Sketch of the context The specifics Summary Imprecise probabilistic models for inference in exponential families Erik Quaeghebeur Gert de Cooman SYSTeMS reseach group Ghent University 2 June 2006 Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families
Sketch of the context The specifics Summary Sketch of the context There is some stochastic process generating samples. Make inferences about ‘things’ depending on this process. Parametric: depending on the value of its parameters. Predictive: depending on the next sample(s). Inferences are typically expressed using probabilities of events, or previsions of gambles. The sample size is possibly small, so instead use lower & upper probabilities of events, or lower & upper previsions of gambles. A classical inference model structure is used: Choose a prior model, and update it with the sample data using (generalized) Bayes’s rule: “prior and likelihood combine into a posterior”. Generate inferences with the posterior. Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families
Sketch of the context The specifics Summary Sketch of the context There is some stochastic process generating samples. Make inferences about ‘things’ depending on this process. Parametric: depending on the value of its parameters. Predictive: depending on the next sample(s). Inferences are typically expressed using probabilities of events, or previsions of gambles. The sample size is possibly small, so instead use lower & upper probabilities of events, or lower & upper previsions of gambles. A classical inference model structure is used: Choose a prior model, and update it with the sample data using (generalized) Bayes’s rule: “prior and likelihood combine into a posterior”. Generate inferences with the posterior. Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families
Sketch of the context The specifics Summary Sketch of the context There is some stochastic process generating samples. Make inferences about ‘things’ depending on this process. Parametric: depending on the value of its parameters. Predictive: depending on the next sample(s). Inferences are typically expressed using probabilities of events, or previsions of gambles. The sample size is possibly small, so instead use lower & upper probabilities of events, or lower & upper previsions of gambles. A classical inference model structure is used: Choose a prior model, and update it with the sample data using (generalized) Bayes’s rule: “prior and likelihood combine into a posterior”. Generate inferences with the posterior. Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families
Sketch of the context The specifics Summary Sketch of the context There is some stochastic process generating samples. Make inferences about ‘things’ depending on this process. Parametric: depending on the value of its parameters. Predictive: depending on the next sample(s). Inferences are typically expressed using probabilities of events, or previsions of gambles. The sample size is possibly small, so instead use lower & upper probabilities of events, or lower & upper previsions of gambles. A classical inference model structure is used: Choose a prior model, and update it with the sample data using (generalized) Bayes’s rule: “prior and likelihood combine into a posterior”. Generate inferences with the posterior. Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families
Sketch of the context The specifics Summary Sketch of the context There is some stochastic process generating samples. Make inferences about ‘things’ depending on this process. Parametric: depending on the value of its parameters. Predictive: depending on the next sample(s). Inferences are typically expressed using probabilities of events, or previsions of gambles. The sample size is possibly small, so instead use lower & upper probabilities of events, or lower & upper previsions of gambles. A classical inference model structure is used: Choose a prior model, and update it with the sample data using (generalized) Bayes’s rule: “prior and likelihood combine into a posterior”. Generate inferences with the posterior. Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families
Sketch of the context The specifics Summary Sketch of the context There is some stochastic process generating samples. Make inferences about ‘things’ depending on this process. Parametric: depending on the value of its parameters. Predictive: depending on the next sample(s). Inferences are typically expressed using probabilities of events, or previsions of gambles. The sample size is possibly small, so instead use lower & upper probabilities of events, or lower & upper previsions of gambles. A classical inference model structure is used: Choose a prior model, and update it with the sample data using (generalized) Bayes’s rule: “prior and likelihood combine into a posterior”. Generate inferences with the posterior. Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families
Sketch of the context The specifics Summary Sketch of the context There is some stochastic process generating samples. Make inferences about ‘things’ depending on this process. Parametric: depending on the value of its parameters. Predictive: depending on the next sample(s). Inferences are typically expressed using probabilities of events, or previsions of gambles. The sample size is possibly small, so instead use lower & upper probabilities of events, or lower & upper previsions of gambles. A classical inference model structure is used: Choose a prior model, and update it with the sample data using (generalized) Bayes’s rule: “prior and likelihood combine into a posterior”. Generate inferences with the posterior. Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families
Sketch of the context Exponential family sampling models The specifics The fundaments: conjugate & predictive distributions Summary The parametric & predictive inference models Exponential family sampling models Stochastic processes we look at: exponential family sampling models. Exponential families: Normal, Poisson, Exponential, Bernoulli,. . . Typical exponential family form: For a sequence x of m samples, Ef ψ ( x ) = a ( x ) exp m ( � ψ, τ ( x ) � − b ( ψ )) . Other concepts: SEf x ( ψ ) = Ef ψ ( x ) , Sufficient statistic ( m , τ ( x )) , and The likelihood function LEf m ,τ ( x ) ( ψ ) = exp m ( � ψ, τ ( x ) � − b ( ψ )) . Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families
Sketch of the context Exponential family sampling models The specifics The fundaments: conjugate & predictive distributions Summary The parametric & predictive inference models Exponential family sampling models Stochastic processes we look at: exponential family sampling models. Exponential families: Normal, Poisson, Exponential, Bernoulli,. . . Typical exponential family form: For a sequence x of m samples, Ef ψ ( x ) = a ( x ) exp m ( � ψ, τ ( x ) � − b ( ψ )) . Other concepts: SEf x ( ψ ) = Ef ψ ( x ) , Sufficient statistic ( m , τ ( x )) , and The likelihood function LEf m ,τ ( x ) ( ψ ) = exp m ( � ψ, τ ( x ) � − b ( ψ )) . Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families
Sketch of the context Exponential family sampling models The specifics The fundaments: conjugate & predictive distributions Summary The parametric & predictive inference models Exponential family sampling models Stochastic processes we look at: exponential family sampling models. Exponential families: Normal, Poisson, Exponential, Bernoulli,. . . Typical exponential family form: For one sample x , Ef ψ ( x ) = a ( x ) exp ( � ψ, τ ( x ) � − b ( ψ )) . For a sequence x of m samples, Ef ψ ( x ) = a ( x ) exp m ( � ψ, τ ( x ) � − b ( ψ )) . Other concepts: SEf x ( ψ ) = Ef ψ ( x ) , Sufficient statistic ( m , τ ( x )) , and The likelihood function LEf m ,τ ( x ) ( ψ ) = exp m ( � ψ, τ ( x ) � − b ( ψ )) . Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families
Sketch of the context Exponential family sampling models The specifics The fundaments: conjugate & predictive distributions Summary The parametric & predictive inference models Exponential family sampling models Stochastic processes we look at: exponential family sampling models. Exponential families: Normal, Poisson, Exponential, Bernoulli,. . . Typical exponential family form: For a sequence x of m samples, Ef ψ ( x ) = a ( x ) exp m ( � ψ, τ ( x ) � − b ( ψ )) . Other concepts: SEf x ( ψ ) = Ef ψ ( x ) , Sufficient statistic ( m , τ ( x )) , and The likelihood function LEf m ,τ ( x ) ( ψ ) = exp m ( � ψ, τ ( x ) � − b ( ψ )) . Erik Quaeghebeur, Gert de Cooman Imprecise prob. models for inference in exponential families
Recommend
More recommend
