa2tb2xupbd3qiqIJIUKDXg6y5RwJmAimaQz2UQHyXQ83tXY39Wh+kYoG41YMQmj7pCtZhlGgjtdIHTp9I7YEm2GmbOyELOvEQpf1QeC7o1Y6Y+fsCfA8yckg2KUW+kvpx3QyAehKSdKNfJ2qJtDk8Qoh1HKiRSEhPZIFxqGCuKDag4nvxnhQ6O0cSeQ5gmNJ+rvjSHxlRr4rpn0ifbUX28s/uc1It05bw6ZCMNgk6DOhHOsDjanCbSaCaDwhVDJzK6YekYRqU+BMyr3nHftM0cIoNenmYozTnybmSbWQyxdzxZuTOkybimJ9tA+yqI8OkMldI3KqIoekBP6Bm9WI/Wq/VmvU9HE1a8s4tmYH1+A8bspKE=</latexit> <latexit sha1_base64="VlY6LCwIOMGS1v8DN+h8EKiNslg=">ACGHicbVDLTgIxFO3gC/GFujFx0hMcCGZAeNjR3TjEhN5GCkUy5MpdOZtB0SQvBDXLvVb3Bn3LrzE/wLO0CMiCdpc3LOfbTHDTlT2rY/rcTC4tLySnI1tba+sbmV3t6pqCSFMo04IGsuUQBZwLKmkOtVAC8V0OVbd3FfvVPkjFAnGrByE0fdIVrMo0UZqpfcafSKz8aU90AQ3uqwPAt8dtdIZO2ePgeJMyUZNEWplf5qtAMa+SA05USpumOHujk0gxnlMEo1IgUhoT3ShbqhgvigmsPxD0b40Cht3AmkOULjsfq7Y0h8pQa+ayp9oj314vF/7x6pDvnzSETYaRB0MmiTsSxDnAcB24zCVTzgSGESmbeiqlHJKHahDaz5d7zjn2maH6UGmdzEeP0J4l5UsnEKucHOSKV5OU0qifXSAshBZ6iIrlEJlRFD+gJPaMX69F6td6s90lpwpr27KIZWB/fkgmf1w=</latexit> <latexit sha1_base64="ugYbrCBJi1MVkecYuNzHImxZ5iY=">ACGXicbVDLSsNAFJ3UV62vqitxEyxCXViSVnzsim5cVrAPaUKZTKfN2EkmzNwUSgh+iGu3+g3uxK0rP8G/ME2LWOuBC4dz7uXe5yAMwWG8alFhaXleyq7m19Y3Nrfz2TkOJUBJaJ4IL2XKwopz5tA4MOG0FkmLP4bTpDK7GfnNIpWLCv4VRQG0P93WYwRDInXye9YQSyuyXHAp4OJdJ3KEgPjIijv5glEyUujzxJySApqi1sl/WV1BQo/6QDhWqm0aAdgRlsAIp3HOChUNMBngPm0n1MceVXaUvhDrh4nS1XtCJuWDnq/JyLsKTXynKTw+Cqv95Y/M9rh9A7tyPmByFQn0wW9UKug9DHehdJikBPkoIJpIlt+rExRITSFKb2XLvusceU6Qc59JsLsY4/UlinjTKJbNSqtycFKqX05SyaB8doCIy0RmqomtUQ3VE0AN6Qs/oRXvUXrU37X3SmtGmM7toBtrHN5wRoQY=</latexit> <latexit sha1_base64="KzdQFVq673SFWj5k8uqMmhH39g=">ACLnicbVDLSgMxFM34rPVdekmWARdOEyt+NiVunFZwT6kU0omc9uJZjJDkimUoX/h7h2q98guBC3+hem0yK+DgROzrmXkxwv5kxpx3mxZmbn5hcWc0v5ZXVtfXCxmZDRYmkUKcRj2TLIwo4E1DXTHNoxRJI6HFoerfnY785AKlYJK70MIZOSPqC9Rgl2kjdgu0OiNQBaNJNq9URdn2Tid2YZdc9N7Ow2cDEPh6v1soOraTAf8lpSkpoilq3cKH60c0CUFoyolS7ZIT605qMhnlMq7iYKY0FvSh7ahgoSgOmn2rxHeNYqPe5E0R2icqd83UhIqNQw9MxkSHajf3lj8z2snunfaSZmIEw2CToJ6Cc6wuOSsM8kUM2HhAqmXkrpgGRhGpT5Y+UmyA4CJmih6N81s3ZGMdfTfwljUO7VLbLl0fFSnXaUg5tox20h0roBFXQBaqhOqLoDj2gR/Rk3VvP1qv1NhmdsaY7W+gHrPdPy+iouw=</latexit> BayesBag incorporates model- and sampling-based uncertainty Var { ˆ θ ( Y boot ) } Bootstrap variance: sampling uncertainty point estimate Sample from posterior: ϑ ∼ π ( θ | Y ) Var( ϑ | Y ) Posterior variance: model-based uncertainty ϑ BB ∼ π BB ( θ | Y ) Sample from BayesBag posterior: � � Var( ϑ BB | Y ) = E Var( ϑ BB | Y boot ) + Var E ( ϑ BB | Y boot ) | {z } | {z } expected posterior variance of variance posterior mean 9 [ H & Miller 2019]

  32. <latexit sha1_base64="mCRhlLKG1JGk8ncnxH3wIYkImLY=">ADp3icnVJdaxNBFJ1k/ajR1raCPvgyGpQGMSpVH0QiXgi1jBpJFMCLOzd3fH7s4sM7OhcRjwb/oL/BtOtrGYtip4H5az95659yPsMi4Np3O91o9uHb9xs2NW43bdza37m7v7I60LBWDIZOZVOQasi4gKHhJoNxoYDmYQbH4cnbZfx4DkpzKT6ZRQHTnCaCx5xR412zndo3Mqdqb/kxKRg6s4eHDpOEz0Hgzy389A0mpYhAhYoysIRGkTaUnS0mHR7hZmSwYCEPCGWPpbpkNpTSuVXGdm1miy7DKZImBU6NjS4RUOc0/woYTgtgBiJcSG1Acal8KoKvovp6nAoGzjn87F9iPXmldjC4IPU/lZJ5kVJhZG4L90sKlvEf5Z43hHOgwouebTc7U5l+DLorkATrezI72yLRJKVOQjDMqr1pNvxvVnfDGcZuAYpNReL01g4qGgOeiprW7F4SfeE+HY14+lMLjy/v7C0lzrR56Zk5Nqi/Gls6rYpPSxK+mlouiNCDYWaG4zLCReHl4OLKrzRbeECZ4l4rZin1i/LjWK/yJU2f51yznmus+aslgEj8qfsmq7/3XkajmuDrpR2cz+syGPXa3f32/scXzX5/NcsN9BA9Rnuoi16iPnqHjtAQsdqP+mb9fv1B0Ao+BKNgfEat1Zv7qE1C+hPNcY3BQ=</latexit> <latexit sha1_base64="C2LlLSP6VsdCkr9/JlGFoG10P1c=">ACJHicbVDLTgIxFO3gC/GFunTiCa4kAxgfOyIblxiIg/DENIpF6bS6UzaDgkh/IAf4tqtfoM748KNe/CAhMj4kmanJxzb87tcUPOlLbtDyuxsLi0vJcTa2tb2xupbd3qiqIJIUKDXg6y5RwJmAimaQz2UQHyXQ83tXY39Wh+kYoG41YMQmj7pCtZhlGgjtdIHTp9I7YEm2GmbOyELOvEQpf1QeC7o1Y6Y+fsCfA8yckg2KUW+kvpx3QyAehKSdKNfJ2qJtDk8Qoh1HKiRSEhPZIFxqGCuKDag4nvxnhQ6O0cSeQ5gmNJ+rvjSHxlRr4rpn0ifbUX28s/uc1It05bw6ZCMNgk6DOhHOsDjanCbSaCaDwhVDJzK6YekYRqU+BMyr3nHftM0cIoNenmYozTnybmSbWQyxdzxZuTOkybimJ9tA+yqI8OkMldI3KqIoekBP6Bm9WI/Wq/VmvU9HE1a8s4tmYH1+A8bspKE=</latexit> <latexit sha1_base64="KzdQFVq673SFWj5k8uqMmhH39g=">ACLnicbVDLSgMxFM34rPVdekmWARdOEyt+NiVunFZwT6kU0omc9uJZjJDkimUoX/h7h2q98guBC3+hem0yK+DgROzrmXkxwv5kxpx3mxZmbn5hcWc0v5ZXVtfXCxmZDRYmkUKcRj2TLIwo4E1DXTHNoxRJI6HFoerfnY785AKlYJK70MIZOSPqC9Rgl2kjdgu0OiNQBaNJNq9URdn2Tid2YZdc9N7Ow2cDEPh6v1soOraTAf8lpSkpoilq3cKH60c0CUFoyolS7ZIT605qMhnlMq7iYKY0FvSh7ahgoSgOmn2rxHeNYqPe5E0R2icqd83UhIqNQw9MxkSHajf3lj8z2snunfaSZmIEw2CToJ6Cc6wuOSsM8kUM2HhAqmXkrpgGRhGpT5Y+UmyA4CJmih6N81s3ZGMdfTfwljUO7VLbLl0fFSnXaUg5tox20h0roBFXQBaqhOqLoDj2gR/Rk3VvP1qv1NhmdsaY7W+gHrPdPy+iouw=</latexit> <latexit sha1_base64="VlY6LCwIOMGS1v8DN+h8EKiNslg=">ACGHicbVDLTgIxFO3gC/GFujFx0hMcCGZAeNjR3TjEhN5GCkUy5MpdOZtB0SQvBDXLvVb3Bn3LrzE/wLO0CMiCdpc3LOfbTHDTlT2rY/rcTC4tLySnI1tba+sbmV3t6pqCSFMo04IGsuUQBZwLKmkOtVAC8V0OVbd3FfvVPkjFAnGrByE0fdIVrMo0UZqpfcafSKz8aU90AQ3uqwPAt8dtdIZO2ePgeJMyUZNEWplf5qtAMa+SA05USpumOHujk0gxnlMEo1IgUhoT3ShbqhgvigmsPxD0b40Cht3AmkOULjsfq7Y0h8pQa+ayp9oj314vF/7x6pDvnzSETYaRB0MmiTsSxDnAcB24zCVTzgSGESmbeiqlHJKHahDaz5d7zjn2maH6UGmdzEeP0J4l5UsnEKucHOSKV5OU0qifXSAshBZ6iIrlEJlRFD+gJPaMX69F6td6s90lpwpr27KIZWB/fkgmf1w=</latexit> <latexit sha1_base64="ugYbrCBJi1MVkecYuNzHImxZ5iY=">ACGXicbVDLSsNAFJ3UV62vqitxEyxCXViSVnzsim5cVrAPaUKZTKfN2EkmzNwUSgh+iGu3+g3uxK0rP8G/ME2LWOuBC4dz7uXe5yAMwWG8alFhaXleyq7m19Y3Nrfz2TkOJUBJaJ4IL2XKwopz5tA4MOG0FkmLP4bTpDK7GfnNIpWLCv4VRQG0P93WYwRDInXye9YQSyuyXHAp4OJdJ3KEgPjIijv5glEyUujzxJySApqi1sl/WV1BQo/6QDhWqm0aAdgRlsAIp3HOChUNMBngPm0n1MceVXaUvhDrh4nS1XtCJuWDnq/JyLsKTXynKTw+Cqv95Y/M9rh9A7tyPmByFQn0wW9UKug9DHehdJikBPkoIJpIlt+rExRITSFKb2XLvusceU6Qc59JsLsY4/UlinjTKJbNSqtycFKqX05SyaB8doCIy0RmqomtUQ3VE0AN6Qs/oRXvUXrU37X3SmtGmM7toBtrHN5wRoQY=</latexit> BayesBag incorporates model- and sampling-based uncertainty Var { ˆ θ ( Y boot ) } Bootstrap variance: sampling uncertainty point estimate Sample from posterior: ϑ ∼ π ( θ | Y ) Var( ϑ | Y ) Posterior variance: model-based uncertainty ϑ BB ∼ π BB ( θ | Y ) Sample from BayesBag posterior: BayesBag posterior variance: � � Var( ϑ BB | Y ) = E Var( ϑ BB | Y boot ) + Var E ( ϑ BB | Y boot ) | {z } | {z } expected posterior variance of variance posterior mean 9 [ H & Miller 2019]

  33. <latexit sha1_base64="mCRhlLKG1JGk8ncnxH3wIYkImLY=">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</latexit> <latexit sha1_base64="C2LlLSP6VsdCkr9/JlGFoG10P1c=">ACJHicbVDLTgIxFO3gC/GFunTiCa4kAxgfOyIblxiIg/DENIpF6bS6UzaDgkh/IAf4tqtfoM748KNe/CAhMj4kmanJxzb87tcUPOlLbtDyuxsLi0vJcTa2tb2xupbd3qiqIJIUKDXg6y5RwJmAimaQz2UQHyXQ83tXY39Wh+kYoG41YMQmj7pCtZhlGgjtdIHTp9I7YEm2GmbOyELOvEQpf1QeC7o1Y6Y+fsCfA8yckg2KUW+kvpx3QyAehKSdKNfJ2qJtDk8Qoh1HKiRSEhPZIFxqGCuKDag4nvxnhQ6O0cSeQ5gmNJ+rvjSHxlRr4rpn0ifbUX28s/uc1It05bw6ZCMNgk6DOhHOsDjanCbSaCaDwhVDJzK6YekYRqU+BMyr3nHftM0cIoNenmYozTnybmSbWQyxdzxZuTOkybimJ9tA+yqI8OkMldI3KqIoekBP6Bm9WI/Wq/VmvU9HE1a8s4tmYH1+A8bspKE=</latexit> <latexit sha1_base64="KzdQFVq673SFWj5k8uqMmhH39g=">ACLnicbVDLSgMxFM34rPVdekmWARdOEyt+NiVunFZwT6kU0omc9uJZjJDkimUoX/h7h2q98guBC3+hem0yK+DgROzrmXkxwv5kxpx3mxZmbn5hcWc0v5ZXVtfXCxmZDRYmkUKcRj2TLIwo4E1DXTHNoxRJI6HFoerfnY785AKlYJK70MIZOSPqC9Rgl2kjdgu0OiNQBaNJNq9URdn2Tid2YZdc9N7Ow2cDEPh6v1soOraTAf8lpSkpoilq3cKH60c0CUFoyolS7ZIT605qMhnlMq7iYKY0FvSh7ahgoSgOmn2rxHeNYqPe5E0R2icqd83UhIqNQw9MxkSHajf3lj8z2snunfaSZmIEw2CToJ6Cc6wuOSsM8kUM2HhAqmXkrpgGRhGpT5Y+UmyA4CJmih6N81s3ZGMdfTfwljUO7VLbLl0fFSnXaUg5tox20h0roBFXQBaqhOqLoDj2gR/Rk3VvP1qv1NhmdsaY7W+gHrPdPy+iouw=</latexit> <latexit sha1_base64="VlY6LCwIOMGS1v8DN+h8EKiNslg=">ACGHicbVDLTgIxFO3gC/GFujFx0hMcCGZAeNjR3TjEhN5GCkUy5MpdOZtB0SQvBDXLvVb3Bn3LrzE/wLO0CMiCdpc3LOfbTHDTlT2rY/rcTC4tLySnI1tba+sbmV3t6pqCSFMo04IGsuUQBZwLKmkOtVAC8V0OVbd3FfvVPkjFAnGrByE0fdIVrMo0UZqpfcafSKz8aU90AQ3uqwPAt8dtdIZO2ePgeJMyUZNEWplf5qtAMa+SA05USpumOHujk0gxnlMEo1IgUhoT3ShbqhgvigmsPxD0b40Cht3AmkOULjsfq7Y0h8pQa+ayp9oj314vF/7x6pDvnzSETYaRB0MmiTsSxDnAcB24zCVTzgSGESmbeiqlHJKHahDaz5d7zjn2maH6UGmdzEeP0J4l5UsnEKucHOSKV5OU0qifXSAshBZ6iIrlEJlRFD+gJPaMX69F6td6s90lpwpr27KIZWB/fkgmf1w=</latexit> <latexit sha1_base64="ugYbrCBJi1MVkecYuNzHImxZ5iY=">ACGXicbVDLSsNAFJ3UV62vqitxEyxCXViSVnzsim5cVrAPaUKZTKfN2EkmzNwUSgh+iGu3+g3uxK0rP8G/ME2LWOuBC4dz7uXe5yAMwWG8alFhaXleyq7m19Y3Nrfz2TkOJUBJaJ4IL2XKwopz5tA4MOG0FkmLP4bTpDK7GfnNIpWLCv4VRQG0P93WYwRDInXye9YQSyuyXHAp4OJdJ3KEgPjIijv5glEyUujzxJySApqi1sl/WV1BQo/6QDhWqm0aAdgRlsAIp3HOChUNMBngPm0n1MceVXaUvhDrh4nS1XtCJuWDnq/JyLsKTXynKTw+Cqv95Y/M9rh9A7tyPmByFQn0wW9UKug9DHehdJikBPkoIJpIlt+rExRITSFKb2XLvusceU6Qc59JsLsY4/UlinjTKJbNSqtycFKqX05SyaB8doCIy0RmqomtUQ3VE0AN6Qs/oRXvUXrU37X3SmtGmM7toBtrHN5wRoQY=</latexit> BayesBag incorporates model- and sampling-based uncertainty Var { ˆ θ ( Y boot ) } Bootstrap variance: sampling uncertainty point estimate Sample from posterior: ϑ ∼ π ( θ | Y ) Var( ϑ | Y ) Posterior variance: model-based uncertainty ϑ BB ∼ π BB ( θ | Y ) Sample from BayesBag posterior: BayesBag posterior variance: � � Var( ϑ BB | Y ) = E Var( ϑ BB | Y boot ) + Var E ( ϑ BB | Y boot ) | {z } | {z } expected posterior variance of variance posterior mean 9 [ H & Miller 2019]

  34. <latexit sha1_base64="mCRhlLKG1JGk8ncnxH3wIYkImLY=">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</latexit> <latexit sha1_base64="C2LlLSP6VsdCkr9/JlGFoG10P1c=">ACJHicbVDLTgIxFO3gC/GFunTiCa4kAxgfOyIblxiIg/DENIpF6bS6UzaDgkh/IAf4tqtfoM748KNe/CAhMj4kmanJxzb87tcUPOlLbtDyuxsLi0vJcTa2tb2xupbd3qiqIJIUKDXg6y5RwJmAimaQz2UQHyXQ83tXY39Wh+kYoG41YMQmj7pCtZhlGgjtdIHTp9I7YEm2GmbOyELOvEQpf1QeC7o1Y6Y+fsCfA8yckg2KUW+kvpx3QyAehKSdKNfJ2qJtDk8Qoh1HKiRSEhPZIFxqGCuKDag4nvxnhQ6O0cSeQ5gmNJ+rvjSHxlRr4rpn0ifbUX28s/uc1It05bw6ZCMNgk6DOhHOsDjanCbSaCaDwhVDJzK6YekYRqU+BMyr3nHftM0cIoNenmYozTnybmSbWQyxdzxZuTOkybimJ9tA+yqI8OkMldI3KqIoekBP6Bm9WI/Wq/VmvU9HE1a8s4tmYH1+A8bspKE=</latexit> <latexit sha1_base64="KzdQFVq673SFWj5k8uqMmhH39g=">ACLnicbVDLSgMxFM34rPVdekmWARdOEyt+NiVunFZwT6kU0omc9uJZjJDkimUoX/h7h2q98guBC3+hem0yK+DgROzrmXkxwv5kxpx3mxZmbn5hcWc0v5ZXVtfXCxmZDRYmkUKcRj2TLIwo4E1DXTHNoxRJI6HFoerfnY785AKlYJK70MIZOSPqC9Rgl2kjdgu0OiNQBaNJNq9URdn2Tid2YZdc9N7Ow2cDEPh6v1soOraTAf8lpSkpoilq3cKH60c0CUFoyolS7ZIT605qMhnlMq7iYKY0FvSh7ahgoSgOmn2rxHeNYqPe5E0R2icqd83UhIqNQw9MxkSHajf3lj8z2snunfaSZmIEw2CToJ6Cc6wuOSsM8kUM2HhAqmXkrpgGRhGpT5Y+UmyA4CJmih6N81s3ZGMdfTfwljUO7VLbLl0fFSnXaUg5tox20h0roBFXQBaqhOqLoDj2gR/Rk3VvP1qv1NhmdsaY7W+gHrPdPy+iouw=</latexit> <latexit sha1_base64="VlY6LCwIOMGS1v8DN+h8EKiNslg=">ACGHicbVDLTgIxFO3gC/GFujFx0hMcCGZAeNjR3TjEhN5GCkUy5MpdOZtB0SQvBDXLvVb3Bn3LrzE/wLO0CMiCdpc3LOfbTHDTlT2rY/rcTC4tLySnI1tba+sbmV3t6pqCSFMo04IGsuUQBZwLKmkOtVAC8V0OVbd3FfvVPkjFAnGrByE0fdIVrMo0UZqpfcafSKz8aU90AQ3uqwPAt8dtdIZO2ePgeJMyUZNEWplf5qtAMa+SA05USpumOHujk0gxnlMEo1IgUhoT3ShbqhgvigmsPxD0b40Cht3AmkOULjsfq7Y0h8pQa+ayp9oj314vF/7x6pDvnzSETYaRB0MmiTsSxDnAcB24zCVTzgSGESmbeiqlHJKHahDaz5d7zjn2maH6UGmdzEeP0J4l5UsnEKucHOSKV5OU0qifXSAshBZ6iIrlEJlRFD+gJPaMX69F6td6s90lpwpr27KIZWB/fkgmf1w=</latexit> <latexit sha1_base64="ugYbrCBJi1MVkecYuNzHImxZ5iY=">ACGXicbVDLSsNAFJ3UV62vqitxEyxCXViSVnzsim5cVrAPaUKZTKfN2EkmzNwUSgh+iGu3+g3uxK0rP8G/ME2LWOuBC4dz7uXe5yAMwWG8alFhaXleyq7m19Y3Nrfz2TkOJUBJaJ4IL2XKwopz5tA4MOG0FkmLP4bTpDK7GfnNIpWLCv4VRQG0P93WYwRDInXye9YQSyuyXHAp4OJdJ3KEgPjIijv5glEyUujzxJySApqi1sl/WV1BQo/6QDhWqm0aAdgRlsAIp3HOChUNMBngPm0n1MceVXaUvhDrh4nS1XtCJuWDnq/JyLsKTXynKTw+Cqv95Y/M9rh9A7tyPmByFQn0wW9UKug9DHehdJikBPkoIJpIlt+rExRITSFKb2XLvusceU6Qc59JsLsY4/UlinjTKJbNSqtycFKqX05SyaB8doCIy0RmqomtUQ3VE0AN6Qs/oRXvUXrU37X3SmtGmM7toBtrHN5wRoQY=</latexit> BayesBag incorporates model- and sampling-based uncertainty Var { ˆ θ ( Y boot ) } Bootstrap variance: sampling uncertainty point estimate Sample from posterior: ϑ ∼ π ( θ | Y ) Var( ϑ | Y ) Posterior variance: model-based uncertainty ϑ BB ∼ π BB ( θ | Y ) Sample from BayesBag posterior: BayesBag posterior variance: � � Var( ϑ BB | Y ) = E Var( ϑ BB | Y boot ) + Var E ( ϑ BB | Y boot ) | {z } | {z } expected posterior variance of variance posterior mean 9 [ H & Miller 2019]

  35. <latexit sha1_base64="mCRhlLKG1JGk8ncnxH3wIYkImLY=">ADp3icnVJdaxNBFJ1k/ajR1raCPvgyGpQGMSpVH0QiXgi1jBpJFMCLOzd3fH7s4sM7OhcRjwb/oL/BtOtrGYtip4H5az95659yPsMi4Np3O91o9uHb9xs2NW43bdza37m7v7I60LBWDIZOZVOQasi4gKHhJoNxoYDmYQbH4cnbZfx4DkpzKT6ZRQHTnCaCx5xR412zndo3Mqdqb/kxKRg6s4eHDpOEz0Hgzy389A0mpYhAhYoysIRGkTaUnS0mHR7hZmSwYCEPCGWPpbpkNpTSuVXGdm1miy7DKZImBU6NjS4RUOc0/woYTgtgBiJcSG1Acal8KoKvovp6nAoGzjn87F9iPXmldjC4IPU/lZJ5kVJhZG4L90sKlvEf5Z43hHOgwouebTc7U5l+DLorkATrezI72yLRJKVOQjDMqr1pNvxvVnfDGcZuAYpNReL01g4qGgOeiprW7F4SfeE+HY14+lMLjy/v7C0lzrR56Zk5Nqi/Gls6rYpPSxK+mlouiNCDYWaG4zLCReHl4OLKrzRbeECZ4l4rZin1i/LjWK/yJU2f51yznmus+aslgEj8qfsmq7/3XkajmuDrpR2cz+syGPXa3f32/scXzX5/NcsN9BA9Rnuoi16iPnqHjtAQsdqP+mb9fv1B0Ao+BKNgfEat1Zv7qE1C+hPNcY3BQ=</latexit> <latexit sha1_base64="C2LlLSP6VsdCkr9/JlGFoG10P1c=">ACJHicbVDLTgIxFO3gC/GFunTiCa4kAxgfOyIblxiIg/DENIpF6bS6UzaDgkh/IAf4tqtfoM748KNe/CAhMj4kmanJxzb87tcUPOlLbtDyuxsLi0vJcTa2tb2xupbd3qiqIJIUKDXg6y5RwJmAimaQz2UQHyXQ83tXY39Wh+kYoG41YMQmj7pCtZhlGgjtdIHTp9I7YEm2GmbOyELOvEQpf1QeC7o1Y6Y+fsCfA8yckg2KUW+kvpx3QyAehKSdKNfJ2qJtDk8Qoh1HKiRSEhPZIFxqGCuKDag4nvxnhQ6O0cSeQ5gmNJ+rvjSHxlRr4rpn0ifbUX28s/uc1It05bw6ZCMNgk6DOhHOsDjanCbSaCaDwhVDJzK6YekYRqU+BMyr3nHftM0cIoNenmYozTnybmSbWQyxdzxZuTOkybimJ9tA+yqI8OkMldI3KqIoekBP6Bm9WI/Wq/VmvU9HE1a8s4tmYH1+A8bspKE=</latexit> <latexit sha1_base64="KzdQFVq673SFWj5k8uqMmhH39g=">ACLnicbVDLSgMxFM34rPVdekmWARdOEyt+NiVunFZwT6kU0omc9uJZjJDkimUoX/h7h2q98guBC3+hem0yK+DgROzrmXkxwv5kxpx3mxZmbn5hcWc0v5ZXVtfXCxmZDRYmkUKcRj2TLIwo4E1DXTHNoxRJI6HFoerfnY785AKlYJK70MIZOSPqC9Rgl2kjdgu0OiNQBaNJNq9URdn2Tid2YZdc9N7Ow2cDEPh6v1soOraTAf8lpSkpoilq3cKH60c0CUFoyolS7ZIT605qMhnlMq7iYKY0FvSh7ahgoSgOmn2rxHeNYqPe5E0R2icqd83UhIqNQw9MxkSHajf3lj8z2snunfaSZmIEw2CToJ6Cc6wuOSsM8kUM2HhAqmXkrpgGRhGpT5Y+UmyA4CJmih6N81s3ZGMdfTfwljUO7VLbLl0fFSnXaUg5tox20h0roBFXQBaqhOqLoDj2gR/Rk3VvP1qv1NhmdsaY7W+gHrPdPy+iouw=</latexit> <latexit sha1_base64="VlY6LCwIOMGS1v8DN+h8EKiNslg=">ACGHicbVDLTgIxFO3gC/GFujFx0hMcCGZAeNjR3TjEhN5GCkUy5MpdOZtB0SQvBDXLvVb3Bn3LrzE/wLO0CMiCdpc3LOfbTHDTlT2rY/rcTC4tLySnI1tba+sbmV3t6pqCSFMo04IGsuUQBZwLKmkOtVAC8V0OVbd3FfvVPkjFAnGrByE0fdIVrMo0UZqpfcafSKz8aU90AQ3uqwPAt8dtdIZO2ePgeJMyUZNEWplf5qtAMa+SA05USpumOHujk0gxnlMEo1IgUhoT3ShbqhgvigmsPxD0b40Cht3AmkOULjsfq7Y0h8pQa+ayp9oj314vF/7x6pDvnzSETYaRB0MmiTsSxDnAcB24zCVTzgSGESmbeiqlHJKHahDaz5d7zjn2maH6UGmdzEeP0J4l5UsnEKucHOSKV5OU0qifXSAshBZ6iIrlEJlRFD+gJPaMX69F6td6s90lpwpr27KIZWB/fkgmf1w=</latexit> <latexit sha1_base64="ugYbrCBJi1MVkecYuNzHImxZ5iY=">ACGXicbVDLSsNAFJ3UV62vqitxEyxCXViSVnzsim5cVrAPaUKZTKfN2EkmzNwUSgh+iGu3+g3uxK0rP8G/ME2LWOuBC4dz7uXe5yAMwWG8alFhaXleyq7m19Y3Nrfz2TkOJUBJaJ4IL2XKwopz5tA4MOG0FkmLP4bTpDK7GfnNIpWLCv4VRQG0P93WYwRDInXye9YQSyuyXHAp4OJdJ3KEgPjIijv5glEyUujzxJySApqi1sl/WV1BQo/6QDhWqm0aAdgRlsAIp3HOChUNMBngPm0n1MceVXaUvhDrh4nS1XtCJuWDnq/JyLsKTXynKTw+Cqv95Y/M9rh9A7tyPmByFQn0wW9UKug9DHehdJikBPkoIJpIlt+rExRITSFKb2XLvusceU6Qc59JsLsY4/UlinjTKJbNSqtycFKqX05SyaB8doCIy0RmqomtUQ3VE0AN6Qs/oRXvUXrU37X3SmtGmM7toBtrHN5wRoQY=</latexit> BayesBag incorporates model- and sampling-based uncertainty Var { ˆ θ ( Y boot ) } Bootstrap variance: sampling uncertainty point estimate Sample from posterior: ϑ ∼ π ( θ | Y ) Var( ϑ | Y ) Posterior variance: model-based uncertainty ϑ BB ∼ π BB ( θ | Y ) Sample from BayesBag posterior: BayesBag posterior variance: � � Var( ϑ BB | Y ) = E Var( ϑ BB | Y boot ) + Var E ( ϑ BB | Y boot ) | {z } | {z } expected posterior variance of variance posterior mean 9 [ H & Miller 2019]

  36. <latexit sha1_base64="mCRhlLKG1JGk8ncnxH3wIYkImLY=">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</latexit> <latexit sha1_base64="C2LlLSP6VsdCkr9/JlGFoG10P1c=">ACJHicbVDLTgIxFO3gC/GFunTiCa4kAxgfOyIblxiIg/DENIpF6bS6UzaDgkh/IAf4tqtfoM748KNe/CAhMj4kmanJxzb87tcUPOlLbtDyuxsLi0vJcTa2tb2xupbd3qiqIJIUKDXg6y5RwJmAimaQz2UQHyXQ83tXY39Wh+kYoG41YMQmj7pCtZhlGgjtdIHTp9I7YEm2GmbOyELOvEQpf1QeC7o1Y6Y+fsCfA8yckg2KUW+kvpx3QyAehKSdKNfJ2qJtDk8Qoh1HKiRSEhPZIFxqGCuKDag4nvxnhQ6O0cSeQ5gmNJ+rvjSHxlRr4rpn0ifbUX28s/uc1It05bw6ZCMNgk6DOhHOsDjanCbSaCaDwhVDJzK6YekYRqU+BMyr3nHftM0cIoNenmYozTnybmSbWQyxdzxZuTOkybimJ9tA+yqI8OkMldI3KqIoekBP6Bm9WI/Wq/VmvU9HE1a8s4tmYH1+A8bspKE=</latexit> <latexit sha1_base64="KzdQFVq673SFWj5k8uqMmhH39g=">ACLnicbVDLSgMxFM34rPVdekmWARdOEyt+NiVunFZwT6kU0omc9uJZjJDkimUoX/h7h2q98guBC3+hem0yK+DgROzrmXkxwv5kxpx3mxZmbn5hcWc0v5ZXVtfXCxmZDRYmkUKcRj2TLIwo4E1DXTHNoxRJI6HFoerfnY785AKlYJK70MIZOSPqC9Rgl2kjdgu0OiNQBaNJNq9URdn2Tid2YZdc9N7Ow2cDEPh6v1soOraTAf8lpSkpoilq3cKH60c0CUFoyolS7ZIT605qMhnlMq7iYKY0FvSh7ahgoSgOmn2rxHeNYqPe5E0R2icqd83UhIqNQw9MxkSHajf3lj8z2snunfaSZmIEw2CToJ6Cc6wuOSsM8kUM2HhAqmXkrpgGRhGpT5Y+UmyA4CJmih6N81s3ZGMdfTfwljUO7VLbLl0fFSnXaUg5tox20h0roBFXQBaqhOqLoDj2gR/Rk3VvP1qv1NhmdsaY7W+gHrPdPy+iouw=</latexit> <latexit sha1_base64="VlY6LCwIOMGS1v8DN+h8EKiNslg=">ACGHicbVDLTgIxFO3gC/GFujFx0hMcCGZAeNjR3TjEhN5GCkUy5MpdOZtB0SQvBDXLvVb3Bn3LrzE/wLO0CMiCdpc3LOfbTHDTlT2rY/rcTC4tLySnI1tba+sbmV3t6pqCSFMo04IGsuUQBZwLKmkOtVAC8V0OVbd3FfvVPkjFAnGrByE0fdIVrMo0UZqpfcafSKz8aU90AQ3uqwPAt8dtdIZO2ePgeJMyUZNEWplf5qtAMa+SA05USpumOHujk0gxnlMEo1IgUhoT3ShbqhgvigmsPxD0b40Cht3AmkOULjsfq7Y0h8pQa+ayp9oj314vF/7x6pDvnzSETYaRB0MmiTsSxDnAcB24zCVTzgSGESmbeiqlHJKHahDaz5d7zjn2maH6UGmdzEeP0J4l5UsnEKucHOSKV5OU0qifXSAshBZ6iIrlEJlRFD+gJPaMX69F6td6s90lpwpr27KIZWB/fkgmf1w=</latexit> <latexit sha1_base64="ugYbrCBJi1MVkecYuNzHImxZ5iY=">ACGXicbVDLSsNAFJ3UV62vqitxEyxCXViSVnzsim5cVrAPaUKZTKfN2EkmzNwUSgh+iGu3+g3uxK0rP8G/ME2LWOuBC4dz7uXe5yAMwWG8alFhaXleyq7m19Y3Nrfz2TkOJUBJaJ4IL2XKwopz5tA4MOG0FkmLP4bTpDK7GfnNIpWLCv4VRQG0P93WYwRDInXye9YQSyuyXHAp4OJdJ3KEgPjIijv5glEyUujzxJySApqi1sl/WV1BQo/6QDhWqm0aAdgRlsAIp3HOChUNMBngPm0n1MceVXaUvhDrh4nS1XtCJuWDnq/JyLsKTXynKTw+Cqv95Y/M9rh9A7tyPmByFQn0wW9UKug9DHehdJikBPkoIJpIlt+rExRITSFKb2XLvusceU6Qc59JsLsY4/UlinjTKJbNSqtycFKqX05SyaB8doCIy0RmqomtUQ3VE0AN6Qs/oRXvUXrU37X3SmtGmM7toBtrHN5wRoQY=</latexit> BayesBag incorporates model- and sampling-based uncertainty Var { ˆ θ ( Y boot ) } Bootstrap variance: sampling uncertainty point estimate Sample from posterior: ϑ ∼ π ( θ | Y ) Var( ϑ | Y ) Posterior variance: model-based uncertainty ϑ BB ∼ π BB ( θ | Y ) Sample from BayesBag posterior: BayesBag posterior variance: � � Var( ϑ BB | Y ) = E Var( ϑ BB | Y boot ) + Var E ( ϑ BB | Y boot ) | {z } | {z } expected posterior variance of variance posterior mean 9 [ H & Miller 2019]

  37. <latexit sha1_base64="mCRhlLKG1JGk8ncnxH3wIYkImLY=">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</latexit> <latexit sha1_base64="C2LlLSP6VsdCkr9/JlGFoG10P1c=">ACJHicbVDLTgIxFO3gC/GFunTiCa4kAxgfOyIblxiIg/DENIpF6bS6UzaDgkh/IAf4tqtfoM748KNe/CAhMj4kmanJxzb87tcUPOlLbtDyuxsLi0vJcTa2tb2xupbd3qiqIJIUKDXg6y5RwJmAimaQz2UQHyXQ83tXY39Wh+kYoG41YMQmj7pCtZhlGgjtdIHTp9I7YEm2GmbOyELOvEQpf1QeC7o1Y6Y+fsCfA8yckg2KUW+kvpx3QyAehKSdKNfJ2qJtDk8Qoh1HKiRSEhPZIFxqGCuKDag4nvxnhQ6O0cSeQ5gmNJ+rvjSHxlRr4rpn0ifbUX28s/uc1It05bw6ZCMNgk6DOhHOsDjanCbSaCaDwhVDJzK6YekYRqU+BMyr3nHftM0cIoNenmYozTnybmSbWQyxdzxZuTOkybimJ9tA+yqI8OkMldI3KqIoekBP6Bm9WI/Wq/VmvU9HE1a8s4tmYH1+A8bspKE=</latexit> <latexit sha1_base64="KzdQFVq673SFWj5k8uqMmhH39g=">ACLnicbVDLSgMxFM34rPVdekmWARdOEyt+NiVunFZwT6kU0omc9uJZjJDkimUoX/h7h2q98guBC3+hem0yK+DgROzrmXkxwv5kxpx3mxZmbn5hcWc0v5ZXVtfXCxmZDRYmkUKcRj2TLIwo4E1DXTHNoxRJI6HFoerfnY785AKlYJK70MIZOSPqC9Rgl2kjdgu0OiNQBaNJNq9URdn2Tid2YZdc9N7Ow2cDEPh6v1soOraTAf8lpSkpoilq3cKH60c0CUFoyolS7ZIT605qMhnlMq7iYKY0FvSh7ahgoSgOmn2rxHeNYqPe5E0R2icqd83UhIqNQw9MxkSHajf3lj8z2snunfaSZmIEw2CToJ6Cc6wuOSsM8kUM2HhAqmXkrpgGRhGpT5Y+UmyA4CJmih6N81s3ZGMdfTfwljUO7VLbLl0fFSnXaUg5tox20h0roBFXQBaqhOqLoDj2gR/Rk3VvP1qv1NhmdsaY7W+gHrPdPy+iouw=</latexit> <latexit sha1_base64="VlY6LCwIOMGS1v8DN+h8EKiNslg=">ACGHicbVDLTgIxFO3gC/GFujFx0hMcCGZAeNjR3TjEhN5GCkUy5MpdOZtB0SQvBDXLvVb3Bn3LrzE/wLO0CMiCdpc3LOfbTHDTlT2rY/rcTC4tLySnI1tba+sbmV3t6pqCSFMo04IGsuUQBZwLKmkOtVAC8V0OVbd3FfvVPkjFAnGrByE0fdIVrMo0UZqpfcafSKz8aU90AQ3uqwPAt8dtdIZO2ePgeJMyUZNEWplf5qtAMa+SA05USpumOHujk0gxnlMEo1IgUhoT3ShbqhgvigmsPxD0b40Cht3AmkOULjsfq7Y0h8pQa+ayp9oj314vF/7x6pDvnzSETYaRB0MmiTsSxDnAcB24zCVTzgSGESmbeiqlHJKHahDaz5d7zjn2maH6UGmdzEeP0J4l5UsnEKucHOSKV5OU0qifXSAshBZ6iIrlEJlRFD+gJPaMX69F6td6s90lpwpr27KIZWB/fkgmf1w=</latexit> <latexit sha1_base64="ugYbrCBJi1MVkecYuNzHImxZ5iY=">ACGXicbVDLSsNAFJ3UV62vqitxEyxCXViSVnzsim5cVrAPaUKZTKfN2EkmzNwUSgh+iGu3+g3uxK0rP8G/ME2LWOuBC4dz7uXe5yAMwWG8alFhaXleyq7m19Y3Nrfz2TkOJUBJaJ4IL2XKwopz5tA4MOG0FkmLP4bTpDK7GfnNIpWLCv4VRQG0P93WYwRDInXye9YQSyuyXHAp4OJdJ3KEgPjIijv5glEyUujzxJySApqi1sl/WV1BQo/6QDhWqm0aAdgRlsAIp3HOChUNMBngPm0n1MceVXaUvhDrh4nS1XtCJuWDnq/JyLsKTXynKTw+Cqv95Y/M9rh9A7tyPmByFQn0wW9UKug9DHehdJikBPkoIJpIlt+rExRITSFKb2XLvusceU6Qc59JsLsY4/UlinjTKJbNSqtycFKqX05SyaB8doCIy0RmqomtUQ3VE0AN6Qs/oRXvUXrU37X3SmtGmM7toBtrHN5wRoQY=</latexit> BayesBag incorporates model- and sampling-based uncertainty Var { ˆ θ ( Y boot ) } Bootstrap variance: sampling uncertainty point estimate Sample from posterior: ϑ ∼ π ( θ | Y ) Var( ϑ | Y ) Posterior variance: model-based uncertainty ϑ BB ∼ π BB ( θ | Y ) Sample from BayesBag posterior: BayesBag posterior variance: point estimate � � Var( ϑ BB | Y ) = E Var( ϑ BB | Y boot ) + Var E ( ϑ BB | Y boot ) | {z } | {z } expected posterior variance of variance posterior mean 9 [ H & Miller 2019]

  38. <latexit sha1_base64="mCRhlLKG1JGk8ncnxH3wIYkImLY=">ADp3icnVJdaxNBFJ1k/ajR1raCPvgyGpQGMSpVH0QiXgi1jBpJFMCLOzd3fH7s4sM7OhcRjwb/oL/BtOtrGYtip4H5az95659yPsMi4Np3O91o9uHb9xs2NW43bdza37m7v7I60LBWDIZOZVOQasi4gKHhJoNxoYDmYQbH4cnbZfx4DkpzKT6ZRQHTnCaCx5xR412zndo3Mqdqb/kxKRg6s4eHDpOEz0Hgzy389A0mpYhAhYoysIRGkTaUnS0mHR7hZmSwYCEPCGWPpbpkNpTSuVXGdm1miy7DKZImBU6NjS4RUOc0/woYTgtgBiJcSG1Acal8KoKvovp6nAoGzjn87F9iPXmldjC4IPU/lZJ5kVJhZG4L90sKlvEf5Z43hHOgwouebTc7U5l+DLorkATrezI72yLRJKVOQjDMqr1pNvxvVnfDGcZuAYpNReL01g4qGgOeiprW7F4SfeE+HY14+lMLjy/v7C0lzrR56Zk5Nqi/Gls6rYpPSxK+mlouiNCDYWaG4zLCReHl4OLKrzRbeECZ4l4rZin1i/LjWK/yJU2f51yznmus+aslgEj8qfsmq7/3XkajmuDrpR2cz+syGPXa3f32/scXzX5/NcsN9BA9Rnuoi16iPnqHjtAQsdqP+mb9fv1B0Ao+BKNgfEat1Zv7qE1C+hPNcY3BQ=</latexit> <latexit sha1_base64="C2LlLSP6VsdCkr9/JlGFoG10P1c=">ACJHicbVDLTgIxFO3gC/GFunTiCa4kAxgfOyIblxiIg/DENIpF6bS6UzaDgkh/IAf4tqtfoM748KNe/CAhMj4kmanJxzb87tcUPOlLbtDyuxsLi0vJcTa2tb2xupbd3qiqIJIUKDXg6y5RwJmAimaQz2UQHyXQ83tXY39Wh+kYoG41YMQmj7pCtZhlGgjtdIHTp9I7YEm2GmbOyELOvEQpf1QeC7o1Y6Y+fsCfA8yckg2KUW+kvpx3QyAehKSdKNfJ2qJtDk8Qoh1HKiRSEhPZIFxqGCuKDag4nvxnhQ6O0cSeQ5gmNJ+rvjSHxlRr4rpn0ifbUX28s/uc1It05bw6ZCMNgk6DOhHOsDjanCbSaCaDwhVDJzK6YekYRqU+BMyr3nHftM0cIoNenmYozTnybmSbWQyxdzxZuTOkybimJ9tA+yqI8OkMldI3KqIoekBP6Bm9WI/Wq/VmvU9HE1a8s4tmYH1+A8bspKE=</latexit> <latexit sha1_base64="KzdQFVq673SFWj5k8uqMmhH39g=">ACLnicbVDLSgMxFM34rPVdekmWARdOEyt+NiVunFZwT6kU0omc9uJZjJDkimUoX/h7h2q98guBC3+hem0yK+DgROzrmXkxwv5kxpx3mxZmbn5hcWc0v5ZXVtfXCxmZDRYmkUKcRj2TLIwo4E1DXTHNoxRJI6HFoerfnY785AKlYJK70MIZOSPqC9Rgl2kjdgu0OiNQBaNJNq9URdn2Tid2YZdc9N7Ow2cDEPh6v1soOraTAf8lpSkpoilq3cKH60c0CUFoyolS7ZIT605qMhnlMq7iYKY0FvSh7ahgoSgOmn2rxHeNYqPe5E0R2icqd83UhIqNQw9MxkSHajf3lj8z2snunfaSZmIEw2CToJ6Cc6wuOSsM8kUM2HhAqmXkrpgGRhGpT5Y+UmyA4CJmih6N81s3ZGMdfTfwljUO7VLbLl0fFSnXaUg5tox20h0roBFXQBaqhOqLoDj2gR/Rk3VvP1qv1NhmdsaY7W+gHrPdPy+iouw=</latexit> <latexit sha1_base64="VlY6LCwIOMGS1v8DN+h8EKiNslg=">ACGHicbVDLTgIxFO3gC/GFujFx0hMcCGZAeNjR3TjEhN5GCkUy5MpdOZtB0SQvBDXLvVb3Bn3LrzE/wLO0CMiCdpc3LOfbTHDTlT2rY/rcTC4tLySnI1tba+sbmV3t6pqCSFMo04IGsuUQBZwLKmkOtVAC8V0OVbd3FfvVPkjFAnGrByE0fdIVrMo0UZqpfcafSKz8aU90AQ3uqwPAt8dtdIZO2ePgeJMyUZNEWplf5qtAMa+SA05USpumOHujk0gxnlMEo1IgUhoT3ShbqhgvigmsPxD0b40Cht3AmkOULjsfq7Y0h8pQa+ayp9oj314vF/7x6pDvnzSETYaRB0MmiTsSxDnAcB24zCVTzgSGESmbeiqlHJKHahDaz5d7zjn2maH6UGmdzEeP0J4l5UsnEKucHOSKV5OU0qifXSAshBZ6iIrlEJlRFD+gJPaMX69F6td6s90lpwpr27KIZWB/fkgmf1w=</latexit> <latexit sha1_base64="ugYbrCBJi1MVkecYuNzHImxZ5iY=">ACGXicbVDLSsNAFJ3UV62vqitxEyxCXViSVnzsim5cVrAPaUKZTKfN2EkmzNwUSgh+iGu3+g3uxK0rP8G/ME2LWOuBC4dz7uXe5yAMwWG8alFhaXleyq7m19Y3Nrfz2TkOJUBJaJ4IL2XKwopz5tA4MOG0FkmLP4bTpDK7GfnNIpWLCv4VRQG0P93WYwRDInXye9YQSyuyXHAp4OJdJ3KEgPjIijv5glEyUujzxJySApqi1sl/WV1BQo/6QDhWqm0aAdgRlsAIp3HOChUNMBngPm0n1MceVXaUvhDrh4nS1XtCJuWDnq/JyLsKTXynKTw+Cqv95Y/M9rh9A7tyPmByFQn0wW9UKug9DHehdJikBPkoIJpIlt+rExRITSFKb2XLvusceU6Qc59JsLsY4/UlinjTKJbNSqtycFKqX05SyaB8doCIy0RmqomtUQ3VE0AN6Qs/oRXvUXrU37X3SmtGmM7toBtrHN5wRoQY=</latexit> BayesBag incorporates model- and sampling-based uncertainty Var { ˆ θ ( Y boot ) } Bootstrap variance: sampling uncertainty point estimate Sample from posterior: ϑ ∼ π ( θ | Y ) Var( ϑ | Y ) Posterior variance: model-based uncertainty ϑ BB ∼ π BB ( θ | Y ) Sample from BayesBag posterior: BayesBag posterior variance: point estimate � � Var( ϑ BB | Y ) = E Var( ϑ BB | Y boot ) + Var E ( ϑ BB | Y boot ) | {z } | {z } expected posterior variance of variance posterior mean 9 [ H & Miller 2019]

  39. BayesBag incorporates model- and sampling-based uncertainty 10 [ H & Miller 2019]

  40. BayesBag incorporates model- and sampling-based uncertainty • Summarizing the previous slide… 10 [ H & Miller 2019]

  41. BayesBag incorporates model- and sampling-based uncertainty • Summarizing the previous slide… Posterior variance = model-based uncertainty 10 [ H & Miller 2019]

  42. BayesBag incorporates model- and sampling-based uncertainty • Summarizing the previous slide… Posterior variance = model-based uncertainty Bootstrap variance = sampling-based uncertainty 10 [ H & Miller 2019]

  43. BayesBag incorporates model- and sampling-based uncertainty • Summarizing the previous slide… Posterior variance = model-based uncertainty Bootstrap variance = sampling-based uncertainty BayesBag variance = model-based + sampling-based uncertainty 10 [ H & Miller 2019]

  44. <latexit sha1_base64="hfvMN8RcV2F5Kqwp0jIDAS7QeA=">ACFnicbVDLSgNBEJz1bXxFBS9eFoMQD4bdKOhR8OJRwZhIEsLspDcZMzO7zPSKYd3/8OxVv8GbePXqJ/gXTmIOJrGgoajqpoKYsENet6XMzM7N7+wuLScW1ldW9/Ib27dmCjRDCosEpGuBdSA4AoqyFALdZAZSCgGvTOB371HrThkbrGfgxNSTuKh5xRtFIrv9NAeEATphKoyoq3rTQKTHbQyhe8kjeEO038ESmQES5b+e9GO2KJBIVMUGPqvhdjM6UaOROQ5RqJgZiyHu1A3VJFJZhmOvw/c/et0nbDSNtR6A7Vvxcplcb0ZWA3JcWumfQG4n9ePcHwtJlyFScIiv0GhYlwMXIHZbhtroGh6FtCmeb2V5d1qaYMbWVjKXfd7qHkhpWznO3Gn2ximtyUS/5RqXx1XDg7HbW0RHbJHikSn5yQM3JBLkmFMPJInskLeXWenDfn3fn4XZ1xRjfbZAzO5w9LiJ+f</latexit> BayesBag incorporates model- and sampling-based uncertainty • Summarizing the previous slide… Posterior variance = model-based uncertainty Bootstrap variance = sampling-based uncertainty BayesBag variance = model-based + sampling-based uncertainty uncertainty about mean ( Y obs ) standard 
 bootstrap posterior distribution bagged posterior 10 [ H & Miller 2019]

  45. <latexit sha1_base64="hfvMN8RcV2F5Kqwp0jIDAS7QeA=">ACFnicbVDLSgNBEJz1bXxFBS9eFoMQD4bdKOhR8OJRwZhIEsLspDcZMzO7zPSKYd3/8OxVv8GbePXqJ/gXTmIOJrGgoajqpoKYsENet6XMzM7N7+wuLScW1ldW9/Ib27dmCjRDCosEpGuBdSA4AoqyFALdZAZSCgGvTOB371HrThkbrGfgxNSTuKh5xRtFIrv9NAeEATphKoyoq3rTQKTHbQyhe8kjeEO038ESmQES5b+e9GO2KJBIVMUGPqvhdjM6UaOROQ5RqJgZiyHu1A3VJFJZhmOvw/c/et0nbDSNtR6A7Vvxcplcb0ZWA3JcWumfQG4n9ePcHwtJlyFScIiv0GhYlwMXIHZbhtroGh6FtCmeb2V5d1qaYMbWVjKXfd7qHkhpWznO3Gn2ximtyUS/5RqXx1XDg7HbW0RHbJHikSn5yQM3JBLkmFMPJInskLeXWenDfn3fn4XZ1xRjfbZAzO5w9LiJ+f</latexit> BayesBag incorporates model- and sampling-based uncertainty • Summarizing the previous slide… Posterior variance = model-based uncertainty Bootstrap variance = sampling-based uncertainty BayesBag variance = model-based + sampling-based uncertainty • Model correct: model-based uncertainty = sampling-based uncertainty uncertainty about mean ( Y obs ) standard 
 bootstrap posterior distribution bagged posterior 10 [ H & Miller 2019]

  46. <latexit sha1_base64="hfvMN8RcV2F5Kqwp0jIDAS7QeA=">ACFnicbVDLSgNBEJz1bXxFBS9eFoMQD4bdKOhR8OJRwZhIEsLspDcZMzO7zPSKYd3/8OxVv8GbePXqJ/gXTmIOJrGgoajqpoKYsENet6XMzM7N7+wuLScW1ldW9/Ib27dmCjRDCosEpGuBdSA4AoqyFALdZAZSCgGvTOB371HrThkbrGfgxNSTuKh5xRtFIrv9NAeEATphKoyoq3rTQKTHbQyhe8kjeEO038ESmQES5b+e9GO2KJBIVMUGPqvhdjM6UaOROQ5RqJgZiyHu1A3VJFJZhmOvw/c/et0nbDSNtR6A7Vvxcplcb0ZWA3JcWumfQG4n9ePcHwtJlyFScIiv0GhYlwMXIHZbhtroGh6FtCmeb2V5d1qaYMbWVjKXfd7qHkhpWznO3Gn2ximtyUS/5RqXx1XDg7HbW0RHbJHikSn5yQM3JBLkmFMPJInskLeXWenDfn3fn4XZ1xRjfbZAzO5w9LiJ+f</latexit> BayesBag incorporates model- and sampling-based uncertainty • Summarizing the previous slide… Posterior variance = model-based uncertainty Bootstrap variance = sampling-based uncertainty BayesBag variance = model-based + sampling-based uncertainty • Model correct: model-based uncertainty = sampling-based uncertainty • Posterior and bootstrap variances correct uncertainty about mean ( Y obs ) standard 
 bootstrap posterior distribution bagged posterior 10 [ H & Miller 2019]

  47. <latexit sha1_base64="hfvMN8RcV2F5Kqwp0jIDAS7QeA=">ACFnicbVDLSgNBEJz1bXxFBS9eFoMQD4bdKOhR8OJRwZhIEsLspDcZMzO7zPSKYd3/8OxVv8GbePXqJ/gXTmIOJrGgoajqpoKYsENet6XMzM7N7+wuLScW1ldW9/Ib27dmCjRDCosEpGuBdSA4AoqyFALdZAZSCgGvTOB371HrThkbrGfgxNSTuKh5xRtFIrv9NAeEATphKoyoq3rTQKTHbQyhe8kjeEO038ESmQES5b+e9GO2KJBIVMUGPqvhdjM6UaOROQ5RqJgZiyHu1A3VJFJZhmOvw/c/et0nbDSNtR6A7Vvxcplcb0ZWA3JcWumfQG4n9ePcHwtJlyFScIiv0GhYlwMXIHZbhtroGh6FtCmeb2V5d1qaYMbWVjKXfd7qHkhpWznO3Gn2ximtyUS/5RqXx1XDg7HbW0RHbJHikSn5yQM3JBLkmFMPJInskLeXWenDfn3fn4XZ1xRjfbZAzO5w9LiJ+f</latexit> BayesBag incorporates model- and sampling-based uncertainty • Summarizing the previous slide… Posterior variance = model-based uncertainty Bootstrap variance = sampling-based uncertainty BayesBag variance = model-based + sampling-based uncertainty • Model correct: model-based uncertainty = sampling-based uncertainty • Posterior and bootstrap variances correct • BayesBag variance double-counts true uncertainty (conservative) uncertainty about mean ( Y obs ) standard 
 bootstrap posterior distribution bagged posterior 10 [ H & Miller 2019]

  48. <latexit sha1_base64="hfvMN8RcV2F5Kqwp0jIDAS7QeA=">ACFnicbVDLSgNBEJz1bXxFBS9eFoMQD4bdKOhR8OJRwZhIEsLspDcZMzO7zPSKYd3/8OxVv8GbePXqJ/gXTmIOJrGgoajqpoKYsENet6XMzM7N7+wuLScW1ldW9/Ib27dmCjRDCosEpGuBdSA4AoqyFALdZAZSCgGvTOB371HrThkbrGfgxNSTuKh5xRtFIrv9NAeEATphKoyoq3rTQKTHbQyhe8kjeEO038ESmQES5b+e9GO2KJBIVMUGPqvhdjM6UaOROQ5RqJgZiyHu1A3VJFJZhmOvw/c/et0nbDSNtR6A7Vvxcplcb0ZWA3JcWumfQG4n9ePcHwtJlyFScIiv0GhYlwMXIHZbhtroGh6FtCmeb2V5d1qaYMbWVjKXfd7qHkhpWznO3Gn2ximtyUS/5RqXx1XDg7HbW0RHbJHikSn5yQM3JBLkmFMPJInskLeXWenDfn3fn4XZ1xRjfbZAzO5w9LiJ+f</latexit> BayesBag incorporates model- and sampling-based uncertainty • Summarizing the previous slide… Posterior variance = model-based uncertainty Bootstrap variance = sampling-based uncertainty BayesBag variance = model-based + sampling-based uncertainty • Model correct: model-based uncertainty = sampling-based uncertainty • Posterior and bootstrap variances correct • BayesBag variance double-counts true uncertainty (conservative) • Model incorrect: model-based uncertainty ≪ sampling-based uncertainty uncertainty about mean ( Y obs ) standard 
 bootstrap posterior distribution bagged posterior 10 [ H & Miller 2019]

  49. <latexit sha1_base64="hfvMN8RcV2F5Kqwp0jIDAS7QeA=">ACFnicbVDLSgNBEJz1bXxFBS9eFoMQD4bdKOhR8OJRwZhIEsLspDcZMzO7zPSKYd3/8OxVv8GbePXqJ/gXTmIOJrGgoajqpoKYsENet6XMzM7N7+wuLScW1ldW9/Ib27dmCjRDCosEpGuBdSA4AoqyFALdZAZSCgGvTOB371HrThkbrGfgxNSTuKh5xRtFIrv9NAeEATphKoyoq3rTQKTHbQyhe8kjeEO038ESmQES5b+e9GO2KJBIVMUGPqvhdjM6UaOROQ5RqJgZiyHu1A3VJFJZhmOvw/c/et0nbDSNtR6A7Vvxcplcb0ZWA3JcWumfQG4n9ePcHwtJlyFScIiv0GhYlwMXIHZbhtroGh6FtCmeb2V5d1qaYMbWVjKXfd7qHkhpWznO3Gn2ximtyUS/5RqXx1XDg7HbW0RHbJHikSn5yQM3JBLkmFMPJInskLeXWenDfn3fn4XZ1xRjfbZAzO5w9LiJ+f</latexit> BayesBag incorporates model- and sampling-based uncertainty • Summarizing the previous slide… Posterior variance = model-based uncertainty Bootstrap variance = sampling-based uncertainty BayesBag variance = model-based + sampling-based uncertainty • Model correct: model-based uncertainty = sampling-based uncertainty • Posterior and bootstrap variances correct • BayesBag variance double-counts true uncertainty (conservative) • Model incorrect: model-based uncertainty ≪ sampling-based uncertainty uncertainty about mean ( Y obs ) • Posterior variance far 
 standard 
 too small bootstrap posterior distribution bagged posterior 10 [ H & Miller 2019]

  50. <latexit sha1_base64="hfvMN8RcV2F5Kqwp0jIDAS7QeA=">ACFnicbVDLSgNBEJz1bXxFBS9eFoMQD4bdKOhR8OJRwZhIEsLspDcZMzO7zPSKYd3/8OxVv8GbePXqJ/gXTmIOJrGgoajqpoKYsENet6XMzM7N7+wuLScW1ldW9/Ib27dmCjRDCosEpGuBdSA4AoqyFALdZAZSCgGvTOB371HrThkbrGfgxNSTuKh5xRtFIrv9NAeEATphKoyoq3rTQKTHbQyhe8kjeEO038ESmQES5b+e9GO2KJBIVMUGPqvhdjM6UaOROQ5RqJgZiyHu1A3VJFJZhmOvw/c/et0nbDSNtR6A7Vvxcplcb0ZWA3JcWumfQG4n9ePcHwtJlyFScIiv0GhYlwMXIHZbhtroGh6FtCmeb2V5d1qaYMbWVjKXfd7qHkhpWznO3Gn2ximtyUS/5RqXx1XDg7HbW0RHbJHikSn5yQM3JBLkmFMPJInskLeXWenDfn3fn4XZ1xRjfbZAzO5w9LiJ+f</latexit> BayesBag incorporates model- and sampling-based uncertainty • Summarizing the previous slide… Posterior variance = model-based uncertainty Bootstrap variance = sampling-based uncertainty BayesBag variance = model-based + sampling-based uncertainty • Model correct: model-based uncertainty = sampling-based uncertainty • Posterior and bootstrap variances correct • BayesBag variance double-counts true uncertainty (conservative) • Model incorrect: model-based uncertainty ≪ sampling-based uncertainty uncertainty about mean ( Y obs ) • Posterior variance far 
 standard 
 too small bootstrap posterior distribution • BayesBag variance 
 appropriately calibrated bagged posterior 10 [ H & Miller 2019]

  51. Diagnosing model–data mismatch Posterior variance = model-based uncertainty BayesBag variance = model-based + sampling-based uncertainty 11 [ H & Miller 2019]

  52. Diagnosing model–data mismatch Posterior variance = model-based uncertainty BayesBag variance = model-based + sampling-based uncertainty • Mismatch index I can diagnose when 
 data disagrees with assumed model 11 [ H & Miller 2019]

  53. Diagnosing model–data mismatch Posterior variance = model-based uncertainty BayesBag variance = model-based + sampling-based uncertainty • Mismatch index I can diagnose when 
 data disagrees with assumed model • - 1 < I < 1 11 [ H & Miller 2019]

  54. Diagnosing model–data mismatch Posterior variance = model-based uncertainty BayesBag variance = model-based + sampling-based uncertainty • Mismatch index I can diagnose when 
 data disagrees with assumed model • - 1 < I < 1 • I ≈ 0 : no disagreement 11 [ H & Miller 2019]

  55. Diagnosing model–data mismatch Posterior variance = model-based uncertainty BayesBag variance = model-based + sampling-based uncertainty • Mismatch index I can diagnose when 
 data disagrees with assumed model • - 1 < I < 1 • I ≈ 0 : no disagreement • I > 0 : posterior overconfident 11 [ H & Miller 2019]

  56. Diagnosing model–data mismatch Posterior variance = model-based uncertainty BayesBag variance = model-based + sampling-based uncertainty • Mismatch index I can diagnose when 
 data disagrees with assumed model • - 1 < I < 1 • I ≈ 0 : no disagreement • I > 0 : posterior overconfident • I < 0 : posterior under-confident 11 [ H & Miller 2019]

  57. Diagnosing model–data mismatch Posterior variance = model-based uncertainty BayesBag variance = model-based + sampling-based uncertainty • Mismatch index I can diagnose when 
 data disagrees with assumed model mismatch index ≈ 1 • - 1 < I < 1 true standard 
 amount posterior • I ≈ 0 : no disagreement bagged • I > 0 : posterior overconfident posterior • I < 0 : posterior under-confident 11 [ H & Miller 2019]

  58. Diagnosing model–data mismatch Posterior variance = model-based uncertainty BayesBag variance = model-based + sampling-based uncertainty • Mismatch index I can diagnose when 
 data disagrees with assumed model mismatch index ≈ 1 • - 1 < I < 1 true standard 
 amount posterior • I ≈ 0 : no disagreement bagged • I > 0 : posterior overconfident posterior • I < 0 : posterior under-confident • Model criticism: mismatch index 
 indicates when model needs improvement 11 [ H & Miller 2019]

  59. Diagnosing model–data mismatch Posterior variance = model-based uncertainty BayesBag variance = model-based + sampling-based uncertainty • Mismatch index I can diagnose when 
 data disagrees with assumed model mismatch index ≈ 1 • - 1 < I < 1 true standard 
 amount posterior • I ≈ 0 : no disagreement bagged • I > 0 : posterior overconfident posterior • I < 0 : posterior under-confident • Model criticism: mismatch index 
 indicates when model needs improvement • Mismatch index can also detect problems with the prior 11 [ H & Miller 2019]

  60. BayesBag in practice 1) compute standard posterior π ( · | Y ) – e.g., use MCMC to get approximate samples θ (1) , . . . , θ ( T ) from π ( · | Y ) 2) compute bagged posterior π BB ( · | Y ) using B ≈ 50 bootstrap datasets – e.g., use MCMC to get approximate samples θ ∗ ( b,T ) from π ( · | Y ∗ ( b ) ) ( b, 1) , . . . , θ ∗ for b = 1 , . . . , B if Gaussian approximation to standard and bagged posteriors decent then 3a) compute mismatch index I 3b) if I ' . 2 , consider refining the model and returning to step 1 4) output bagged posterior computed in step 2 12 [ H & Miller 2019]

  61. BayesBag in practice 1) compute standard posterior π ( · | Y ) – e.g., use MCMC to get approximate samples θ (1) , . . . , θ ( T ) from π ( · | Y ) 2) compute bagged posterior π BB ( · | Y ) using B ≈ 50 bootstrap datasets – e.g., use MCMC to get approximate samples θ ∗ ( b,T ) from π ( · | Y ∗ ( b ) ) ( b, 1) , . . . , θ ∗ for b = 1 , . . . , B if Gaussian approximation to standard and bagged posteriors decent then 3a) compute mismatch index I 3b) if I ' . 2 , consider refining the model and returning to step 1 4) output bagged posterior computed in step 2 12 [ H & Miller 2019]

  62. BayesBag in practice 1) compute standard posterior π ( · | Y ) – e.g., use MCMC to get approximate samples θ (1) , . . . , θ ( T ) from π ( · | Y ) 2) compute bagged posterior π BB ( · | Y ) using B ≈ 50 bootstrap datasets – e.g., use MCMC to get approximate samples θ ∗ ( b,T ) from π ( · | Y ∗ ( b ) ) ( b, 1) , . . . , θ ∗ for b = 1 , . . . , B if Gaussian approximation to standard and bagged posteriors decent then 3a) compute mismatch index I 3b) if I ' . 2 , consider refining the model and returning to step 1 4) output bagged posterior computed in step 2 12 [ H & Miller 2019]

  63. BayesBag in practice 1) compute standard posterior π ( · | Y ) – e.g., use MCMC to get approximate samples θ (1) , . . . , θ ( T ) from π ( · | Y ) 2) compute bagged posterior π BB ( · | Y ) using B ≈ 50 bootstrap datasets – e.g., use MCMC to get approximate samples θ ∗ ( b,T ) from π ( · | Y ∗ ( b ) ) ( b, 1) , . . . , θ ∗ for b = 1 , . . . , B if Gaussian approximation to standard and bagged posteriors decent then 3a) compute mismatch index I 3b) if I ' . 2 , consider refining the model and returning to step 1 4) output bagged posterior computed in step 2 12 [ H & Miller 2019]

  64. BayesBag in practice 1) compute standard posterior π ( · | Y ) – e.g., use MCMC to get approximate samples θ (1) , . . . , θ ( T ) from π ( · | Y ) 2) compute bagged posterior π BB ( · | Y ) using B ≈ 50 bootstrap datasets – e.g., use MCMC to get approximate samples θ ∗ ( b,T ) from π ( · | Y ∗ ( b ) ) ( b, 1) , . . . , θ ∗ for b = 1 , . . . , B if Gaussian approximation to standard and bagged posteriors decent then 3a) compute mismatch index I 3b) if I ' . 2 , consider refining the model and returning to step 1 4) output bagged posterior computed in step 2 12 [ H & Miller 2019]

  65. BayesBag in practice 1) compute standard posterior π ( · | Y ) – e.g., use MCMC to get approximate samples θ (1) , . . . , θ ( T ) from π ( · | Y ) 2) compute bagged posterior π BB ( · | Y ) using B ≈ 50 bootstrap datasets – e.g., use MCMC to get approximate samples θ ∗ ( b,T ) from π ( · | Y ∗ ( b ) ) ( b, 1) , . . . , θ ∗ for b = 1 , . . . , B if Gaussian approximation to standard and bagged posteriors decent then 3a) compute mismatch index I 3b) if I ' . 2 , consider refining the model and returning to step 1 4) output bagged posterior computed in step 2 12 [ H & Miller 2019]

  66. BayesBag in practice 1) compute standard posterior π ( · | Y ) – e.g., use MCMC to get approximate samples θ (1) , . . . , θ ( T ) from π ( · | Y ) 2) compute bagged posterior π BB ( · | Y ) using B ≈ 50 bootstrap datasets – e.g., use MCMC to get approximate samples θ ∗ ( b,T ) from π ( · | Y ∗ ( b ) ) ( b, 1) , . . . , θ ∗ for b = 1 , . . . , B if Gaussian approximation to standard and bagged posteriors decent then 3a) compute mismatch index I 3b) if I ' . 2 , consider refining the model and returning to step 1 4) output bagged posterior computed in step 2 12 [ H & Miller 2019]

  67. BayesBag in practice 1) compute standard posterior π ( · | Y ) – e.g., use MCMC to get approximate samples θ (1) , . . . , θ ( T ) from π ( · | Y ) 2) compute bagged posterior π BB ( · | Y ) using B ≈ 50 bootstrap datasets – e.g., use MCMC to get approximate samples θ ∗ ( b,T ) from π ( · | Y ∗ ( b ) ) ( b, 1) , . . . , θ ∗ for b = 1 , . . . , B if Gaussian approximation to standard and bagged posteriors decent then 3a) compute mismatch index I 3b) if I ' . 2 , consider refining the model and returning to step 1 4) output bagged posterior computed in step 2 12 [ H & Miller 2019]

  68. BayesBag in practice 1) compute standard posterior π ( · | Y ) – e.g., use MCMC to get approximate samples θ (1) , . . . , θ ( T ) from π ( · | Y ) 2) compute bagged posterior π BB ( · | Y ) using B ≈ 50 bootstrap datasets – e.g., use MCMC to get approximate samples θ ∗ ( b,T ) from π ( · | Y ∗ ( b ) ) ( b, 1) , . . . , θ ∗ for b = 1 , . . . , B if Gaussian approximation to standard and bagged posteriors decent then 3a) compute mismatch index I 3b) if I ' . 2 , consider refining the model and returning to step 1 4) output bagged posterior computed in step 2 12 [ H & Miller 2019]

  69. Agenda • BayesBag for parameter inference (and prediction) • BayesBag theory and methodology ➡ BayesBag for model selection 13

  70. Bayesian model selection 14

  71. Bayesian model selection • Goal: based on data Y , select between 
 a (finite or countable) set of models 
 M = { m 1 , m 2 , …} 14

  72. Bayesian model selection • Goal: based on data Y , select between 
 a (finite or countable) set of models 
 M = { m 1 , m 2 , …} • Example: systematics 14

  73. <latexit sha1_base64="psq2jvV0jDNdqjwPulyjlu1ZWqw=">ACAXicbVDLSgNBEOz1GeMr6tHLYBC8GHajoMeAF48RzQOSJcxOZrNjZmaXmVkhLDl59qrf4E28+iV+gn/hJNmDSxoKq6e4KEs60cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqQhsk5rFqB1hTziRtGY4bSeKYhFw2gqGNxO/9USVZrF8MKOE+gIPJAsZwcZK96Ln9Uplt+JOgZaJl5My5Kj3Sj/dfkxSQaUhHGvd8dzE+BlWhFOx8VuqmCyRAPaMdSiQXVfjY9dYxOrdJHYaxsSYOm6t+JDAutRyKwnQKbSC96E/E/r5Oa8NrPmExSQyWZLQpTjkyMJn+jPlOUGD6yBPF7K2IRFhYmw6c1seo+hcME2q46LNxltMYpk0qxXvolK9uyzXanlKBTiGEzgD6gBrdQhwYQGMALvMKb8+y8Ox/O56x1xclnjmAOztcvFi2W3Q=</latexit> <latexit sha1_base64="fhdmpcEdfAZWx7deDX0E4BpTnA=">ACAXicbVDLSgNBEOz1GeMr6tHLYBC8GHajoMeAF48RzQOSJcxOZrNjZmaXmVkhLDl59qrf4E28+iV+gn/hJNmDSxoKq6e4KEs60cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqQhsk5rFqB1hTziRtGY4bSeKYhFw2gqGNxO/9USVZrF8MKOE+gIPJAsZwcZK96JX7ZXKbsWdAi0TLydlyFHvlX6/ZikgkpDONa647mJ8TOsDCOcjovdVNMEkyEe0I6lEguq/Wx6hidWqWPwljZkgZN1b8TGRZaj0RgOwU2kV70JuJ/Xic14bWfMZmkhkoyWxSmHJkYTf5GfaYoMXxkCSaK2VsRibDCxNh05rY8RtG5YJpUx0WbjbeYxDJpViveRaV6d1mu1fKUCnAMJ3AGHlxBDW6hDg0gMIAXeIU359l5dz6cz1nripPHMEcnK9fF8iW3g=</latexit> Bayesian model selection • Goal: based on data Y , select between 
 Fin whale a (finite or countable) set of models 
 m 1 M = { m 1 , m 2 , …} Blue whale Grey whale • Example: systematics Minke whale • Goal: learn about evolutionary history of a set of species [e.g. whales] Fin whale m 2 Blue whale Grey whale Minke whale . . . 14

  74. <latexit sha1_base64="Dna1Mf4nI3VDSu1KPRrf8wbeGoE=">ACFnicbVDLSgMxFM3UV62vUcGNm2AR6sJhpgp2IxTcuKxgH9IOQyZNO7FJZkgyhVL7H67d6je4E7du/QT/wvSxsK0HLhzOuZdzOWHCqNKu+21lVlbX1jeym7mt7Z3dPXv/oKbiVGJSxTGLZSNEijAqSFVTzUgjkQTxkJF62LsZ+/U+kYrG4l4PEuJz1BW0QzHSRgrso1ZCzwYKtL+0TAhzN4DZ1SYOdx50ALhNvRvJghkpg/7TaMU45ERozpFTcxPtD5HUFDMyrVSRKEe6hLmoYKxInyh5P/R/DUKG3YiaUZoeFE/XsxRFypAQ/NJkc6UoveWPzPa6a6U/KHVCSpJgJPgzopgzqG4zJgm0qCNRsYgrCk5leIyQR1qayuZTHKDrnVOHiKGe68RabWCa1ouNdOMW7y3y5PGspC47BCSgAD1yBMrgFVAFGDyBF/AK3qxn6936sD6nqxlrdnMI5mB9/QJjBp1D</latexit> <latexit sha1_base64="vicNYVxka5dmJD8Dq9XVcawYc0=">ACFnicbVDLSgMxFM3UV62vquDGTbAIdeEwUwXdCAU3LivYh3RKyaSZTmySGZJMoYz9D9du9RvciVu3foJ/YdrOwloPXDicy/ncvyYUaUd58vKLS2vrK7l1wsbm1vbO8XdvYaKEolJHUcski0fKcKoIHVNSOtWBLEfUa/uB64jeHRCoaiTs9ikmHo76gAcVIG6lbPBiWubdCvT6dEgEvD+BV9B2u8WSYztTwEXiZqQEMtS6xW+vF+GE6ExQ0q1XSfWnRJTEj4KXKBIjPEB90jZUIE5UJ53+P4bHRunBIJmhIZT9fdFirhSI+6bTY50qP56E/E/r53o4LKTUhEnmg8CwoSBnUEJ2XAHpUEazYyBGFJza8Qh0girE1lcykPYXjKqcKVcF04/5tYpE0KrZ7Zlduz0vVatZSHhyCI1AGLrgAVXADaqAOMHgEz+AFvFpP1pv1bn3MVnNWdrMP5mB9/gBZc509</latexit> <latexit sha1_base64="fhdmpcEdfAZWx7deDX0E4BpTnA=">ACAXicbVDLSgNBEOz1GeMr6tHLYBC8GHajoMeAF48RzQOSJcxOZrNjZmaXmVkhLDl59qrf4E28+iV+gn/hJNmDSxoKq6e4KEs60cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqQhsk5rFqB1hTziRtGY4bSeKYhFw2gqGNxO/9USVZrF8MKOE+gIPJAsZwcZK96JX7ZXKbsWdAi0TLydlyFHvlX6/ZikgkpDONa647mJ8TOsDCOcjovdVNMEkyEe0I6lEguq/Wx6hidWqWPwljZkgZN1b8TGRZaj0RgOwU2kV70JuJ/Xic14bWfMZmkhkoyWxSmHJkYTf5GfaYoMXxkCSaK2VsRibDCxNh05rY8RtG5YJpUx0WbjbeYxDJpViveRaV6d1mu1fKUCnAMJ3AGHlxBDW6hDg0gMIAXeIU359l5dz6cz1nripPHMEcnK9fF8iW3g=</latexit> <latexit sha1_base64="psq2jvV0jDNdqjwPulyjlu1ZWqw=">ACAXicbVDLSgNBEOz1GeMr6tHLYBC8GHajoMeAF48RzQOSJcxOZrNjZmaXmVkhLDl59qrf4E28+iV+gn/hJNmDSxoKq6e4KEs60cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqQhsk5rFqB1hTziRtGY4bSeKYhFw2gqGNxO/9USVZrF8MKOE+gIPJAsZwcZK96Ln9Uplt+JOgZaJl5My5Kj3Sj/dfkxSQaUhHGvd8dzE+BlWhFOx8VuqmCyRAPaMdSiQXVfjY9dYxOrdJHYaxsSYOm6t+JDAutRyKwnQKbSC96E/E/r5Oa8NrPmExSQyWZLQpTjkyMJn+jPlOUGD6yBPF7K2IRFhYmw6c1seo+hcME2q46LNxltMYpk0qxXvolK9uyzXanlKBTiGEzgD6gBrdQhwYQGMALvMKb8+y8Ox/O56x1xclnjmAOztcvFi2W3Q=</latexit> Bayesian model selection π ( m 1 | Y ) = . 8 • Goal: based on data Y , select between 
 Fin whale a (finite or countable) set of models 
 m 1 M = { m 1 , m 2 , …} Blue whale Grey whale • Example: systematics Minke whale • Goal: learn about evolutionary history π ( m 2 | Y ) = . 1 of a set of species [e.g. whales] Fin whale m 2 • Approach: infer which phylogenetic Blue whale trees are consistent with observed Grey whale species characteristics Y 
 Minke whale [e.g. genetic data, physical features such as coloring] . . . 14

  75. <latexit sha1_base64="psq2jvV0jDNdqjwPulyjlu1ZWqw=">ACAXicbVDLSgNBEOz1GeMr6tHLYBC8GHajoMeAF48RzQOSJcxOZrNjZmaXmVkhLDl59qrf4E28+iV+gn/hJNmDSxoKq6e4KEs60cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqQhsk5rFqB1hTziRtGY4bSeKYhFw2gqGNxO/9USVZrF8MKOE+gIPJAsZwcZK96Ln9Uplt+JOgZaJl5My5Kj3Sj/dfkxSQaUhHGvd8dzE+BlWhFOx8VuqmCyRAPaMdSiQXVfjY9dYxOrdJHYaxsSYOm6t+JDAutRyKwnQKbSC96E/E/r5Oa8NrPmExSQyWZLQpTjkyMJn+jPlOUGD6yBPF7K2IRFhYmw6c1seo+hcME2q46LNxltMYpk0qxXvolK9uyzXanlKBTiGEzgD6gBrdQhwYQGMALvMKb8+y8Ox/O56x1xclnjmAOztcvFi2W3Q=</latexit> <latexit sha1_base64="vicNYVxka5dmJD8Dq9XVcawYc0=">ACFnicbVDLSgMxFM3UV62vquDGTbAIdeEwUwXdCAU3LivYh3RKyaSZTmySGZJMoYz9D9du9RvciVu3foJ/YdrOwloPXDicy/ncvyYUaUd58vKLS2vrK7l1wsbm1vbO8XdvYaKEolJHUcski0fKcKoIHVNSOtWBLEfUa/uB64jeHRCoaiTs9ikmHo76gAcVIG6lbPBiWubdCvT6dEgEvD+BV9B2u8WSYztTwEXiZqQEMtS6xW+vF+GE6ExQ0q1XSfWnRJTEj4KXKBIjPEB90jZUIE5UJ53+P4bHRunBIJmhIZT9fdFirhSI+6bTY50qP56E/E/r53o4LKTUhEnmg8CwoSBnUEJ2XAHpUEazYyBGFJza8Qh0girE1lcykPYXjKqcKVcF04/5tYpE0KrZ7Zlduz0vVatZSHhyCI1AGLrgAVXADaqAOMHgEz+AFvFpP1pv1bn3MVnNWdrMP5mB9/gBZc509</latexit> <latexit sha1_base64="Dna1Mf4nI3VDSu1KPRrf8wbeGoE=">ACFnicbVDLSgMxFM3UV62vUcGNm2AR6sJhpgp2IxTcuKxgH9IOQyZNO7FJZkgyhVL7H67d6je4E7du/QT/wvSxsK0HLhzOuZdzOWHCqNKu+21lVlbX1jeym7mt7Z3dPXv/oKbiVGJSxTGLZSNEijAqSFVTzUgjkQTxkJF62LsZ+/U+kYrG4l4PEuJz1BW0QzHSRgrso1ZCzwYKtL+0TAhzN4DZ1SYOdx50ALhNvRvJghkpg/7TaMU45ERozpFTcxPtD5HUFDMyrVSRKEe6hLmoYKxInyh5P/R/DUKG3YiaUZoeFE/XsxRFypAQ/NJkc6UoveWPzPa6a6U/KHVCSpJgJPgzopgzqG4zJgm0qCNRsYgrCk5leIyQR1qayuZTHKDrnVOHiKGe68RabWCa1ouNdOMW7y3y5PGspC47BCSgAD1yBMrgFVAFGDyBF/AK3qxn6936sD6nqxlrdnMI5mB9/QJjBp1D</latexit> <latexit sha1_base64="fhdmpcEdfAZWx7deDX0E4BpTnA=">ACAXicbVDLSgNBEOz1GeMr6tHLYBC8GHajoMeAF48RzQOSJcxOZrNjZmaXmVkhLDl59qrf4E28+iV+gn/hJNmDSxoKq6e4KEs60cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqQhsk5rFqB1hTziRtGY4bSeKYhFw2gqGNxO/9USVZrF8MKOE+gIPJAsZwcZK96JX7ZXKbsWdAi0TLydlyFHvlX6/ZikgkpDONa647mJ8TOsDCOcjovdVNMEkyEe0I6lEguq/Wx6hidWqWPwljZkgZN1b8TGRZaj0RgOwU2kV70JuJ/Xic14bWfMZmkhkoyWxSmHJkYTf5GfaYoMXxkCSaK2VsRibDCxNh05rY8RtG5YJpUx0WbjbeYxDJpViveRaV6d1mu1fKUCnAMJ3AGHlxBDW6hDg0gMIAXeIU359l5dz6cz1nripPHMEcnK9fF8iW3g=</latexit> Bayesian model selection π ( m 1 | Y ) = . 8 • Goal: based on data Y , select between 
 Fin whale a (finite or countable) set of models 
 m 1 M = { m 1 , m 2 , …} Blue whale Grey whale • Example: systematics Minke whale • Goal: learn about evolutionary history π ( m 2 | Y ) = . 1 of a set of species [e.g. whales] Fin whale m 2 • Approach: infer which phylogenetic Blue whale trees are consistent with observed Grey whale species characteristics Y 
 Minke whale [e.g. genetic data, physical features such as coloring] . • Problem: Bayesian model selection still . . assumes some model in M is correct 14

  76. Illustration: two normal models • Models are m 1 = N ( − 1 , 1) and m 2 = N (1 , 1) • True distribution is P true = N (0 , 1) • Generate datasets Y (1) , Y (2) , . . . of size n = 1000 , where Y ( i ) ∼ N (0 , 1) . j 15 [ H & Miller 2019]

  77. Illustration: two normal models • Models are m 1 = N ( − 1 , 1) and m 2 = N (1 , 1) • True distribution is P true = N (0 , 1) • Generate datasets Y (1) , Y (2) , . . . of size n = 1000 , where Y ( i ) ∼ N (0 , 1) . j 15 [ H & Miller 2019]

  78. <latexit sha1_base64="fhdmpcEdfAZWx7deDX0E4BpTnA=">ACAXicbVDLSgNBEOz1GeMr6tHLYBC8GHajoMeAF48RzQOSJcxOZrNjZmaXmVkhLDl59qrf4E28+iV+gn/hJNmDSxoKq6e4KEs60cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqQhsk5rFqB1hTziRtGY4bSeKYhFw2gqGNxO/9USVZrF8MKOE+gIPJAsZwcZK96JX7ZXKbsWdAi0TLydlyFHvlX6/ZikgkpDONa647mJ8TOsDCOcjovdVNMEkyEe0I6lEguq/Wx6hidWqWPwljZkgZN1b8TGRZaj0RgOwU2kV70JuJ/Xic14bWfMZmkhkoyWxSmHJkYTf5GfaYoMXxkCSaK2VsRibDCxNh05rY8RtG5YJpUx0WbjbeYxDJpViveRaV6d1mu1fKUCnAMJ3AGHlxBDW6hDg0gMIAXeIU359l5dz6cz1nripPHMEcnK9fF8iW3g=</latexit> <latexit sha1_base64="psq2jvV0jDNdqjwPulyjlu1ZWqw=">ACAXicbVDLSgNBEOz1GeMr6tHLYBC8GHajoMeAF48RzQOSJcxOZrNjZmaXmVkhLDl59qrf4E28+iV+gn/hJNmDSxoKq6e4KEs60cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqQhsk5rFqB1hTziRtGY4bSeKYhFw2gqGNxO/9USVZrF8MKOE+gIPJAsZwcZK96Ln9Uplt+JOgZaJl5My5Kj3Sj/dfkxSQaUhHGvd8dzE+BlWhFOx8VuqmCyRAPaMdSiQXVfjY9dYxOrdJHYaxsSYOm6t+JDAutRyKwnQKbSC96E/E/r5Oa8NrPmExSQyWZLQpTjkyMJn+jPlOUGD6yBPF7K2IRFhYmw6c1seo+hcME2q46LNxltMYpk0qxXvolK9uyzXanlKBTiGEzgD6gBrdQhwYQGMALvMKb8+y8Ox/O56x1xclnjmAOztcvFi2W3Q=</latexit> <latexit sha1_base64="kZ0OSsl3umUaxihkyoGtKj21hWI=">ACD3icbVDLSgNBEJyNrxgfiXr0MhgEL4bdKOgx4MVjBPOAZFlmJ73JmNkHM71iWPIRnr3qN3gTr36Cn+BfOHkcTGJBQ1HVTXlJ1JotO1vK7e2vrG5ld8u7Ozu7RdLB4dNHaeKQ4PHMlZtn2mQIoIGCpTQThSw0JfQ8oc3E7/1CEqLOLrHUQJuyPqRCARnaCSvVKx7XYQn1EGKoWxVyrbFXsKukqcOSmTOepe6afbi3kaQoRcMq07jp2gmzGFgksYF7qphoTxIetDx9CIhaDdbPr4mJ4apUeDWJmJkE7VvxcZC7Uehb7ZDBkO9LI3Ef/zOikG124moiRFiPgsKEglxZhOWqA9oYCjHBnCuBLmV8oHTDGOpquFlIfB4DwUmlfHBdONs9zEKmlWK85FpXp3Wa7V5i3lyTE5IWfEIVekRm5JnTQIJyl5Ia/kzXq23q0P63O2mrPmN0dkAdbXLx7JnOo=</latexit> Illustration: two normal models P true • Models are m 1 = N ( − 1 , 1) and m 2 = N (1 , 1) m 1 m 2 • True distribution is P true = N (0 , 1) • Generate datasets Y (1) , Y (2) , . . . of size n = 1000 , where Y ( i ) ∼ N (0 , 1) . j 15 [ H & Miller 2019]

  79. <latexit sha1_base64="fhdmpcEdfAZWx7deDX0E4BpTnA=">ACAXicbVDLSgNBEOz1GeMr6tHLYBC8GHajoMeAF48RzQOSJcxOZrNjZmaXmVkhLDl59qrf4E28+iV+gn/hJNmDSxoKq6e4KEs60cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqQhsk5rFqB1hTziRtGY4bSeKYhFw2gqGNxO/9USVZrF8MKOE+gIPJAsZwcZK96JX7ZXKbsWdAi0TLydlyFHvlX6/ZikgkpDONa647mJ8TOsDCOcjovdVNMEkyEe0I6lEguq/Wx6hidWqWPwljZkgZN1b8TGRZaj0RgOwU2kV70JuJ/Xic14bWfMZmkhkoyWxSmHJkYTf5GfaYoMXxkCSaK2VsRibDCxNh05rY8RtG5YJpUx0WbjbeYxDJpViveRaV6d1mu1fKUCnAMJ3AGHlxBDW6hDg0gMIAXeIU359l5dz6cz1nripPHMEcnK9fF8iW3g=</latexit> <latexit sha1_base64="psq2jvV0jDNdqjwPulyjlu1ZWqw=">ACAXicbVDLSgNBEOz1GeMr6tHLYBC8GHajoMeAF48RzQOSJcxOZrNjZmaXmVkhLDl59qrf4E28+iV+gn/hJNmDSxoKq6e4KEs60cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqQhsk5rFqB1hTziRtGY4bSeKYhFw2gqGNxO/9USVZrF8MKOE+gIPJAsZwcZK96Ln9Uplt+JOgZaJl5My5Kj3Sj/dfkxSQaUhHGvd8dzE+BlWhFOx8VuqmCyRAPaMdSiQXVfjY9dYxOrdJHYaxsSYOm6t+JDAutRyKwnQKbSC96E/E/r5Oa8NrPmExSQyWZLQpTjkyMJn+jPlOUGD6yBPF7K2IRFhYmw6c1seo+hcME2q46LNxltMYpk0qxXvolK9uyzXanlKBTiGEzgD6gBrdQhwYQGMALvMKb8+y8Ox/O56x1xclnjmAOztcvFi2W3Q=</latexit> <latexit sha1_base64="kZ0OSsl3umUaxihkyoGtKj21hWI=">ACD3icbVDLSgNBEJyNrxgfiXr0MhgEL4bdKOgx4MVjBPOAZFlmJ73JmNkHM71iWPIRnr3qN3gTr36Cn+BfOHkcTGJBQ1HVTXlJ1JotO1vK7e2vrG5ld8u7Ozu7RdLB4dNHaeKQ4PHMlZtn2mQIoIGCpTQThSw0JfQ8oc3E7/1CEqLOLrHUQJuyPqRCARnaCSvVKx7XYQn1EGKoWxVyrbFXsKukqcOSmTOepe6afbi3kaQoRcMq07jp2gmzGFgksYF7qphoTxIetDx9CIhaDdbPr4mJ4apUeDWJmJkE7VvxcZC7Uehb7ZDBkO9LI3Ef/zOikG124moiRFiPgsKEglxZhOWqA9oYCjHBnCuBLmV8oHTDGOpquFlIfB4DwUmlfHBdONs9zEKmlWK85FpXp3Wa7V5i3lyTE5IWfEIVekRm5JnTQIJyl5Ia/kzXq23q0P63O2mrPmN0dkAdbXLx7JnOo=</latexit> Illustration: two normal models P true • Models are m 1 = N ( − 1 , 1) and m 2 = N (1 , 1) m 1 m 2 • True distribution is P true = N (0 , 1) • Generate datasets Y (1) , Y (2) , . . . of size n = 1000 , where Y ( i ) ∼ N (0 , 1) . j 15 [ H & Miller 2019]

  80. <latexit sha1_base64="kZ0OSsl3umUaxihkyoGtKj21hWI=">ACD3icbVDLSgNBEJyNrxgfiXr0MhgEL4bdKOgx4MVjBPOAZFlmJ73JmNkHM71iWPIRnr3qN3gTr36Cn+BfOHkcTGJBQ1HVTXlJ1JotO1vK7e2vrG5ld8u7Ozu7RdLB4dNHaeKQ4PHMlZtn2mQIoIGCpTQThSw0JfQ8oc3E7/1CEqLOLrHUQJuyPqRCARnaCSvVKx7XYQn1EGKoWxVyrbFXsKukqcOSmTOepe6afbi3kaQoRcMq07jp2gmzGFgksYF7qphoTxIetDx9CIhaDdbPr4mJ4apUeDWJmJkE7VvxcZC7Uehb7ZDBkO9LI3Ef/zOikG124moiRFiPgsKEglxZhOWqA9oYCjHBnCuBLmV8oHTDGOpquFlIfB4DwUmlfHBdONs9zEKmlWK85FpXp3Wa7V5i3lyTE5IWfEIVekRm5JnTQIJyl5Ia/kzXq23q0P63O2mrPmN0dkAdbXLx7JnOo=</latexit> <latexit sha1_base64="Ivi0gmU1upmZsqXA7NUQCvEpOTc=">ACRnicdVDLSgMxFL1T3/VdekmWJS6sMxUwW6EUkFcKlgfdOqQSdM2NskMSUYow/yRH+LanegPuHEnbk1rFz4PBA7nMu9OWHMmTau+jkJianpmdm5/LzC4tLy4WV1XMdJYrQBol4pC5DrClnkjYM5xexopiEXJ6EfYPh/7FLVWaRfLMDGLaErgrWYcRbKwUFI6QH7OSCDzkd9ktlejqOi1529k2jpAVvTzw0CQ1uvZfym3XK0EhaJbdkdAv4k3JkUY4yQovPjtiCSCSkM41rpubFpVgZRjN8n6iaYxJH3dp01KJBdWtdPTfDG1apY06kbJPGjRSv06kWGg9EKFNCmx6+qc3FP/ymonpVFspk3FiqCSfizoJRyZCw/JQmylKDB9Ygoli9lZEelhYmzF37bc9Ho7gmlSyfK2G+9nE7/JeaXs7ZYrp3vFWnXc0iyswaUwIN9qMExnEADCNzBAzBs3PvDpvzvtnNOeMZ9bgG3LwAcFJq7Y=</latexit> <latexit sha1_base64="fhdmpcEdfAZWx7deDX0E4BpTnA=">ACAXicbVDLSgNBEOz1GeMr6tHLYBC8GHajoMeAF48RzQOSJcxOZrNjZmaXmVkhLDl59qrf4E28+iV+gn/hJNmDSxoKq6e4KEs60cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqQhsk5rFqB1hTziRtGY4bSeKYhFw2gqGNxO/9USVZrF8MKOE+gIPJAsZwcZK96JX7ZXKbsWdAi0TLydlyFHvlX6/ZikgkpDONa647mJ8TOsDCOcjovdVNMEkyEe0I6lEguq/Wx6hidWqWPwljZkgZN1b8TGRZaj0RgOwU2kV70JuJ/Xic14bWfMZmkhkoyWxSmHJkYTf5GfaYoMXxkCSaK2VsRibDCxNh05rY8RtG5YJpUx0WbjbeYxDJpViveRaV6d1mu1fKUCnAMJ3AGHlxBDW6hDg0gMIAXeIU359l5dz6cz1nripPHMEcnK9fF8iW3g=</latexit> <latexit sha1_base64="psq2jvV0jDNdqjwPulyjlu1ZWqw=">ACAXicbVDLSgNBEOz1GeMr6tHLYBC8GHajoMeAF48RzQOSJcxOZrNjZmaXmVkhLDl59qrf4E28+iV+gn/hJNmDSxoKq6e4KEs60cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqQhsk5rFqB1hTziRtGY4bSeKYhFw2gqGNxO/9USVZrF8MKOE+gIPJAsZwcZK96Ln9Uplt+JOgZaJl5My5Kj3Sj/dfkxSQaUhHGvd8dzE+BlWhFOx8VuqmCyRAPaMdSiQXVfjY9dYxOrdJHYaxsSYOm6t+JDAutRyKwnQKbSC96E/E/r5Oa8NrPmExSQyWZLQpTjkyMJn+jPlOUGD6yBPF7K2IRFhYmw6c1seo+hcME2q46LNxltMYpk0qxXvolK9uyzXanlKBTiGEzgD6gBrdQhwYQGMALvMKb8+y8Ox/O56x1xclnjmAOztcvFi2W3Q=</latexit> Illustration: two normal models P true • Models are m 1 = N ( − 1 , 1) and m 2 = N (1 , 1) m 1 m 2 • True distribution is P true = N (0 , 1) • Generate datasets Y (1) , Y (2) , . . . of size n = 1000 , where Y ( i ) ∼ N (0 , 1) . j π ( m 1 | Y (1) ) = 1 π BB ( m 1 | Y (1) ) = 0 . 82 15 [ H & Miller 2019]

  81. <latexit sha1_base64="kZ0OSsl3umUaxihkyoGtKj21hWI=">ACD3icbVDLSgNBEJyNrxgfiXr0MhgEL4bdKOgx4MVjBPOAZFlmJ73JmNkHM71iWPIRnr3qN3gTr36Cn+BfOHkcTGJBQ1HVTXlJ1JotO1vK7e2vrG5ld8u7Ozu7RdLB4dNHaeKQ4PHMlZtn2mQIoIGCpTQThSw0JfQ8oc3E7/1CEqLOLrHUQJuyPqRCARnaCSvVKx7XYQn1EGKoWxVyrbFXsKukqcOSmTOepe6afbi3kaQoRcMq07jp2gmzGFgksYF7qphoTxIetDx9CIhaDdbPr4mJ4apUeDWJmJkE7VvxcZC7Uehb7ZDBkO9LI3Ef/zOikG124moiRFiPgsKEglxZhOWqA9oYCjHBnCuBLmV8oHTDGOpquFlIfB4DwUmlfHBdONs9zEKmlWK85FpXp3Wa7V5i3lyTE5IWfEIVekRm5JnTQIJyl5Ia/kzXq23q0P63O2mrPmN0dkAdbXLx7JnOo=</latexit> <latexit sha1_base64="TNEudSfRl895hmo28QCwhwGM0FI=">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</latexit> <latexit sha1_base64="fhdmpcEdfAZWx7deDX0E4BpTnA=">ACAXicbVDLSgNBEOz1GeMr6tHLYBC8GHajoMeAF48RzQOSJcxOZrNjZmaXmVkhLDl59qrf4E28+iV+gn/hJNmDSxoKq6e4KEs60cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqQhsk5rFqB1hTziRtGY4bSeKYhFw2gqGNxO/9USVZrF8MKOE+gIPJAsZwcZK96JX7ZXKbsWdAi0TLydlyFHvlX6/ZikgkpDONa647mJ8TOsDCOcjovdVNMEkyEe0I6lEguq/Wx6hidWqWPwljZkgZN1b8TGRZaj0RgOwU2kV70JuJ/Xic14bWfMZmkhkoyWxSmHJkYTf5GfaYoMXxkCSaK2VsRibDCxNh05rY8RtG5YJpUx0WbjbeYxDJpViveRaV6d1mu1fKUCnAMJ3AGHlxBDW6hDg0gMIAXeIU359l5dz6cz1nripPHMEcnK9fF8iW3g=</latexit> <latexit sha1_base64="psq2jvV0jDNdqjwPulyjlu1ZWqw=">ACAXicbVDLSgNBEOz1GeMr6tHLYBC8GHajoMeAF48RzQOSJcxOZrNjZmaXmVkhLDl59qrf4E28+iV+gn/hJNmDSxoKq6e4KEs60cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqQhsk5rFqB1hTziRtGY4bSeKYhFw2gqGNxO/9USVZrF8MKOE+gIPJAsZwcZK96Ln9Uplt+JOgZaJl5My5Kj3Sj/dfkxSQaUhHGvd8dzE+BlWhFOx8VuqmCyRAPaMdSiQXVfjY9dYxOrdJHYaxsSYOm6t+JDAutRyKwnQKbSC96E/E/r5Oa8NrPmExSQyWZLQpTjkyMJn+jPlOUGD6yBPF7K2IRFhYmw6c1seo+hcME2q46LNxltMYpk0qxXvolK9uyzXanlKBTiGEzgD6gBrdQhwYQGMALvMKb8+y8Ox/O56x1xclnjmAOztcvFi2W3Q=</latexit> <latexit sha1_base64="Ivi0gmU1upmZsqXA7NUQCvEpOTc=">ACRnicdVDLSgMxFL1T3/VdekmWJS6sMxUwW6EUkFcKlgfdOqQSdM2NskMSUYow/yRH+LanegPuHEnbk1rFz4PBA7nMu9OWHMmTau+jkJianpmdm5/LzC4tLy4WV1XMdJYrQBol4pC5DrClnkjYM5xexopiEXJ6EfYPh/7FLVWaRfLMDGLaErgrWYcRbKwUFI6QH7OSCDzkd9ktlejqOi1529k2jpAVvTzw0CQ1uvZfym3XK0EhaJbdkdAv4k3JkUY4yQovPjtiCSCSkM41rpubFpVgZRjN8n6iaYxJH3dp01KJBdWtdPTfDG1apY06kbJPGjRSv06kWGg9EKFNCmx6+qc3FP/ymonpVFspk3FiqCSfizoJRyZCw/JQmylKDB9Ygoli9lZEelhYmzF37bc9Ho7gmlSyfK2G+9nE7/JeaXs7ZYrp3vFWnXc0iyswaUwIN9qMExnEADCNzBAzBs3PvDpvzvtnNOeMZ9bgG3LwAcFJq7Y=</latexit> Illustration: two normal models P true • Models are m 1 = N ( − 1 , 1) and m 2 = N (1 , 1) m 1 m 2 • True distribution is P true = N (0 , 1) • Generate datasets Y (1) , Y (2) , . . . of size n = 1000 , where Y ( i ) ∼ N (0 , 1) . j π ( m 1 | Y (1) ) = 1 π ( m 1 | Y (2) ) = 10 − 5 π BB ( m 1 | Y (1) ) = 0 . 82 π BB ( m 1 | Y (2) ) = 0 . 38 15 [ H & Miller 2019]

  82. <latexit sha1_base64="kZ0OSsl3umUaxihkyoGtKj21hWI=">ACD3icbVDLSgNBEJyNrxgfiXr0MhgEL4bdKOgx4MVjBPOAZFlmJ73JmNkHM71iWPIRnr3qN3gTr36Cn+BfOHkcTGJBQ1HVTXlJ1JotO1vK7e2vrG5ld8u7Ozu7RdLB4dNHaeKQ4PHMlZtn2mQIoIGCpTQThSw0JfQ8oc3E7/1CEqLOLrHUQJuyPqRCARnaCSvVKx7XYQn1EGKoWxVyrbFXsKukqcOSmTOepe6afbi3kaQoRcMq07jp2gmzGFgksYF7qphoTxIetDx9CIhaDdbPr4mJ4apUeDWJmJkE7VvxcZC7Uehb7ZDBkO9LI3Ef/zOikG124moiRFiPgsKEglxZhOWqA9oYCjHBnCuBLmV8oHTDGOpquFlIfB4DwUmlfHBdONs9zEKmlWK85FpXp3Wa7V5i3lyTE5IWfEIVekRm5JnTQIJyl5Ia/kzXq23q0P63O2mrPmN0dkAdbXLx7JnOo=</latexit> <latexit sha1_base64="Ivi0gmU1upmZsqXA7NUQCvEpOTc=">ACRnicdVDLSgMxFL1T3/VdekmWJS6sMxUwW6EUkFcKlgfdOqQSdM2NskMSUYow/yRH+LanegPuHEnbk1rFz4PBA7nMu9OWHMmTau+jkJianpmdm5/LzC4tLy4WV1XMdJYrQBol4pC5DrClnkjYM5xexopiEXJ6EfYPh/7FLVWaRfLMDGLaErgrWYcRbKwUFI6QH7OSCDzkd9ktlejqOi1529k2jpAVvTzw0CQ1uvZfym3XK0EhaJbdkdAv4k3JkUY4yQovPjtiCSCSkM41rpubFpVgZRjN8n6iaYxJH3dp01KJBdWtdPTfDG1apY06kbJPGjRSv06kWGg9EKFNCmx6+qc3FP/ymonpVFspk3FiqCSfizoJRyZCw/JQmylKDB9Ygoli9lZEelhYmzF37bc9Ho7gmlSyfK2G+9nE7/JeaXs7ZYrp3vFWnXc0iyswaUwIN9qMExnEADCNzBAzBs3PvDpvzvtnNOeMZ9bgG3LwAcFJq7Y=</latexit> <latexit sha1_base64="fhdmpcEdfAZWx7deDX0E4BpTnA=">ACAXicbVDLSgNBEOz1GeMr6tHLYBC8GHajoMeAF48RzQOSJcxOZrNjZmaXmVkhLDl59qrf4E28+iV+gn/hJNmDSxoKq6e4KEs60cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqQhsk5rFqB1hTziRtGY4bSeKYhFw2gqGNxO/9USVZrF8MKOE+gIPJAsZwcZK96JX7ZXKbsWdAi0TLydlyFHvlX6/ZikgkpDONa647mJ8TOsDCOcjovdVNMEkyEe0I6lEguq/Wx6hidWqWPwljZkgZN1b8TGRZaj0RgOwU2kV70JuJ/Xic14bWfMZmkhkoyWxSmHJkYTf5GfaYoMXxkCSaK2VsRibDCxNh05rY8RtG5YJpUx0WbjbeYxDJpViveRaV6d1mu1fKUCnAMJ3AGHlxBDW6hDg0gMIAXeIU359l5dz6cz1nripPHMEcnK9fF8iW3g=</latexit> <latexit sha1_base64="psq2jvV0jDNdqjwPulyjlu1ZWqw=">ACAXicbVDLSgNBEOz1GeMr6tHLYBC8GHajoMeAF48RzQOSJcxOZrNjZmaXmVkhLDl59qrf4E28+iV+gn/hJNmDSxoKq6e4KEs60cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqQhsk5rFqB1hTziRtGY4bSeKYhFw2gqGNxO/9USVZrF8MKOE+gIPJAsZwcZK96Ln9Uplt+JOgZaJl5My5Kj3Sj/dfkxSQaUhHGvd8dzE+BlWhFOx8VuqmCyRAPaMdSiQXVfjY9dYxOrdJHYaxsSYOm6t+JDAutRyKwnQKbSC96E/E/r5Oa8NrPmExSQyWZLQpTjkyMJn+jPlOUGD6yBPF7K2IRFhYmw6c1seo+hcME2q46LNxltMYpk0qxXvolK9uyzXanlKBTiGEzgD6gBrdQhwYQGMALvMKb8+y8Ox/O56x1xclnjmAOztcvFi2W3Q=</latexit> <latexit sha1_base64="A9avW824nbyUBeyHQCbXB9hytI=">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</latexit> <latexit sha1_base64="TNEudSfRl895hmo28QCwhwGM0FI=">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</latexit> Illustration: two normal models P true • Models are m 1 = N ( − 1 , 1) and m 2 = N (1 , 1) m 1 m 2 • True distribution is P true = N (0 , 1) • Generate datasets Y (1) , Y (2) , . . . of size n = 1000 , where Y ( i ) ∼ N (0 , 1) . j π ( m 1 | Y (1) ) = 1 π ( m 1 | Y (2) ) = 10 − 5 π ( m 1 | Y (3) ) = 1 π BB ( m 1 | Y (1) ) = 0 . 82 π BB ( m 1 | Y (2) ) = 0 . 38 π BB ( m 1 | Y (3) ) = 0 . 90 15 [ H & Miller 2019]

  83. BayesBag stabilizes posterior probabilities of similar models 16 [Yang & Zhu 2018, H & Miller 2019]

  84. BayesBag stabilizes posterior probabilities of similar models • Assume two models m 1 and m 2 [e.g. two possible trees] 16 [Yang & Zhu 2018, H & Miller 2019]

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