PPT-Exercise 2.1 Posterior inference: Suppose

you have a Beta4 4 prior distribution on the probability θ that a coin will yield a head when spun in a specified manner The coin is independently spun ten times and heads appear fewer than 3 times . Chris . Mathys. Wellcome Trust Centre for Neuroimaging. UCL. SPM Course (M/EEG). London, May 14, 2013. Thanks to Jean . Daunizeau. and . Jérémie. . Mattout. for previous versions of this talk. A spectacular piece of information. Bayes. Approach to Sample Survey Inference. Roderick . Little. Department of Biostatistics, University of Michigan. Associate Director for Research & Methodology, Bureau of Census. Learning Objectives. . EM/. Posterior Regularization . (. Ganchev. et al, 10). E-step:. M. -step: . argmax. w. . E. q. . log . P . (. x. , . y. ; . w. ). Hard EM/. Constraint driven-learning (Chang et al, 07). E-step. Chris . Mathys. Wellcome Trust Centre for Neuroimaging. UCL. SPM Course. London, May 11, 2015. Thanks to Jean . Daunizeau. and . Jérémie. . Mattout. for previous versions of this talk. A spectacular piece of information. Source: “Topic models”, David . Blei. , MLSS ‘09. Topic modeling - Motivation. Discover topics from a corpus . Model connections between topics . Model the evolution of topics over time . Image annotation. EGU 2012, Vienna. Michail Vrettas. 1. , Dan Cornford. 1. , Manfred Opper. 2. 1. NCRG, Computer Science, Aston University, UK. 2. Technical University of Berlin, Germany. Why do data assimilation?. Aim of data assimilation is to estimate the posterior distribution of the state of a dynamical model (X) given observations (Y). A general scenario:. Query . variables:. . X. Evidence . (. observed. ) . variables and their values: . E. = . e. . Unobserved . variables: . Y. . Inference problem. : answer questions about the query variables given the evidence variables. Posterior inference: Suppose . you have a Beta(4, 4) prior distribution on the probability . θ that . a coin will yield a ‘head’ when spun in a specified manner. . The . coin is . independently spun . A general scenario:. Query . variables:. . X. Evidence . (. observed. ) . variables and their values: . E. = . e. . Unobserved . variables: . Y. . Inference problem. : answer questions about the query variables given the evidence variables. Problem statement. Objective is to estimate or infer unknown parameter . q . based on observations y. Result is given by probability distribution.. Identify parameter . q . that we’d like to estimate.. Mathys. Wellcome Trust Centre for Neuroimaging. UCL. SPM Course. London, May 12, 2014. Thanks to Jean . Daunizeau. and . Jérémie. . Mattout. for previous versions of this talk. A spectacular piece of information. Mathys. Wellcome Trust Centre for Neuroimaging. UCL. London SPM Course. Thanks to Jean . Daunizeau. and . Jérémie. . Mattout. for previous versions of this talk. A spectacular piece of information.  . With Bayesian conditional density estimation. Problem. Analytic expressions for likelihood of parameters is not available with simulation based models. Approximate Bayesian Computation (ABC). Provides likelihood free inference. Bayesian . Networks. agenda. Probabilistic . inference . queries. Top-down . inference. Variable elimination. Probability Queries. Given: some probabilistic model over variables . X. Find: distribution over .