finite approximations of discrete random measures
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Finite Approximations of Discrete Random Measures Jonathan H. - PowerPoint PPT Presentation

Finite Approximations of Discrete Random Measures Jonathan H. Huggins Postdoctoral Research Fellow Department of Biostatistics, Harvard with: Trevor Campbell, Jonathan How, Lorenzo Masoero, Lester Mackey, Tamara Broderick Bayesian


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[Prunster 2014] [Yang 2015] [Yu 2012] [Ozaki 2008] [Kottas 2008] [Gopalan 2014]

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2010] [Prunster 2014] [Yang 2015] [Yu 2012] [Ozaki 2008] [Kottas 2008] [Gopalan 2014]

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sha1_base64="GS65xYe7wEz8wPA5Q5WNH9n/e5E=">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</latexit> Bayesian nonparametrics Need models that can e.g. keep learning new topics extract new, useful from a stream of documents information from unbounded streams of data genetics movie text medicine robotics s Bayesian nonparametrics: achieves growing model size via infinite parameters finance astronomy traffic agriculture pathology parameter Π (d Θ | X ) ∝ Θ f ( X | Θ ) Π 0 (d Θ ) data likelihood [Teh 2006] [Huang 2014] [Michini 2015] [Lennox 2010] [Prunster 2014] [Yang 2015] [Yu 2012] [Ozaki 2008] [Kottas 2008] [Gopalan 2014]

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[Lennox 2010] [Prunster 2014] [Yang 2015] [Yu 2012] [Ozaki 2008] [Kottas 2008] [Gopalan 2014]

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[Michini 2015] [Lennox 2010] [Prunster 2014] [Yang 2015] [Yu 2012] [Ozaki 2008] [Kottas 2008] [Gopalan 2014]

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sha1_base64="cNS2H+PALUlqnNrUNjNPSF/RX2g=">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</latexit> <latexit sha1_base64="cNS2H+PALUlqnNrUNjNPSF/RX2g=">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</latexit> Inference in BNP models π (d Θ | X ) ∝ Θ f ( X | Θ ) π 0 (d Θ )

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sha1_base64="DZ40NfcVNd9ARWOoJPQXW3ABc3U=">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</latexit> Inference in BNP models π (d Θ | X ) ∝ Θ f ( X | Θ ) π 0 (d Θ ) • Option #1: Integrate out the parameter (CRP, IBP, etc.) 
 Θ issues: care about the parameters, using certain inference algs. (HMC/VB), distributed computation, discrete latent variables instead

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sha1_base64="DZ40NfcVNd9ARWOoJPQXW3ABc3U=">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</latexit> Inference in BNP models π (d Θ | X ) ∝ Θ f ( X | Θ ) π 0 (d Θ ) • Option #1: Integrate out the parameter (CRP, IBP, etc.) 
 Θ issues: care about the parameters, using certain inference algs. (HMC/VB), distributed computation, discrete latent variables instead • Option #2: use a finite approximation... 
 with e.g. variational inference, HMC [Blei 06; Neal 10]

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sha1_base64="DZ40NfcVNd9ARWOoJPQXW3ABc3U=">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</latexit> Inference in BNP models π (d Θ | X ) ∝ Θ f ( X | Θ ) π 0 (d Θ ) • Option #1: Integrate out the parameter (CRP, IBP, etc.) 
 Θ issues: care about the parameters, using certain inference algs. (HMC/VB), distributed computation, discrete latent variables instead • Option #2: use a finite approximation... 
 with e.g. variational inference, HMC [Blei 06; Neal 10] Problem: Wide variety of priors in BNP with 
 no or poorly understood finite approximation

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sha1_base64="cNS2H+PALUlqnNrUNjNPSF/RX2g=">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</latexit> <latexit sha1_base64="cNS2H+PALUlqnNrUNjNPSF/RX2g=">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</latexit> Inference in BNP models π (d Θ | X ) ∝ Θ f ( X | Θ ) π 0 (d Θ ) • Option #1: Integrate out the parameter (CRP, IBP, etc.) 
 Θ issues: care about the parameters, using certain inference algs. (HMC/VB), distributed computation, discrete latent variables instead • Option #2: use a finite approximation... 
 with e.g. variational inference, HMC [Blei 06; Neal 10] Problem: Wide variety of priors in BNP with 
 no or poorly understood finite approximation In this talk:

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sha1_base64="DZ40NfcVNd9ARWOoJPQXW3ABc3U=">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</latexit> Inference in BNP models π (d Θ | X ) ∝ Θ f ( X | Θ ) π 0 (d Θ ) • Option #1: Integrate out the parameter (CRP, IBP, etc.) 
 Θ issues: care about the parameters, using certain inference algs. (HMC/VB), distributed computation, discrete latent variables instead • Option #2: use a finite approximation... 
 with e.g. variational inference, HMC [Blei 06; Neal 10] Problem: Wide variety of priors in BNP with 
 no or poorly understood finite approximation In this talk: 1) Two finite approximation types: truncated and non-nested

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sha1_base64="DZ40NfcVNd9ARWOoJPQXW3ABc3U=">ACaHicbVHLThtBEBwvkDgQgk0OEeIyworEBWvXFzhFIC45EAkbJBsC/XO9rID81jN9IKslf8h1/Bn+QW+grHxITaUNFKpulrdXZOWSnqK43+NaG1948PH5qfNrc/bX3Za7d2Bt5UT2BdWXedgkclDfZJksLr0iHoVOFVen8+q189oPSmkualDjWcGtkLgVQkAajywIJblqduBvPwd+SZE6bIGLm3bj1yizotJoSCjwfpjEJY1rcCSFwunmqPJYgriHWxwGakCjH9fzdaf8e1AynlsXniE+V/vqEF7P9FpcGqgwq/WZuJ7tWF+cm4lqasCI14HZRXipPls9t5Jh0KUpNAQDgZduWiAeCQkJLU+6K4khL3pLp9RK2wydCaJDg4/Cag0mq0dOZ5hDpWhal8VDGMzPnAS1avP5+7aQf7Ka9lsy6HWTuJv8TjqnPxY/0WT7IAdsoQds1P2k12wPhPsjv1hf9lT4zlqRd+ivVdr1Fj0fGVLiA5eAFubvCo=</latexit> Inference in BNP models π (d Θ | X ) ∝ Θ f ( X | Θ ) π 0 (d Θ ) • Option #1: Integrate out the parameter (CRP, IBP, etc.) 
 Θ issues: care about the parameters, using certain inference algs. (HMC/VB), distributed computation, discrete latent variables instead • Option #2: use a finite approximation... 
 with e.g. variational inference, HMC [Blei 06; Neal 10] Problem: Wide variety of priors in BNP with 
 no or poorly understood finite approximation In this talk: 1) Two finite approximation types: truncated and non-nested 2) Two truncated forms (7 reps total) that allow finite approximation of (normalized) completely random measures [(N)CRMs]

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sha1_base64="cNS2H+PALUlqnNrUNjNPSF/RX2g=">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</latexit> Inference in BNP models π (d Θ | X ) ∝ Θ f ( X | Θ ) π 0 (d Θ ) • Option #1: Integrate out the parameter (CRP, IBP, etc.) 
 Θ issues: care about the parameters, using certain inference algs. (HMC/VB), distributed computation, discrete latent variables instead • Option #2: use a finite approximation... 
 with e.g. variational inference, HMC [Blei 06; Neal 10] Problem: Wide variety of priors in BNP with 
 no or poorly understood finite approximation In this talk: 1) Two finite approximation types: truncated and non-nested 2) Two truncated forms (7 reps total) that allow finite approximation of (normalized) completely random measures [(N)CRMs] 3) Truncation approximation error analysis

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sha1_base64="cNS2H+PALUlqnNrUNjNPSF/RX2g=">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</latexit> Inference in BNP models π (d Θ | X ) ∝ Θ f ( X | Θ ) π 0 (d Θ ) • Option #1: Integrate out the parameter (CRP, IBP, etc.) 
 Θ issues: care about the parameters, using certain inference algs. (HMC/VB), distributed computation, discrete latent variables instead • Option #2: use a finite approximation... 
 with e.g. variational inference, HMC [Blei 06; Neal 10] Problem: Wide variety of priors in BNP with 
 no or poorly understood finite approximation In this talk: 1) Two finite approximation types: truncated and non-nested 2) Two truncated forms (7 reps total) that allow finite approximation of (normalized) completely random measures [(N)CRMs] 3) Truncation approximation error analysis 4) One non-nested form for (N)CRMs

  15. Outline ➡ Tractable priors in BNP • Truncated approximations • Two forms for sequential representations • Truncation and error analysis • Non-nested approximations

  16. The Standard Model in BNP (By Example ) politics s t r d o o … p o s f 343 189 Doc 1 (532 words) 210 Doc 2 (210 words) 854 Doc 3 (854 words) 342 584 Doc 4 (926 words) … 0.7 0.5 0.2

  17. The Standard Model in BNP (By Example ) frequency space politics s t r d o o … p o s f 343 189 Doc 1 (532 words) 210 Doc 2 (210 words) 854 Doc 3 (854 words) topic 
 342 584 space Doc 4 (926 words) … 0.7 0.5 0.2

  18. The Standard Model in BNP (By Example ) frequency space politics s t r d o o … p o s f 343 189 Doc 1 (532 words) 210 Doc 2 (210 words) 854 Doc 3 (854 words) topic 
 342 584 space Doc 4 (926 words) … 0.7 0.5 0.2

  19. The Standard Model in BNP (By Example ) frequency space politics s t r d o o … p o s f 343 189 Doc 1 (532 words) 210 Doc 2 (210 words) 854 Doc 3 (854 words) topic 
 342 584 space Doc 4 (926 words) … 0.7 0.5 0.2

  20. The Standard Model in BNP (By Example ) frequency space politics s t r d o o … p o s f 343 189 Doc 1 (532 words) 0.7 210 Doc 2 (210 words) sports 854 Doc 3 (854 words) topic 
 342 584 space Doc 4 (926 words) … 0.7 0.5 0.2

  21. The Standard Model in BNP (By Example ) frequency space politics s t r d o o … p o s f 343 189 Doc 1 (532 words) 0.7 210 Doc 2 (210 words) sports 854 Doc 3 (854 words) topic 
 342 584 space Doc 4 (926 words) … 0.7 0.5 0.2

  22. The Standard Model in BNP (By Example ) frequency space politics s t r d o o … p o s f 343 189 Doc 1 (532 words) 0.7 210 Doc 2 (210 words) sports 854 Doc 3 (854 words) topic 
 342 584 space Doc 4 (926 words) … 0.7 0.5 0.2

  23. The Standard Model in BNP (By Example ) frequency space politics s t r d o o … p o s f 343 189 Doc 1 (532 words) 0.7 210 Doc 2 (210 words) sports 854 Doc 3 (854 words) topic 
 342 584 space Doc 4 (926 words) … 0.7 0.5 0.2

  24. The Standard Model in BNP (By Example ) frequency space politics s t r d o o … p o s f 343 189 Doc 1 (532 words) 0.7 210 Doc 2 (210 words) sports 854 Doc 3 (854 words) topic 
 342 584 space Doc 4 (926 words) … 0.7 0.5 0.2

  25. The Standard Model in BNP (By Example ) frequency space politics s t r d o o … p o Θ s f 343 189 Doc 1 (532 words) 0.7 210 Doc 2 (210 words) sports 854 Doc 3 (854 words) topic 
 342 584 space Doc 4 (926 words) ϴ is a random … discrete measure on the topics 0.7 0.5 0.2

  26. The Standard Model in BNP (By Example ) frequency space politics s t r d o o … p o Θ s f 343 189 Doc 1 (532 words) 0.7 210 Doc 2 (210 words) sports 854 Doc 3 (854 words) topic 
 342 584 space Doc 4 (926 words) ϴ is a random … discrete measure on the topics 0.7 0.5 0.2

  27. The Standard Model in BNP (By Example ) rate 
 space “traits” ψ 1 ψ 2 ψ 3 … Θ 343 189 Obs 1 210 Obs 2 854 Obs 3 trait 
 342 584 space Obs 4 ϴ is a random … discrete measure θ 1 θ 2 θ 3 on the topics topics traits “rates”

  28. The Standard Model in BNP (By Example ) rate 
 space “traits” ψ 1 ψ 2 ψ 3 … Θ 343 189 Obs 1 210 Obs 2 854 Obs 3 trait 
 342 584 space Obs 4 ϴ is a random … discrete measure θ 1 θ 2 θ 3 on the topics topics traits “rates”

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 space θ ψ trait space

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Poisson process with intensity measure µ (d θ × d ψ ) rate 
 = ν (d θ ) H (d ψ ) space θ ψ trait space [Kingman 93]

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sha1_base64="XnLCs7PWYVgb0od3U6a7iJQEaIw=">ACBHicbVDLSgNBEJz1GeMr6tHLYBC8GHaDoMegF48RzAOSJcxOZrOTzMwuM71CWHL17FW/wZt49T/8BP/CSbIHk1jQUFR1090VJIbcN1vZ219Y3Nru7BT3N3bPzgsHR03TZxqyho0FrFuB8QwRVrAfB2olmRAaCtYLR3dRvPTFteKweYZwX5KB4iGnBKzU7ELEgPRKZbfizoBXiZeTMspR75V+uv2YpIpoIY0/HcBPyMaOBUsEmxmxqWEDoiA9axVBHJjJ/Nrp3gc6v0cRhrWwrwTP07kRFpzFgGtlMSiMyNxX/8zophDd+xlWSAlN0vihMBYT1/Hfa4ZBTG2hFDN7a2YRkQTCjaghS3DKLqU3NDqpGiz8ZaTWCXNasVzK97DVbl2m6dUQKfoDF0gD12jGrpHdRAFA3RC3pFb86z8+58OJ/z1jUnzlBC3C+fgHgSphu</latexit> Poisson processes and (N)CRMs ( ψ , θ ) How do we generate infinitely many trait/rate points ? Poisson process with intensity measure µ (d θ × d ψ ) rate 
 = ν (d θ ) H (d ψ ) space θ ψ trait space [Kingman 93]

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Poisson process with intensity measure µ (d θ × d ψ ) rate 
 = ν (d θ ) H (d ψ ) space Θ θ completely random measure (CRM) Θ = P ∞ (e.g. BP, 𝚫 P) k =1 θ k δ ψ k ψ trait space [Kingman 93]

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sha1_base64="h0WJveWrvbC/z9WbpNDreFRJuQM=">ACDHicbVDLSsNAFJ3UV62vqks3g0WoCUpgi6LblxWsA9oYplMJ83YmSTM3Ail9Bdcu9VvcCdu/Qc/wb9w2mZhWw9cOJxzL+dy/ERwDb9beVWVtfWN/Kbha3tnd294v5BU8epoqxBYxGrtk80EzxiDeAgWDtRjEhfsJY/uJn4rSemNI+jexgmzJOkH/GAUwJGei7ieZn2IWQATntFkt2xZ4CLxMnIyWUod4t/ri9mKaSRUAF0brj2Al4I6KAU8HGBTfVLCF0QPqsY2hEJNPeaPr1GJ8YpYeDWJmJAE/VvxcjIrUeSt9sSgKhXvQm4n9eJ4XgyhvxKEmBRXQWFKQCQ4wnFeAeV4yCGBpCqOLmV0xDogFU9RcymMYnkuaXVcMN04i0sk2a14tgV5+6iVLvOWsqjI3SMyshBl6iGblEdNRBFCr2gV/RmPVv1of1OVvNWdnNIZqD9fULxcWbAw=</latexit> Poisson processes and (N)CRMs ( ψ , θ ) How do we generate infinitely many trait/rate points ? Poisson process with intensity measure µ (d θ × d ψ ) rate 
 = ν (d θ ) H (d ψ ) space Θ θ completely random measure (CRM) Θ = P ∞ (e.g. BP, 𝚫 P) k =1 θ k δ ψ k ψ trait space Normalize rates: normalized CRM (NCRM) (e.g. DP) [Kingman 93]

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sha1_base64="h0WJveWrvbC/z9WbpNDreFRJuQM=">ACDHicbVDLSsNAFJ3UV62vqks3g0WoCUpgi6LblxWsA9oYplMJ83YmSTM3Ail9Bdcu9VvcCdu/Qc/wb9w2mZhWw9cOJxzL+dy/ERwDb9beVWVtfWN/Kbha3tnd294v5BU8epoqxBYxGrtk80EzxiDeAgWDtRjEhfsJY/uJn4rSemNI+jexgmzJOkH/GAUwJGei7ieZn2IWQATntFkt2xZ4CLxMnIyWUod4t/ri9mKaSRUAF0brj2Al4I6KAU8HGBTfVLCF0QPqsY2hEJNPeaPr1GJ8YpYeDWJmJAE/VvxcjIrUeSt9sSgKhXvQm4n9eJ4XgyhvxKEmBRXQWFKQCQ4wnFeAeV4yCGBpCqOLmV0xDogFU9RcymMYnkuaXVcMN04i0sk2a14tgV5+6iVLvOWsqjI3SMyshBl6iGblEdNRBFCr2gV/RmPVv1of1OVvNWdnNIZqD9fULxcWbAw=</latexit> Poisson processes and (N)CRMs ( ψ , θ ) How do we generate infinitely many trait/rate points ? Poisson process with intensity measure µ (d θ × d ψ ) rate 
 = ν (d θ ) H (d ψ ) space Θ θ completely random measure (CRM) Θ = P ∞ (e.g. BP, 𝚫 P) k =1 θ k δ ψ k ψ trait space Normalize rates: normalized CRM (NCRM) (e.g. DP) Captures a large class of useful priors in BNP [Kingman 93]

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sha1_base64="h0WJveWrvbC/z9WbpNDreFRJuQM=">ACDHicbVDLSsNAFJ3UV62vqks3g0WoCUpgi6LblxWsA9oYplMJ83YmSTM3Ail9Bdcu9VvcCdu/Qc/wb9w2mZhWw9cOJxzL+dy/ERwDb9beVWVtfWN/Kbha3tnd294v5BU8epoqxBYxGrtk80EzxiDeAgWDtRjEhfsJY/uJn4rSemNI+jexgmzJOkH/GAUwJGei7ieZn2IWQATntFkt2xZ4CLxMnIyWUod4t/ri9mKaSRUAF0brj2Al4I6KAU8HGBTfVLCF0QPqsY2hEJNPeaPr1GJ8YpYeDWJmJAE/VvxcjIrUeSt9sSgKhXvQm4n9eJ4XgyhvxKEmBRXQWFKQCQ4wnFeAeV4yCGBpCqOLmV0xDogFU9RcymMYnkuaXVcMN04i0sk2a14tgV5+6iVLvOWsqjI3SMyshBl6iGblEdNRBFCr2gV/RmPVv1of1OVvNWdnNIZqD9fULxcWbAw=</latexit> Poisson processes and (N)CRMs ( ψ , θ ) How do we generate infinitely many trait/rate points ? Poisson process with intensity measure µ (d θ × d ψ ) rate 
 = ν (d θ ) H (d ψ ) space Θ θ completely random measure (CRM) Θ = P ∞ (e.g. BP, 𝚫 P) k =1 θ k δ ψ k ψ trait space Normalize rates: normalized CRM (NCRM) (e.g. DP) Captures a large class of useful priors in BNP How do we approximate with finite number of atoms? [Kingman 93]

  36. Finite approximation approaches

  37. <latexit sha1_base64="rBStTY2vcGAB+ctQB2Tc2GbuGAY=">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</latexit> <latexit sha1_base64="lIduhWecwSVNePDn/k0GHgIN76M=">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</latexit> <latexit sha1_base64="lIduhWecwSVNePDn/k0GHgIN76M=">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</latexit> <latexit sha1_base64="xKYZKLCFliNIYfHlXeIDsERpHo=">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</latexit> Finite approximation approaches Ω Θ = P ∞ k =1 θ k δ ψ k

  38. <latexit sha1_base64="rBStTY2vcGAB+ctQB2Tc2GbuGAY=">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</latexit> <latexit sha1_base64="lIduhWecwSVNePDn/k0GHgIN76M=">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</latexit> <latexit sha1_base64="lIduhWecwSVNePDn/k0GHgIN76M=">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</latexit> <latexit sha1_base64="xKYZKLCFliNIYfHlXeIDsERpHo=">ACX3icbVBNaxRBFOwdPxJXjaOexEvjJhBl5lc9BIetBjhOwmsL0Ob3rfZDrb3TN0vxGWYX6eP8KjR/Gqd3s2C5rEgoZ6Va94TeW1Vp6S5NsgunX7zt2t7XvD+w8e7jyKHz+Z+qpxEiey0pU7y8GjVhYnpEjWe0QTK7xNF+7/3TL+i8quwJrWqcGzi3qlASKEhZnImTEgn4IRe+MVm7PEy7z61QtqBVJ6j3gtiJBeqeidqrfu6Ge7t/3T5N0OxP+EVFx/AmLXxcjeLR8k4WYPfJOmGjNgGx1n8Qywq2Ri0JDV4P0uTmuYtOFJSYzcUjca5BLOcRaoBYN+3q6L6PheUBa8qFx4lvha/TfRgvF+ZfKwaYBKf93rxf95s4aKt/NW2bohtPLyUNFoThXvW+UL5VCSXgUC0qnwVy5LcCApdH/lykVZvjbKy4NuGLpJrzdxk0wPxmkyTj+lo6N3m5a2XP2gu2zlL1hR+wjO2YTJtlX9pP9Yr8H36OtaCeKL1ejwSbzlF1B9OwPheO5Ew=</latexit> <latexit sha1_base64="wjIn6ySmfsJU3VJTUPYbSkJGB6o=">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</latexit> <latexit sha1_base64="9m/xSNo+P+tj3DfSgwpdKIHimE=">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</latexit> <latexit sha1_base64="9m/xSNo+P+tj3DfSgwpdKIHimE=">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</latexit> <latexit sha1_base64="v6w9gb6bt1vHLAcjKLbCQ1ZQsk=">ACNHicbZDLSsNAFIYn3q23qks3g0VwY0lE0E1BdCN0U8Gq0NQwmZ6asTNJmDkRSsiL+Cu3eobCO7EhRufwUntwtsPAz/fOYdz5g9TKQy67rMzMTk1PTM7N19ZWFxaXqmurp2bJNMc2jyRib4MmQEpYmijQAmXqQamQgkX4eC4rF/cgjYic9wmEJXsetY9AVnaFQ3fPIkAWNGmD+iZTQT5oeMVk/o4vmgoH4PZGn91IgSFEG15tbdkehf41NjYzVCqrvfi/hmYIYuWTGdDw3xW7ONAouoaj4mYGU8QG7ho61MVNguvnodwXdsqRH+4m2L0Y6ot8ncqaMGarQdiqGkfldK+F/tU6G/YNuLuI0Q4j516J+JikmtIyK9oQGjnJoDeNa2Fspj5hmHG2gP7bcRNGOEobvFhWbjfc7ib/mfLfuXv1KsdHo1TmiMbZJNsE4/sk0NyQlqkTi5Iw/kTw5986L8+q8fbVOZdfJDzscndk6rlA=</latexit> Finite approximation approaches Ω Ω Θ K = P K Θ = P ∞ k =1 θ k δ ψ k k =1 θ k δ ψ k

  39. <latexit sha1_base64="wjIn6ySmfsJU3VJTUPYbSkJGB6o=">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</latexit> <latexit 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sha1_base64="xKYZKLCFliNIYfHlXeIDsERpHo=">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</latexit> Finite approximation approaches Ω Ω Θ K = P K Θ = P ∞ k =1 θ k δ ψ k k =1 θ k δ ψ k Truncated finite approx. Θ K = P K k =1 θ k δ ψ k

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sha1_base64="v6w9gb6bt1vHLAcjKLbCQ1ZQsk=">ACNHicbZDLSsNAFIYn3q23qks3g0VwY0lE0E1BdCN0U8Gq0NQwmZ6asTNJmDkRSsiL+Cu3eobCO7EhRufwUntwtsPAz/fOYdz5g9TKQy67rMzMTk1PTM7N19ZWFxaXqmurp2bJNMc2jyRib4MmQEpYmijQAmXqQamQgkX4eC4rF/cgjYic9wmEJXsetY9AVnaFQ3fPIkAWNGmD+iZTQT5oeMVk/o4vmgoH4PZGn91IgSFEG15tbdkehf41NjYzVCqrvfi/hmYIYuWTGdDw3xW7ONAouoaj4mYGU8QG7ho61MVNguvnodwXdsqRH+4m2L0Y6ot8ncqaMGarQdiqGkfldK+F/tU6G/YNuLuI0Q4j516J+JikmtIyK9oQGjnJoDeNa2Fspj5hmHG2gP7bcRNGOEobvFhWbjfc7ib/mfLfuXv1KsdHo1TmiMbZJNsE4/sk0NyQlqkTi5Iw/kTw5986L8+q8fbVOZdfJDzscndk6rlA=</latexit> Finite approximation approaches Ω Ω Θ K = P K Θ = P ∞ k =1 θ k δ ψ k k =1 θ k δ ψ k Truncated finite approx. Non-nested finite approx. Θ K = P K Θ K = P K k =1 θ k δ ψ k k =1 θ K,k δ ψ k

  41. Past work: finite approximations to BNP priors Truncated Truncation Error Non-nested Approximations Bounds Approximations ✓ ✓ ✓ DP ✓ ✓ ✓ BP ✓ BPP ✓ ✓ ✓ 𝚫 P ✓ (N)CRM

  42. Past work: finite approximations to BNP priors Truncated Truncation Error Non-nested Approximations Bounds Approximations ✓ ✓ ✓ DP [Sethuraman 94] [Ishwaran 01] [Ishwaran 02] [Roychowdhury 15] ✓ ✓ ✓ BP [Teh 07] [Doshi-Velez 09] [Paisley 12] [Paisley 16] [Paisley 12] [Thibaux 07] ✓ BPP [Broderick 14] ✓ ✓ ✓ 𝚫 P [Bondesson 82] [Roychowdhury 15] [Titsias 07] [Roychowdhury 15] [Ferguson 72] ✓ (N)CRM [Bondesson 82] [Rosinski 01] [Broderick 14]

  43. Past work: finite approximations to BNP priors Truncated Truncation Error Non-nested Approximations Bounds Approximations ✓ ✓ ✓ DP [Sethuraman 94] [Ishwaran 01] [Ishwaran 02] [Roychowdhury 15] ✓ ✓ ✓ BP [Teh 07] [Doshi-Velez 09] [Paisley 12] [Paisley 16] [Paisley 12] Sparse results for a few [Thibaux 07] ✓ BPP priors in BNP [Broderick 14] ✓ ✓ ✓ 𝚫 P [Bondesson 82] [Roychowdhury 15] [Titsias 07] [Roychowdhury 15] [Ferguson 72] ✓ (N)CRM [Bondesson 82] [Rosinski 01] [Broderick 14]

  44. Past work: finite approximations to BNP priors Truncated Truncation Error Non-nested Approximations Bounds Approximations ✓ ✓ ✓ DP [Sethuraman 94] [Ishwaran 01] [Ishwaran 02] [Roychowdhury 15] ✓ ✓ ✓ BP [Teh 07] [Doshi-Velez 09] [Paisley 12] [Paisley 16] [Paisley 12] Sparse results for a few [Thibaux 07] ✓ BPP priors in BNP [Broderick 14] ✓ ✓ ✓ 𝚫 P [Bondesson 82] [Roychowdhury 15] [Titsias 07] [Roychowdhury 15] [Ferguson 72] ✓ (N)CRM Incomplete general theory [Bondesson 82] [Rosinski 01] [Broderick 14]

  45. Outline • Tractable priors in BNP • Truncated approximations ➡ Two forms for sequential representations • Truncation and error analysis • Non-nested approximations

  46. Ordering of (N)CRM atoms

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sha1_base64="/A2qTVFqBcZqYDJ4tHgLauN2nSI=">ACHicbVA9TwJBEN3zE/ELtdRiIzGBQnJno6XRhITAROkL1l8Fb29i67cyaE0PhDrG31N9gZWxN/gv/CBa4Q8CWTvH1vJrPzgkQKg67Swtr6yurec28ptb2zu7hb39holTzaHOYxnru4AZkEJBHQVKuEs0sCiQ0Az612O/+QjaiFjd4iCBdsTulegJztBKncKRr9KS3wWgPoaArFzNXokR5U6h6FbcCegi8TJSJBlqncKP3415GoFCLpkxLc9NsD1kGgWXMr7qYGE8T67h5alikVg2sPJFSN6YpUu7cXalkI6Uf9ODFlkzCAKbGfEMDTz3lj8z2ul2LtoD4VKUgTFp4t6qaQY03EktCs0cJQDSxjXwv6V8pBpxtEGN7PlIQxPI2H42Shvs/Hmk1gkjbOK51a8G694eZWlCOH5JiUiEfOySWpkhqpE06eyAt5JW/Os/PufDif09YlJ5s5IDNwvn4Bgv6gnQ=</latexit> Ordering of (N)CRM atoms Θ K = P K Θ = P ∞ k =1 θ k δ ψ k k =1 θ k δ ψ k 2 forms for sequential representations ν (d θ ) H (d ψ ) Series representation 
 function of a homogenous 
 Poisson point process 
 (4 versions) [Ferguson and Klass 1972, Bondesson 1982, Rosinski 2001]

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sha1_base64="lIduhWecwSVNePDn/k0GHgIN76M=">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</latexit> <latexit sha1_base64="xKYZKLCFliNIYfHlXeIDsERpHo=">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</latexit> Ordering of (N)CRM atoms Θ K = P K Θ = P ∞ k =1 θ k δ ψ k k =1 θ k δ ψ k 2 forms for sequential representations ν (d θ ) H (d ψ ) Series representation 
 function of a homogenous 
 Poisson point process 
 (4 versions) Γ k [Ferguson and Klass 1972, Bondesson 1982, Rosinski 2001]

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 function of a homogenous 
 Poisson point process 
 (4 versions) Γ k i.i.d. V k ∼ g [Ferguson and Klass 1972, Bondesson 1982, Rosinski 2001]

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sha1_base64="bpgOWvhyb/d+91F9p4e9K1bGV6M=">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</latexit> <latexit sha1_base64="h0c9LWel2bJEmUlu84hXjhPfbQ=">ACiHicbVFdb9MwFHXC1whf5eMNgSyqVkMaVbKXwcOkaTzA45DWblJdKse5aUxtJ7JvgCrKO3+Rn8ALvwEnqwTbuJKlc8/x8bWP0pJh3H8Mwhv3Lx1+87O3eje/QcPHw0eP5m5srYCpqJUpT1PuQMlDUxRoLzygLXqYKzdP2+08+gnWyNKe4qWCh+crIXAqOnloOfozYaQHI6fiQMlfrZbM+TNrPDZMmx03LsBM92bIMVIdY5WTXt3t0TKPRmCF8x+abxKl42j019CfiLzenXdHmUfuNa98tpbo56lLPNPlDKjq+VgGE/ivuh1kGzBkGzrZDn4xbJS1BoMCsWdmydxhYuGW5RCQRux2kHFxZqvYO6h4RrcoukTa+nIMxnNS+uXQdqz/zoarp3b6NTv1BwLd1XryP9p8xrzt4tGmqpGMOJiUF4riXt4qeZtCBQbTzgwkp/VyoKbrlA/0mXpnwpijdaOrHfRj6b5GoS18Fsf5LEk+RTMjw63qa0Q56TV2SXJOSAHJGP5IRMiSC/g2fBi+BlGIVxeBC+u9gaBlvPU3KpwuM/l7nDXQ=</latexit> Ordering of (N)CRM atoms Θ K = P K Θ = P ∞ k =1 θ k δ ψ k k =1 θ k δ ψ k 2 forms for sequential representations ν (d θ ) H (d ψ ) Series representation 
 function of a homogenous 
 Poisson point process 
 (4 versions) Γ k i.i.d. V k ∼ g Θ = P ∞ k =1 θ k δ ψ k [Ferguson and Klass 1972, Bondesson 1982, Rosinski 2001]

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sha1_base64="h0c9LWel2bJEmUlu84hXjhPfbQ=">ACiHicbVFdb9MwFHXC1whf5eMNgSyqVkMaVbKXwcOkaTzA45DWblJdKse5aUxtJ7JvgCrKO3+Rn8ALvwEnqwTbuJKlc8/x8bWP0pJh3H8Mwhv3Lx1+87O3eje/QcPHw0eP5m5srYCpqJUpT1PuQMlDUxRoLzygLXqYKzdP2+08+gnWyNKe4qWCh+crIXAqOnloOfozYaQHI6fiQMlfrZbM+TNrPDZMmx03LsBM92bIMVIdY5WTXt3t0TKPRmCF8x+abxKl42j019CfiLzenXdHmUfuNa98tpbo56lLPNPlDKjq+VgGE/ivuh1kGzBkGzrZDn4xbJS1BoMCsWdmydxhYuGW5RCQRux2kHFxZqvYO6h4RrcoukTa+nIMxnNS+uXQdqz/zoarp3b6NTv1BwLd1XryP9p8xrzt4tGmqpGMOJiUF4riXt4qeZtCBQbTzgwkp/VyoKbrlA/0mXpnwpijdaOrHfRj6b5GoS18Fsf5LEk+RTMjw63qa0Q56TV2SXJOSAHJGP5IRMiSC/g2fBi+BlGIVxeBC+u9gaBlvPU3KpwuM/l7nDXQ=</latexit> Ordering of (N)CRM atoms Θ K = P K Θ = P ∞ k =1 θ k δ ψ k k =1 θ k δ ψ k 2 forms for sequential representations ν (d θ ) H (d ψ ) Series representation 
 Superposition representation 
 function of a homogenous 
 infinite sum of CRMs, 
 Poisson point process 
 each with finite # of atoms 
 (4 versions) (3 versions) Γ k i.i.d. V k ∼ g Θ = P ∞ k =1 θ k δ ψ k [Ferguson and Klass 1972, Bondesson 1982, Rosinski 2001] [James 2014, Broderick et al 2017]

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sha1_base64="xKYZKLCFliNIYfHlXeIDsERpHo=">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</latexit> Ordering of (N)CRM atoms Θ K = P K Θ = P ∞ k =1 θ k δ ψ k k =1 θ k δ ψ k 2 forms for sequential representations ν (d θ ) H (d ψ ) Series representation 
 Superposition representation 
 function of a homogenous 
 infinite sum of CRMs, 
 Poisson point process 
 each with finite # of atoms 
 (4 versions) (3 versions) + + + ⋯ Γ k i.i.d. V k ∼ g Θ (1) Θ (2) Θ (3) Θ = P ∞ k =1 θ k δ ψ k [Ferguson and Klass 1972, Bondesson 1982, Rosinski 2001] [James 2014, Broderick et al 2017]

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sha1_base64="xKYZKLCFliNIYfHlXeIDsERpHo=">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</latexit> Ordering of (N)CRM atoms Θ K = P K Θ = P ∞ k =1 θ k δ ψ k k =1 θ k δ ψ k 2 forms for sequential representations ν (d θ ) H (d ψ ) Series representation 
 Superposition representation 
 function of a homogenous 
 infinite sum of CRMs, 
 Poisson point process 
 each with finite # of atoms 
 (4 versions) (3 versions) + + + ⋯ Γ k i.i.d. V k ∼ g Θ (1) Θ (2) Θ (3) Θ = P ∞ Θ = P ∞ k =1 θ k δ ψ k k =1 Θ ( k ) [Ferguson and Klass 1972, Bondesson 1982, Rosinski 2001] [James 2014, Broderick et al 2017]

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sha1_base64="xKYZKLCFliNIYfHlXeIDsERpHo=">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</latexit> Ordering of (N)CRM atoms Θ K = P K Θ = P ∞ k =1 θ k δ ψ k k =1 θ k δ ψ k 2 forms for sequential representations ν (d θ ) H (d ψ ) Series representation 
 Superposition representation 
 function of a homogenous 
 infinite sum of CRMs, 
 Poisson point process 
 each with finite # of atoms 
 (4 versions) (3 versions) Theorem (H., Campbell, How, Broderick). 
 Can generate (N)CRMs using all 7 sequential representations

  58. Sequential representation comparison Why so many representations?

  59. Sequential representation comparison Why so many representations? They’re all useful in different circumstances

  60. Sequential representation comparison Why so many representations? They’re all useful in different circumstances Series Reps Superposition Reps B-Rep IL-Rep R-Rep T-Rep DB-Rep PL-Rep SB-Rep Error ✓ ✓ ✓ / ✗ ✓ ✓ ✗ ✗ Bound Decay Ease of ✓ ✓ ✓ ✗ ✗✗ ✗ ✗ Analysis ✓ ✓ ✓ ✓ ✓ ✓ ✓ Generality Known # ✓ ✓ ✗ ✗ ✗ ✗ ✗ Atoms

  61. Sequential representation example Given Gamma process:

  62. Sequential representation example Given Gamma process: Step 1: compute

  63. Sequential representation example Given Gamma process: Step 1: compute

  64. Sequential representation example Given Gamma process: Step 1: compute Step 2: compute

  65. Sequential representation example Given Gamma process: Step 1: compute Step 2: compute

  66. Sequential representation example Given Gamma process: Step 1: compute Step 2: compute Exponential( 𝜇 ) density!

  67. Sequential representation example Given Gamma process: Step 1: compute Step 2: compute Exponential( 𝜇 ) Step 3: plug in! density!

  68. Outline ✓ Tractable priors in BNP • Truncated approximations ✓ Two forms for sequential representations ➡ Truncation and error analysis • Non-nested approximations

  69. Choosing between the seven representations How close is our finite approximation?

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sha1_base64="GS65xYe7wEz8wPA5Q5WNH9n/e5E=">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</latexit> Choosing between the seven representations How close is our finite approximation? Π (d Θ | X ) ∝ Θ f ( X | Θ ) Π 0 (d Θ )

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sha1_base64="Bj2NVk1+Y0KoAngwQrx4jwhf1m4=">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</latexit> Choosing between the seven representations How close is our finite approximation? Π (d Θ | X ) ∝ Θ f ( X | Θ ) Π 0 (d Θ ) Truncation error: k p N, ∞ � p N,K k 1 = 1 Z | p N, ∞ ( X ) � p N,K ( X ) | d X 2

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Π (d Θ | X ) ∝ Θ f ( X | Θ ) Π 0 (d Θ ) Truncation error: k p N, ∞ � p N,K k 1 = 1 Z | p N, ∞ ( X ) � p N,K ( X ) | d X 2 full infinite truncated Θ K Θ

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sha1_base64="vRWyiJnsWJhOYTXGQycr1/D7+c=">ACBHicbVDLSgNBEJyNrxhfUY9eBoPgxbAbBD0GvXiMkBckS5idzGYnmZldZnqFEHL17FW/wZt49T/8BP/CSbIHk1jQUFR1090VJIbcN1vJ7exubW9k98t7O0fHB4Vj0+aJk41ZQ0ai1i3A2KY4Io1gINg7UQzIgPBWsHofua3npg2PFZ1GCfMl2SgeMgpASs1u/WIAekVS27ZnQOvEy8jJZSh1iv+dPsxTSVTQAUxpuO5CfgToFTwaFbmpYQuiIDFjHUkUkM/5kfu0UX1ilj8NY21KA5+rfiQmRxoxlYDslgcisejPxP6+TQnjrT7hKUmCKLhaFqcAQ49nruM81oyDGlhCqub0V04hoQsEGtLRlGEVXkhtamRZsNt5qEukWSl7btl7vC5V7KU8ugMnaNL5KEbVEUPqIYaiKIhekGv6M15dt6dD+dz0ZpzsplTtATn6xesaphO</latexit> Choosing between the seven representations How close is our finite approximation? Π (d Θ | X ) ∝ Θ f ( X | Θ ) Π 0 (d Θ ) Truncation error: k p N, ∞ � p N,K k 1 = 1 Z | p N, ∞ ( X ) � p N,K ( X ) | d X 2 full infinite truncated Θ K Θ f f data X data X

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Π (d Θ | X ) ∝ Θ f ( X | Θ ) Π 0 (d Θ ) Truncation error: k p N, ∞ � p N,K k 1 = 1 Z | p N, ∞ ( X ) � p N,K ( X ) | d X 2 full infinite truncated Θ K Θ f f data X data X Compare the distribution of the data under full vs. truncated

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sha1_base64="Bj2NVk1+Y0KoAngwQrx4jwhf1m4=">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</latexit> Choosing between the seven representations How close is our finite approximation? Π (d Θ | X ) ∝ Θ f ( X | Θ ) Π 0 (d Θ ) Truncation error: k p N, ∞ � p N,K k 1 = 1 Z | p N, ∞ ( X ) � p N,K ( X ) | d X 2

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sha1_base64="GS65xYe7wEz8wPA5Q5WNH9n/e5E=">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</latexit> Choosing between the seven representations How close is our finite approximation? Π (d Θ | X ) ∝ Θ f ( X | Θ ) Π 0 (d Θ ) Truncation error: k p N, ∞ � p N,K k 1 = 1 Z | p N, ∞ ( X ) � p N,K ( X ) | d X 2 Depends on number of observations N and truncation level K

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sha1_base64="Bj2NVk1+Y0KoAngwQrx4jwhf1m4=">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</latexit> Choosing between the seven representations How close is our finite approximation? Π (d Θ | X ) ∝ Θ f ( X | Θ ) Π 0 (d Θ ) Truncation error: k p N, ∞ � p N,K k 1 = 1 Z | p N, ∞ ( X ) � p N,K ( X ) | d X 2 Depends on number of observations N and truncation level K As N gets larger, error increases

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sha1_base64="GS65xYe7wEz8wPA5Q5WNH9n/e5E=">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</latexit> Choosing between the seven representations How close is our finite approximation? Π (d Θ | X ) ∝ Θ f ( X | Θ ) Π 0 (d Θ ) Truncation error: k p N, ∞ � p N,K k 1 = 1 Z | p N, ∞ ( X ) � p N,K ( X ) | d X 2 Depends on number of observations N and truncation level K As N gets larger, error increases As K gets larger, error decreases

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sha1_base64="Bj2NVk1+Y0KoAngwQrx4jwhf1m4=">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</latexit> <latexit sha1_base64="Bj2NVk1+Y0KoAngwQrx4jwhf1m4=">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</latexit> Choosing between the seven representations How close is our finite approximation? Π (d Θ | X ) ∝ Θ f ( X | Θ ) Π 0 (d Θ ) Truncation error: k p N, ∞ � p N,K k 1 = 1 Z | p N, ∞ ( X ) � p N,K ( X ) | d X 2 Depends on number of observations N and truncation level K As N gets larger, error increases As K gets larger, error decreases We develop new upper bounds

  80. Protobound Leads to all the other truncation error bounds in this work Lemma (H., Campbell, How, Broderick).

  81. Protobound Leads to all the other truncation error bounds in this work Lemma (H., Campbell, How, Broderick). Proposition (HCHB). The protobound is tight

  82. Protobound Leads to all the other truncation error bounds in this work Lemma (H., Campbell, How, Broderick).

  83. Protobound Leads to all the other truncation error bounds in this work Lemma (H., Campbell, How, Broderick). Theorem (HCHB). The series rep error is bounded by k p N, ∞ � p N,K k 1 R ∞ π ( τ ( V,u + G K )) N ]d u E [¯  1 � e − 0

  84. Protobound Leads to all the other truncation error bounds in this work Lemma (H., Campbell, How, Broderick). Theorem (HCHB). The Theorem (HCHB). The series superposition rep error is rep error is bounded by bounded by k p N, ∞ � p N,K k 1 k p N, ∞ � p N,K k 1 R ∞ π ( τ ( V,u + G K )) N ]d u E [¯  1 � e − 0 R ∞ π ( θ ) N ν + ¯ K (d θ )  1 � e − 0

  85. Error bound example Given Gamma-Poisson process:

  86. Error bound example Given Gamma-Poisson process: Step 1: bound the integral, where : G K ∼ Gamma( K, c )

  87. Error bound example Given Gamma-Poisson process: Step 1: bound the integral, where : G K ∼ Gamma( K, c ) Integration by parts

  88. Error bound example Given Gamma-Poisson process: Step 1: bound the integral, where : G K ∼ Gamma( K, c ) Integration by parts

  89. Error bound example Given Gamma-Poisson process: Step 1: bound the integral, where : G K ∼ Gamma( K, c ) Integration by parts Gamma expectation

  90. Error bound example Given Gamma-Poisson process: Step 1: bound the integral, where : G K ∼ Gamma( K, c ) Integration by parts Gamma expectation Step 2: plug in! ( ◆ K ) ✓ γλ 1 2 k p N, ∞ � p N,K k 1  1 � exp � N γ 1 + γλ ◆ K ✓ γλ ⇠ N γ K ! 1 , 1 + γλ

  91. <latexit sha1_base64="yAan1GPc5t+g+R8MNZurN6dXf8=">ACIHicbZDLSsNAFIYn9VbrLerSzWARXGhJiqDLohtXUsFeoA1lMpm0YyeTMHMiltCtD+LarT6DO3Gpb+BbOL0sbOuBgY/P4cz5/cTwTU4zpeVW1peWV3Lrxc2Nre2d+zdvbqOU0VZjcYiVk2faCa4ZDXgIFgzUYxEvmANv3818hsPTGkeyzsYJMyLSFfykFMCRurY+Aa3IcZtLkMYnBhmj5D5cSqD4dhwO3bRKTnjwovgTqGIplXt2D/tIKZpxCRQbRuU4CXkYUcCrYsNBONUsI7ZMuaxmUJGLay8aXDPGRUQIcxso8CXis/p3ISKT1IPJNZ0Sgp+e9kfif10ohvPAyLpMUmKSTRWEqsDlxFAsOuGIUxMAoYqbv2LaI4pQMOHNbLnv9U4jrml5WDZuPNJLEK9XHKdknt7VqxcTlPKowN0iI6Ri85RBV2jKqohip7QC3pFb9az9W59WJ+T1pw1ndlHM2V9/wJXKx</latexit> <latexit sha1_base64="yAan1GPc5t+g+R8MNZurN6dXf8=">ACIHicbZDLSsNAFIYn9VbrLerSzWARXGhJiqDLohtXUsFeoA1lMpm0YyeTMHMiltCtD+LarT6DO3Gpb+BbOL0sbOuBgY/P4cz5/cTwTU4zpeVW1peWV3Lrxc2Nre2d+zdvbqOU0VZjcYiVk2faCa4ZDXgIFgzUYxEvmANv3818hsPTGkeyzsYJMyLSFfykFMCRurY+Aa3IcZtLkMYnBhmj5D5cSqD4dhwO3bRKTnjwovgTqGIplXt2D/tIKZpxCRQbRuU4CXkYUcCrYsNBONUsI7ZMuaxmUJGLay8aXDPGRUQIcxso8CXis/p3ISKT1IPJNZ0Sgp+e9kfif10ohvPAyLpMUmKSTRWEqsDlxFAsOuGIUxMAoYqbv2LaI4pQMOHNbLnv9U4jrml5WDZuPNJLEK9XHKdknt7VqxcTlPKowN0iI6Ri85RBV2jKqohip7QC3pFb9az9W59WJ+T1pw1ndlHM2V9/wJXKx</latexit> <latexit sha1_base64="yAan1GPc5t+g+R8MNZurN6dXf8=">ACIHicbZDLSsNAFIYn9VbrLerSzWARXGhJiqDLohtXUsFeoA1lMpm0YyeTMHMiltCtD+LarT6DO3Gpb+BbOL0sbOuBgY/P4cz5/cTwTU4zpeVW1peWV3Lrxc2Nre2d+zdvbqOU0VZjcYiVk2faCa4ZDXgIFgzUYxEvmANv3818hsPTGkeyzsYJMyLSFfykFMCRurY+Aa3IcZtLkMYnBhmj5D5cSqD4dhwO3bRKTnjwovgTqGIplXt2D/tIKZpxCRQbRuU4CXkYUcCrYsNBONUsI7ZMuaxmUJGLay8aXDPGRUQIcxso8CXis/p3ISKT1IPJNZ0Sgp+e9kfif10ohvPAyLpMUmKSTRWEqsDlxFAsOuGIUxMAoYqbv2LaI4pQMOHNbLnv9U4jrml5WDZuPNJLEK9XHKdknt7VqxcTlPKowN0iI6Ri85RBV2jKqohip7QC3pFb9az9W59WJ+T1pw1ndlHM2V9/wJXKx</latexit> <latexit sha1_base64="yAan1GPc5t+g+R8MNZurN6dXf8=">ACIHicbZDLSsNAFIYn9VbrLerSzWARXGhJiqDLohtXUsFeoA1lMpm0YyeTMHMiltCtD+LarT6DO3Gpb+BbOL0sbOuBgY/P4cz5/cTwTU4zpeVW1peWV3Lrxc2Nre2d+zdvbqOU0VZjcYiVk2faCa4ZDXgIFgzUYxEvmANv3818hsPTGkeyzsYJMyLSFfykFMCRurY+Aa3IcZtLkMYnBhmj5D5cSqD4dhwO3bRKTnjwovgTqGIplXt2D/tIKZpxCRQbRuU4CXkYUcCrYsNBONUsI7ZMuaxmUJGLay8aXDPGRUQIcxso8CXis/p3ISKT1IPJNZ0Sgp+e9kfif10ohvPAyLpMUmKSTRWEqsDlxFAsOuGIUxMAoYqbv2LaI4pQMOHNbLnv9U4jrml5WDZuPNJLEK9XHKdknt7VqxcTlPKowN0iI6Ri85RBV2jKqohip7QC3pFb9az9W59WJ+T1pw1ndlHM2V9/wJXKx</latexit> Error bound example Given Gamma-Poisson process: Step 1: bound the integral, where : G K ∼ Gamma( K, c ) Integration by parts Gamma expectation Step 2: plug in! ( ◆ K ) ✓ γλ 1 2 k p N, ∞ � p N,K k 1  1 � exp � N γ 1 + γλ ◆ K ✓ γλ N → ∞ , bound → 1 ⇠ N γ K ! 1 , 1 + γλ

  92. <latexit sha1_base64="H7CHnG3lTbSPVZSXlgjwWpQlbds=">ACIHicbZDLSsNAFIYn9VbrLerSzWARXGhJiqDLohvBTQV7gTaUyWTSjp1MwsyJWEK3Pohrt/oM7sSlvoFv4fSysK0HBj7+/xzOnN9PBNfgOF9Wbml5ZXUtv17Y2Nza3rF39+o6ThVlNRqLWDV9opngktWAg2DNRDES+YI1/P7VyG8MKV5LO9gkDAvIl3JQ04JGKlj4xvchi3uQxhcGKYPULmx6kMhmPD6dhFp+SMCy+CO4Uimla1Y/+0g5imEZNABdG65ToJeBlRwKlgw0I71SwhtE+6rGVQkohpLxtfMsRHRglwGCvzJOCx+nciI5HWg8g3nRGBnp73RuJ/XiuF8MLuExSYJOFoWpwObEUSw4IpREAMDhCpu/opjyhCwYQ3s+W+1zuNuKblYcFk484nsQj1csl1Su7tWbFyOU0pjw7QITpGLjpHFXSNqiGKHpCL+gVvVnP1rv1YX1OWnPWdGYfzZT1/QtQk6Kt</latexit> <latexit sha1_base64="H7CHnG3lTbSPVZSXlgjwWpQlbds=">ACIHicbZDLSsNAFIYn9VbrLerSzWARXGhJiqDLohvBTQV7gTaUyWTSjp1MwsyJWEK3Pohrt/oM7sSlvoFv4fSysK0HBj7+/xzOnN9PBNfgOF9Wbml5ZXUtv17Y2Nza3rF39+o6ThVlNRqLWDV9opngktWAg2DNRDES+YI1/P7VyG8MKV5LO9gkDAvIl3JQ04JGKlj4xvchi3uQxhcGKYPULmx6kMhmPD6dhFp+SMCy+CO4Uimla1Y/+0g5imEZNABdG65ToJeBlRwKlgw0I71SwhtE+6rGVQkohpLxtfMsRHRglwGCvzJOCx+nciI5HWg8g3nRGBnp73RuJ/XiuF8MLuExSYJOFoWpwObEUSw4IpREAMDhCpu/opjyhCwYQ3s+W+1zuNuKblYcFk484nsQj1csl1Su7tWbFyOU0pjw7QITpGLjpHFXSNqiGKHpCL+gVvVnP1rv1YX1OWnPWdGYfzZT1/QtQk6Kt</latexit> <latexit sha1_base64="H7CHnG3lTbSPVZSXlgjwWpQlbds=">ACIHicbZDLSsNAFIYn9VbrLerSzWARXGhJiqDLohvBTQV7gTaUyWTSjp1MwsyJWEK3Pohrt/oM7sSlvoFv4fSysK0HBj7+/xzOnN9PBNfgOF9Wbml5ZXUtv17Y2Nza3rF39+o6ThVlNRqLWDV9opngktWAg2DNRDES+YI1/P7VyG8MKV5LO9gkDAvIl3JQ04JGKlj4xvchi3uQxhcGKYPULmx6kMhmPD6dhFp+SMCy+CO4Uimla1Y/+0g5imEZNABdG65ToJeBlRwKlgw0I71SwhtE+6rGVQkohpLxtfMsRHRglwGCvzJOCx+nciI5HWg8g3nRGBnp73RuJ/XiuF8MLuExSYJOFoWpwObEUSw4IpREAMDhCpu/opjyhCwYQ3s+W+1zuNuKblYcFk484nsQj1csl1Su7tWbFyOU0pjw7QITpGLjpHFXSNqiGKHpCL+gVvVnP1rv1YX1OWnPWdGYfzZT1/QtQk6Kt</latexit> <latexit sha1_base64="H7CHnG3lTbSPVZSXlgjwWpQlbds=">ACIHicbZDLSsNAFIYn9VbrLerSzWARXGhJiqDLohvBTQV7gTaUyWTSjp1MwsyJWEK3Pohrt/oM7sSlvoFv4fSysK0HBj7+/xzOnN9PBNfgOF9Wbml5ZXUtv17Y2Nza3rF39+o6ThVlNRqLWDV9opngktWAg2DNRDES+YI1/P7VyG8MKV5LO9gkDAvIl3JQ04JGKlj4xvchi3uQxhcGKYPULmx6kMhmPD6dhFp+SMCy+CO4Uimla1Y/+0g5imEZNABdG65ToJeBlRwKlgw0I71SwhtE+6rGVQkohpLxtfMsRHRglwGCvzJOCx+nciI5HWg8g3nRGBnp73RuJ/XiuF8MLuExSYJOFoWpwObEUSw4IpREAMDhCpu/opjyhCwYQ3s+W+1zuNuKblYcFk484nsQj1csl1Su7tWbFyOU0pjw7QITpGLjpHFXSNqiGKHpCL+gVvVnP1rv1YX1OWnPWdGYfzZT1/QtQk6Kt</latexit> <latexit sha1_base64="yAan1GPc5t+g+R8MNZurN6dXf8=">ACIHicbZDLSsNAFIYn9VbrLerSzWARXGhJiqDLohtXUsFeoA1lMpm0YyeTMHMiltCtD+LarT6DO3Gpb+BbOL0sbOuBgY/P4cz5/cTwTU4zpeVW1peWV3Lrxc2Nre2d+zdvbqOU0VZjcYiVk2faCa4ZDXgIFgzUYxEvmANv3818hsPTGkeyzsYJMyLSFfykFMCRurY+Aa3IcZtLkMYnBhmj5D5cSqD4dhwO3bRKTnjwovgTqGIplXt2D/tIKZpxCRQbRuU4CXkYUcCrYsNBONUsI7ZMuaxmUJGLay8aXDPGRUQIcxso8CXis/p3ISKT1IPJNZ0Sgp+e9kfif10ohvPAyLpMUmKSTRWEqsDlxFAsOuGIUxMAoYqbv2LaI4pQMOHNbLnv9U4jrml5WDZuPNJLEK9XHKdknt7VqxcTlPKowN0iI6Ri85RBV2jKqohip7QC3pFb9az9W59WJ+T1pw1ndlHM2V9/wJXKx</latexit> <latexit sha1_base64="yAan1GPc5t+g+R8MNZurN6dXf8=">ACIHicbZDLSsNAFIYn9VbrLerSzWARXGhJiqDLohtXUsFeoA1lMpm0YyeTMHMiltCtD+LarT6DO3Gpb+BbOL0sbOuBgY/P4cz5/cTwTU4zpeVW1peWV3Lrxc2Nre2d+zdvbqOU0VZjcYiVk2faCa4ZDXgIFgzUYxEvmANv3818hsPTGkeyzsYJMyLSFfykFMCRurY+Aa3IcZtLkMYnBhmj5D5cSqD4dhwO3bRKTnjwovgTqGIplXt2D/tIKZpxCRQbRuU4CXkYUcCrYsNBONUsI7ZMuaxmUJGLay8aXDPGRUQIcxso8CXis/p3ISKT1IPJNZ0Sgp+e9kfif10ohvPAyLpMUmKSTRWEqsDlxFAsOuGIUxMAoYqbv2LaI4pQMOHNbLnv9U4jrml5WDZuPNJLEK9XHKdknt7VqxcTlPKowN0iI6Ri85RBV2jKqohip7QC3pFb9az9W59WJ+T1pw1ndlHM2V9/wJXKx</latexit> <latexit sha1_base64="yAan1GPc5t+g+R8MNZurN6dXf8=">ACIHicbZDLSsNAFIYn9VbrLerSzWARXGhJiqDLohtXUsFeoA1lMpm0YyeTMHMiltCtD+LarT6DO3Gpb+BbOL0sbOuBgY/P4cz5/cTwTU4zpeVW1peWV3Lrxc2Nre2d+zdvbqOU0VZjcYiVk2faCa4ZDXgIFgzUYxEvmANv3818hsPTGkeyzsYJMyLSFfykFMCRurY+Aa3IcZtLkMYnBhmj5D5cSqD4dhwO3bRKTnjwovgTqGIplXt2D/tIKZpxCRQbRuU4CXkYUcCrYsNBONUsI7ZMuaxmUJGLay8aXDPGRUQIcxso8CXis/p3ISKT1IPJNZ0Sgp+e9kfif10ohvPAyLpMUmKSTRWEqsDlxFAsOuGIUxMAoYqbv2LaI4pQMOHNbLnv9U4jrml5WDZuPNJLEK9XHKdknt7VqxcTlPKowN0iI6Ri85RBV2jKqohip7QC3pFb9az9W59WJ+T1pw1ndlHM2V9/wJXKx</latexit> <latexit sha1_base64="yAan1GPc5t+g+R8MNZurN6dXf8=">ACIHicbZDLSsNAFIYn9VbrLerSzWARXGhJiqDLohtXUsFeoA1lMpm0YyeTMHMiltCtD+LarT6DO3Gpb+BbOL0sbOuBgY/P4cz5/cTwTU4zpeVW1peWV3Lrxc2Nre2d+zdvbqOU0VZjcYiVk2faCa4ZDXgIFgzUYxEvmANv3818hsPTGkeyzsYJMyLSFfykFMCRurY+Aa3IcZtLkMYnBhmj5D5cSqD4dhwO3bRKTnjwovgTqGIplXt2D/tIKZpxCRQbRuU4CXkYUcCrYsNBONUsI7ZMuaxmUJGLay8aXDPGRUQIcxso8CXis/p3ISKT1IPJNZ0Sgp+e9kfif10ohvPAyLpMUmKSTRWEqsDlxFAsOuGIUxMAoYqbv2LaI4pQMOHNbLnv9U4jrml5WDZuPNJLEK9XHKdknt7VqxcTlPKowN0iI6Ri85RBV2jKqohip7QC3pFb9az9W59WJ+T1pw1ndlHM2V9/wJXKx</latexit> Error bound example Given Gamma-Poisson process: Step 1: bound the integral, where : G K ∼ Gamma( K, c ) Integration by parts Gamma expectation Step 2: plug in! ( ◆ K ) ✓ γλ 1 2 k p N, ∞ � p N,K k 1  1 � exp � N γ 1 + γλ ◆ K ✓ γλ N → ∞ , bound → 1 K → ∞ , bound → 0 ⇠ N γ K ! 1 , 1 + γλ

  93. Outline ✓ Tractable priors in BNP ✓ Truncated approximations ✓ Two forms for sequential representations ✓ Truncation and error analysis ➡ Non-nested approximations

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