DOCSLIB.ORG

Bayesian Inference

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Bayesian Inference Bayesian Inference Thomas Nichols With thanks Lee Harrison Bayesian segmentation Spatial priors Posterior probability Dynamic Causal and normalisation on activation extent maps (PPMs) Modelling Attention to Motion Paradigm Results SPC V3A V5+ Attention – No attention Büchel & Friston 1997, Cereb. Cortex Büchel et al. 1998, Brain - fixation only - observe static dots + photic V1 - observe moving dots + motion V5 - task on moving dots + attention V5 + parietal cortex Attention to Motion Paradigm Dynamic Causal Models Model 1 (forward): Model 2 (backward): attentional modulation attentional modulation of V1→V5: forward of SPC→V5: backward Photic SPC Attention Photic SPC V1 V1 - fixation only V5 - observe static dots V5 - observe moving dots Motion Motion - task on moving dots Attention Bayesian model selection: Which model is optimal? Responses to Uncertainty Long term memory Short term memory Responses to Uncertainty Paradigm Stimuli sequence of randomly sampled discrete events Model simple computational model of an observers response to uncertainty based on the number of past events (extent of memory) 1 2 3 4 Question which regions are best explained by short / long term memory model? … 1 2 40 trials ? ? Overview • Introductory remarks • Some probability densities/distributions • Probabilistic (generative) models • Bayesian inference • A simple example – Bayesian linear regression • SPM applications – Segmentation – Dynamic causal modeling – Spatial models of fMRI time series Probability distributions and densities k=2 Probability distributions and densities k=2 Probability distributions and densities k=2 Probability distributions and densities k=2 Probability distributions and densities k=2 Probability distributions and densities k=2 Probability distributions and densities k=2 Generative models estimation space space generation time Bayesian statistics new data prior knowledge posterior ∝ likelihood ∙ prior Bayes theorem allows one to The “posterior” probability of the formally incorporate prior parameters given the data is an knowledge into computing optimal combination of prior statistical probabilities. knowledge and new data, weighted by their relative precision. Bayes’ rule Given data y and parameters θ, their joint probability can be written in 2 ways: Eliminating p(y,θ) gives Bayes’ rule: Likelihood Prior Posterior Evidence Principles of Bayesian inference Formulation of a generative model likelihood p(y|θ) prior distribution p(θ) Observation of data y Update of beliefs based upon observations, given a prior state of knowledge Univariate Gaussian Normal densities Posterior mean = precision-weighted combination of prior mean and data mean Bayesian GLM: univariate case Normal densities Bayesian GLM: multivariate case Normal densities 2 β One step if Ce and Cp are known. Otherwise iterative estimation. β1 Approximate inference: optimization True posterior mean-field approximation iteratively improve Approximate posterior free energy Objective function Value of parameter Simple example – linear regression Data Ordinary least squares Simple example – linear regression Data and model fit Ordinary least squares Bases (explanatory variables) Sum of squared errors Simple example – linear regression Data and model fit Ordinary least squares Bases (explanatory variables) Sum of squared errors Simple example – linear regression Data and model fit Ordinary least squares Bases (explanatory variables) Sum of squared errors Simple example – linear regression Data and model fit Ordinary least squares Over-fitting: model fits noise Inadequate cost function: blind to overly complex models Solution: include uncertainty in model parameters Bases (explanatory variables) Sum of squared errors Bayesian linear regression: priors and likelihood Model: Bayesian linear regression: priors and likelihood Model: Prior: Bayesian linear regression: priors and likelihood Model: Prior: Sample curves from prior (before observing any data) Mean curve Bayesian linear regression: priors and likelihood Model: Prior: Likelihood: Bayesian linear regression: priors and likelihood Model: Prior: Likelihood: Bayesian linear regression: priors and likelihood Model: Prior: Likelihood: Bayesian linear regression: posterior Model: Prior: Likelihood: Bayes Rule: Bayesian linear regression: posterior Model: Prior: Likelihood: Bayes Rule: Posterior: Bayesian linear regression: posterior Model: Prior: Likelihood: Bayes Rule: Posterior: Bayesian linear regression: posterior Model: Prior: Likelihood: Bayes Rule: Posterior: Posterior Probability Maps (PPMs) Posterior distribution: probability of the effect given the data mean: size of effect precision: variability Posterior probability map: images of the probability (confidence) that an activation exceeds some specified threshold sth, given the data y Two thresholds: • activation threshold sth : percentage of whole brain mean signal (physiologically relevant size of effect) • probability pth that voxels must exceed to be displayed (e.g. 95%) Bayesian linear regression: model selection Bayes Rule: normalizing constant Model evidence: aMRI segmentation PPM of belonging to… grey matter white matter CSF Dynamic Causal Modelling: generative model for fMRI and ERPs Hemodynamic Electric/magnetic forward model: forward model: neural activity→BOLD neural activity→EEG MEG LFP Neural state equation: fMRI ERPs Neural model: Neural model: 1 state variable per region 8 state variables per region bilinear state equation nonlinear state equation no propagation delays propagation delays inputs Bayesian Model Selection for fMRI m1 m2 m3 m4 attention attention attention attention PPC PPC PPC PPC stim stim stim stim V1 V5 V1 V5 V1 V5 V1 V5 attention models marginal likelihood estimated 0.10 effective synaptic strengths 15 for best model (m4) PPC 0.39 0.26 10 1.25 0.26 stim V1 0.13 5 V5 0.46 0 m1 m2 m3 m4 [Stephan et al., Neuroimage, 2008] fMRI time series analysis with spatial priors degree of smoothness Spatial precision matrix prior precision prior precision aMRI Smooth Y (RFT) of GLM coeff of AR coeff prior precision of data noise GLM coeff AR coeff (correlated noise) observations ML estimate of β VB estimate of β Penny et al 2005 fMRI time series analysis with spatial priors: posterior probability maps Display only voxels that exceed e.g. 95% activation threshold Probability mass pn Mean (Cbeta_*.img) PPM (spmP_*.img) Posterior density q(βn) probability of getting an effect, given the data mean: size of effect covariance: uncertainty Std dev (SDbeta_*.img) fMRI time series analysis with spatial priors: Bayesian model selection Log-evidence maps subject 1 model 1 subject N model K Compute log-evidence for each model/subject fMRI time series analysis with spatial priors: Bayesian model selection Log-evidence maps BMS maps subject 1 model 1 subject N PPM model K EPM Probability that model k generated data Compute log-evidence model k for each model/subject Joao et al, 2009 Reminder… Long term memory Short term memory Compare two models Short-term memory model long-term memory model IT indices: H,h,I,i IT indices are smoother onsets Missed trials H=entropy; h=surprise; I=mutual information; i=mutual surprise Group data: Bayesian Model Selection maps Regions best Regions best explained explained by by short- long-term term memory memory model model frontal cortex primary visual (executive cortex control) Thank-you .
Load more

Recommended publications
  • Ordinary Least Squares 1 Ordinary Least Squares
    Ordinary least squares 1 Ordinary least squares In statistics, ordinary least squares (OLS) or linear least squares is a method for estimating the unknown parameters in a linear regression model. This method minimizes the sum of squared vertical distances between the observed responses in the dataset and the responses predicted by the linear approximation. The resulting estimator can be expressed by a simple formula, especially in the case of a single regressor on the right-hand side. The OLS estimator is consistent when the regressors are exogenous and there is no Okun's law in macroeconomics states that in an economy the GDP growth should multicollinearity, and optimal in the class of depend linearly on the changes in the unemployment rate. Here the ordinary least squares method is used to construct the regression line describing this law. linear unbiased estimators when the errors are homoscedastic and serially uncorrelated. Under these conditions, the method of OLS provides minimum-variance mean-unbiased estimation when the errors have finite variances. Under the additional assumption that the errors be normally distributed, OLS is the maximum likelihood estimator. OLS is used in economics (econometrics) and electrical engineering (control theory and signal processing), among many areas of application. Linear model Suppose the data consists of n observations { y , x } . Each observation includes a scalar response y and a i i i vector of predictors (or regressors) x . In a linear regression model the response variable is a linear function of the i regressors: where β is a p×1 vector of unknown parameters; ε 's are unobserved scalar random variables (errors) which account i for the discrepancy between the actually observed responses y and the "predicted outcomes" x′ β; and ′ denotes i i matrix transpose, so that x′ β is the dot product between the vectors x and β.
    [Show full text]
  • Ordinary Least Squares: the Univariate Case
    Introduction The OLS method The linear causal model A simulation & applications Conclusion and exercises Ordinary Least Squares: the univariate case Clément de Chaisemartin Majeure Economie September 2011 Clément de Chaisemartin Ordinary Least Squares Introduction The OLS method The linear causal model A simulation & applications Conclusion and exercises 1 Introduction 2 The OLS method Objective and principles of OLS Deriving the OLS estimates Do OLS keep their promises ? 3 The linear causal model Assumptions Identification and estimation Limits 4 A simulation & applications OLS do not always yield good estimates... But things can be improved... Empirical applications 5 Conclusion and exercises Clément de Chaisemartin Ordinary Least Squares Introduction The OLS method The linear causal model A simulation & applications Conclusion and exercises Objectives Objective 1 : to make the best possible guess on a variable Y based on X . Find a function of X which yields good predictions for Y . Given cigarette prices, what will be cigarettes sales in September 2010 in France ? Objective 2 : to determine the causal mechanism by which X influences Y . Cetebus paribus type of analysis. Everything else being equal, how a change in X affects Y ? By how much one more year of education increases an individual’s wage ? By how much the hiring of 1 000 more policemen would decrease the crime rate in Paris ? The tool we use = a data set, in which we have the wages and number of years of education of N individuals. Clément de Chaisemartin Ordinary Least Squares Introduction The OLS method The linear causal model A simulation & applications Conclusion and exercises Objective and principles of OLS What we have and what we want For each individual in our data set we observe his wage and his number of years of education.
    [Show full text]
  • Naïve Bayes Classifier
    CS 60050 Machine Learning Naïve Bayes Classifier Some slides taken from course materials of Tan, Steinbach, Kumar Bayes Classifier ● A probabilistic framework for solving classification problems ● Approach for modeling probabilistic relationships between the attribute set and the class variable – May not be possible to certainly predict class label of a test record even if it has identical attributes to some training records – Reason: noisy data or presence of certain factors that are not included in the analysis Probability Basics ● P(A = a, C = c): joint probability that random variables A and C will take values a and c respectively ● P(A = a | C = c): conditional probability that A will take the value a, given that C has taken value c P(A,C) P(C | A) = P(A) P(A,C) P(A | C) = P(C) Bayes Theorem ● Bayes theorem: P(A | C)P(C) P(C | A) = P(A) ● P(C) known as the prior probability ● P(C | A) known as the posterior probability Example of Bayes Theorem ● Given: – A doctor knows that meningitis causes stiff neck 50% of the time – Prior probability of any patient having meningitis is 1/50,000 – Prior probability of any patient having stiff neck is 1/20 ● If a patient has stiff neck, what’s the probability he/she has meningitis? P(S | M )P(M ) 0.5×1/50000 P(M | S) = = = 0.0002 P(S) 1/ 20 Bayesian Classifiers ● Consider each attribute and class label as random variables ● Given a record with attributes (A1, A2,…,An) – Goal is to predict class C – Specifically, we want to find the value of C that maximizes P(C| A1, A2,…,An ) ● Can we estimate
    [Show full text]
  • The Composite Marginal Likelihood (CML) Inference Approach with Applications to Discrete and Mixed Dependent Variable Models
    Technical Report 101 The Composite Marginal Likelihood (CML) Inference Approach with Applications to Discrete and Mixed Dependent Variable Models Chandra R. Bhat Center for Transportation Research September 2014 Data-Supported Transportation Operations & Planning Center (D-STOP) A Tier 1 USDOT University Transportation Center at The University of Texas at Austin D-STOP is a collaborative initiative by researchers at the Center for Transportation Research and the Wireless Networking and Communications Group at The University of Texas at Austin. DISCLAIMER The contents of this report reflect the views of the authors, who are responsible for the facts and the accuracy of the information presented herein. This document is disseminated under the sponsorship of the U.S. Department of Transportation’s University Transportation Centers Program, in the interest of information exchange. The U.S. Government assumes no liability for the contents or use thereof. Technical Report Documentation Page 1. Report No. 2. Government Accession No. 3. Recipient's Catalog No. D-STOP/2016/101 4. Title and Subtitle 5. Report Date The Composite Marginal Likelihood (CML) Inference Approach September 2014 with Applications to Discrete and Mixed Dependent Variable 6. Performing Organization Code Models 7. Author(s) 8. Performing Organization Report No. Chandra R. Bhat Report 101 9. Performing Organization Name and Address 10. Work Unit No. (TRAIS) Data-Supported Transportation Operations & Planning Center (D- STOP) 11. Contract or Grant No. The University of Texas at Austin DTRT13-G-UTC58 1616 Guadalupe Street, Suite 4.202 Austin, Texas 78701 12. Sponsoring Agency Name and Address 13. Type of Report and Period Covered Data-Supported Transportation Operations & Planning Center (D- STOP) The University of Texas at Austin 14.
    [Show full text]
  • Chapter 2: Ordinary Least Squares Regression
    Chapter 2: Ordinary Least Squares In this chapter: 1. Running a simple regression for weight/height example (UE 2.1.4) 2. Contents of the EViews equation window 3. Creating a workfile for the demand for beef example (UE, Table 2.2, p. 45) 4. Importing data from a spreadsheet file named Beef 2.xls 5. Using EViews to estimate a multiple regression model of beef demand (UE 2.2.3) 6. Exercises Ordinary Least Squares (OLS) regression is the core of econometric analysis. While it is important to calculate estimated regression coefficients without the aid of a regression program one time in order to better understand how OLS works (see UE, Table 2.1, p.41), easy access to regression programs makes it unnecessary for everyday analysis.1 In this chapter, we will estimate simple and multivariate regression models in order to pinpoint where the regression statistics discussed throughout the text are found in the EViews program output. Begin by opening the EViews program and opening the workfile named htwt1.wf1 (this is the file of student height and weight that was created and saved in Chapter 1). Running a simple regression for weight/height example (UE 2.1.4): Regression estimation in EViews is performed using the equation object. To create an equation object in EViews, follow these steps: Step 1. Open the EViews workfile named htwt1.wf1 by selecting File/Open/Workfile on the main menu bar and click on the file name. Step 2. Select Objects/New Object/Equation from the workfile menu.2 Step 3.
    [Show full text]
  • The Bayesian Approach to Statistics
    THE BAYESIAN APPROACH TO STATISTICS ANTHONY O’HAGAN INTRODUCTION the true nature of scientific reasoning. The fi- nal section addresses various features of modern By far the most widely taught and used statisti- Bayesian methods that provide some explanation for the rapid increase in their adoption since the cal methods in practice are those of the frequen- 1980s. tist school. The ideas of frequentist inference, as set out in Chapter 5 of this book, rest on the frequency definition of probability (Chapter 2), BAYESIAN INFERENCE and were developed in the first half of the 20th century. This chapter concerns a radically differ- We first present the basic procedures of Bayesian ent approach to statistics, the Bayesian approach, inference. which depends instead on the subjective defini- tion of probability (Chapter 3). In some respects, Bayesian methods are older than frequentist ones, Bayes’s Theorem and the Nature of Learning having been the basis of very early statistical rea- Bayesian inference is a process of learning soning as far back as the 18th century. Bayesian from data. To give substance to this statement, statistics as it is now understood, however, dates we need to identify who is doing the learning and back to the 1950s, with subsequent development what they are learning about. in the second half of the 20th century. Over that time, the Bayesian approach has steadily gained Terms and Notation ground, and is now recognized as a legitimate al- ternative to the frequentist approach. The person doing the learning is an individual This chapter is organized into three sections.
    [Show full text]
  • Scalable and Robust Bayesian Inference Via the Median Posterior
    Scalable and Robust Bayesian Inference via the Median Posterior CS 584: Big Data Analytics Material adapted from David Dunson’s talk (http://bayesian.org/sites/default/files/Dunson.pdf) & Lizhen Lin’s ICML talk (http://techtalks.tv/talks/scalable-and-robust-bayesian-inference-via-the-median-posterior/61140/) Big Data Analytics • Large (big N) and complex (big P with interactions) data are collected routinely • Both speed & generality of data analysis methods are important • Bayesian approaches offer an attractive general approach for modeling the complexity of big data • Computational intractability of posterior sampling is a major impediment to application of flexible Bayesian methods CS 584 [Spring 2016] - Ho Existing Frequentist Approaches: The Positives • Optimization-based approaches, such as ADMM or glmnet, are currently most popular for analyzing big data • General and computationally efficient • Used orders of magnitude more than Bayes methods • Can exploit distributed & cloud computing platforms • Can borrow some advantages of Bayes methods through penalties and regularization CS 584 [Spring 2016] - Ho Existing Frequentist Approaches: The Drawbacks • Such optimization-based methods do not provide measure of uncertainty • Uncertainty quantification is crucial for most applications • Scalable penalization methods focus primarily on convex optimization — greatly limits scope and puts ceiling on performance • For non-convex problems and data with complex structure, existing optimization algorithms can fail badly CS 584 [Spring 2016] -
    [Show full text]
  • LINEAR REGRESSION MODELS Likelihood and Reference Bayesian
    { LINEAR REGRESSION MODELS { Likeliho o d and Reference Bayesian Analysis Mike West May 3, 1999 1 Straight Line Regression Mo dels We b egin with the simple straight line regression mo del y = + x + i i i where the design p oints x are xed in advance, and the measurement/sampling i 2 errors are indep endent and normally distributed, N 0; for each i = i i 1;::: ;n: In this context, wehavelooked at general mo delling questions, data and the tting of least squares estimates of and : Now we turn to more formal likeliho o d and Bayesian inference. 1.1 Likeliho o d and MLEs The formal parametric inference problem is a multi-parameter problem: we 2 require inferences on the three parameters ; ; : The likeliho o d function has a simple enough form, as wenow show. Throughout, we do not indicate the design p oints in conditioning statements, though they are implicitly conditioned up on. Write Y = fy ;::: ;y g and X = fx ;::: ;x g: Given X and the mo del 1 n 1 n parameters, each y is the corresp onding zero-mean normal random quantity i plus the term + x ; so that y is normal with this term as its mean and i i i 2 variance : Also, since the are indep endent, so are the y : Thus i i 2 2 y j ; ; N + x ; i i with conditional density function 2 2 2 2 1=2 py j ; ; = expfy x =2 g=2 i i i for each i: Also, by indep endence, the joint density function is n Y 2 2 : py j ; ; = pY j ; ; i i=1 1 Given the observed resp onse values Y this provides the likeliho o d function for the three parameters: the joint density ab ove evaluated at the observed and 2 hence now xed values of Y; now viewed as a function of ; ; as they vary across p ossible parameter values.
    [Show full text]
  • 1 Estimation and Beyond in the Bayes Universe
    ISyE8843A, Brani Vidakovic Handout 7 1 Estimation and Beyond in the Bayes Universe. 1.1 Estimation No Bayes estimate can be unbiased but Bayesians are not upset! No Bayes estimate with respect to the squared error loss can be unbiased, except in a trivial case when its Bayes’ risk is 0. Suppose that for a proper prior ¼ the Bayes estimator ±¼(X) is unbiased, Xjθ (8θ)E ±¼(X) = θ: This implies that the Bayes risk is 0. The Bayes risk of ±¼(X) can be calculated as repeated expectation in two ways, θ Xjθ 2 X θjX 2 r(¼; ±¼) = E E (θ ¡ ±¼(X)) = E E (θ ¡ ±¼(X)) : Thus, conveniently choosing either EθEXjθ or EX EθjX and using the properties of conditional expectation we have, θ Xjθ 2 θ Xjθ X θjX X θjX 2 r(¼; ±¼) = E E θ ¡ E E θ±¼(X) ¡ E E θ±¼(X) + E E ±¼(X) θ Xjθ 2 θ Xjθ X θjX X θjX 2 = E E θ ¡ E θ[E ±¼(X)] ¡ E ±¼(X)E θ + E E ±¼(X) θ Xjθ 2 θ X X θjX 2 = E E θ ¡ E θ ¢ θ ¡ E ±¼(X)±¼(X) + E E ±¼(X) = 0: Bayesians are not upset. To check for its unbiasedness, the Bayes estimator is averaged with respect to the model measure (Xjθ), and one of the Bayesian commandments is: Thou shall not average with respect to sample space, unless you have Bayesian design in mind. Even frequentist agree that insisting on unbiasedness can lead to bad estimators, and that in their quest to minimize the risk by trading off between variance and bias-squared a small dosage of bias can help.
    [Show full text]
  • Time-Series Regression and Generalized Least Squares in R*
    Time-Series Regression and Generalized Least Squares in R* An Appendix to An R Companion to Applied Regression, third edition John Fox & Sanford Weisberg last revision: 2018-09-26 Abstract Generalized least-squares (GLS) regression extends ordinary least-squares (OLS) estimation of the normal linear model by providing for possibly unequal error variances and for correlations between different errors. A common application of GLS estimation is to time-series regression, in which it is generally implausible to assume that errors are independent. This appendix to Fox and Weisberg (2019) briefly reviews GLS estimation and demonstrates its application to time-series data using the gls() function in the nlme package, which is part of the standard R distribution. 1 Generalized Least Squares In the standard linear model (for example, in Chapter 4 of the R Companion), E(yjX) = Xβ or, equivalently y = Xβ + " where y is the n×1 response vector; X is an n×k +1 model matrix, typically with an initial column of 1s for the regression constant; β is a k + 1 ×1 vector of regression coefficients to estimate; and " is 2 an n×1 vector of errors. Assuming that " ∼ Nn(0; σ In), or at least that the errors are uncorrelated and equally variable, leads to the familiar ordinary-least-squares (OLS) estimator of β, 0 −1 0 bOLS = (X X) X y with covariance matrix 2 0 −1 Var(bOLS) = σ (X X) More generally, we can assume that " ∼ Nn(0; Σ), where the error covariance matrix Σ is sym- metric and positive-definite. Different diagonal entries in Σ error variances that are not necessarily all equal, while nonzero off-diagonal entries correspond to correlated errors.
    [Show full text]
  • Introduction to Bayesian Inference and Modeling Edps 590BAY
    Introduction to Bayesian Inference and Modeling Edps 590BAY Carolyn J. Anderson Department of Educational Psychology c Board of Trustees, University of Illinois Fall 2019 Introduction What Why Probability Steps Example History Practice Overview ◮ What is Bayes theorem ◮ Why Bayesian analysis ◮ What is probability? ◮ Basic Steps ◮ An little example ◮ History (not all of the 705+ people that influenced development of Bayesian approach) ◮ In class work with probabilities Depending on the book that you select for this course, read either Gelman et al. Chapter 1 or Kruschke Chapters 1 & 2. C.J. Anderson (Illinois) Introduction Fall 2019 2.2/ 29 Introduction What Why Probability Steps Example History Practice Main References for Course Throughout the coures, I will take material from ◮ Gelman, A., Carlin, J.B., Stern, H.S., Dunson, D.B., Vehtari, A., & Rubin, D.B. (20114). Bayesian Data Analysis, 3rd Edition. Boco Raton, FL, CRC/Taylor & Francis.** ◮ Hoff, P.D., (2009). A First Course in Bayesian Statistical Methods. NY: Sringer.** ◮ McElreath, R.M. (2016). Statistical Rethinking: A Bayesian Course with Examples in R and Stan. Boco Raton, FL, CRC/Taylor & Francis. ◮ Kruschke, J.K. (2015). Doing Bayesian Data Analysis: A Tutorial with JAGS and Stan. NY: Academic Press.** ** There are e-versions these of from the UofI library. There is a verson of McElreath, but I couldn’t get if from UofI e-collection. C.J. Anderson (Illinois) Introduction Fall 2019 3.3/ 29 Introduction What Why Probability Steps Example History Practice Bayes Theorem A whole semester on this? p(y|θ)p(θ) p(θ|y)= p(y) where ◮ y is data, sample from some population.
    [Show full text]
  • A Widely Applicable Bayesian Information Criterion
    JournalofMachineLearningResearch14(2013)867-897 Submitted 8/12; Revised 2/13; Published 3/13 A Widely Applicable Bayesian Information Criterion Sumio Watanabe [email protected] Department of Computational Intelligence and Systems Science Tokyo Institute of Technology Mailbox G5-19, 4259 Nagatsuta, Midori-ku Yokohama, Japan 226-8502 Editor: Manfred Opper Abstract A statistical model or a learning machine is called regular if the map taking a parameter to a prob- ability distribution is one-to-one and if its Fisher information matrix is always positive definite. If otherwise, it is called singular. In regular statistical models, the Bayes free energy, which is defined by the minus logarithm of Bayes marginal likelihood, can be asymptotically approximated by the Schwarz Bayes information criterion (BIC), whereas in singular models such approximation does not hold. Recently, it was proved that the Bayes free energy of a singular model is asymptotically given by a generalized formula using a birational invariant, the real log canonical threshold (RLCT), instead of half the number of parameters in BIC. Theoretical values of RLCTs in several statistical models are now being discovered based on algebraic geometrical methodology. However, it has been difficult to estimate the Bayes free energy using only training samples, because an RLCT depends on an unknown true distribution. In the present paper, we define a widely applicable Bayesian information criterion (WBIC) by the average log likelihood function over the posterior distribution with the inverse temperature 1/logn, where n is the number of training samples. We mathematically prove that WBIC has the same asymptotic expansion as the Bayes free energy, even if a statistical model is singular for or unrealizable by a statistical model.
    [Show full text]