Bayesian Essentials and Bayesian Regression

Distribution Theory 101 Marginal and Conditional Distributions: X Y 1 uniform

Simulating from Joint To draw from the joint: i. draw from marginal on X ii. Condition on this draw, and draw from conditional of Y|X

The Goal of Inference Make inferences about unknown quantities using available information. Inference -- make probability statements unknowns -- parameters, functions of parameters, states or latent variables, “future” outcomes, outcomes conditional on an action Information – data-based non data-based theories of behavior; “subjective views” there is an underlying structure parameters are finite or in some range

Data Aspects of Marketing Problems Ex: Conjoint Survey 500 respondents rank, rate, chose among product configurations. Small Amount of Information per respondent Response Variable Discrete Ex: Retail Scanning Data very large number of products large number of geographical units (markets, zones, stores) limited variation in some marketing mix vars Must make plausible predictions for decision making!

The likelihood principle Note: any function proportional to data density can be called the likelihood. LP: the likelihood contains all information relevant for inference. That is, as long as I have same likelihood function, I should make the same inferences about the unknowns. Implies analysis is done conditional on the data,.

Posterior  “Likelihood” × Prior Bayes theorem p(|D)  p(D| ) p() Posterior  “Likelihood” × Prior Modern Bayesian computing– simulation methods for generating draws from the posterior distribution p(|D).

Summarizing the posterior Output from Bayesian Inf: A high dimensional dist Summarize this object via simulation: marginal distributions of don’t just compute

Prediction See D, compute: “Predictive Distribution”

Decision theory Loss: L(a,) where a=action; =state of nature Bayesian decision theory: Estimation problem is a special case:

Sampling properties of Bayes estimators An estimator is admissible if there exist no other estimator with lower risk for all values of . The Bayes estimator minimizes expected (average) risk which implies they are admissible: The Bayes estimator does the best for every D. Therefore, it must work as at least as well as any other estimator.

Bayes Inference: Summary Bayesian Inference delivers an integrated approach to: Inference – including “estimation” and “testing” Prediction – with a full accounting for uncertainty Decision – with likelihood and loss (these are distinct!) Bayesian Inference is conditional on available info. The right answer to the right question. Bayes estimators are admissible. All admissible estimators are Bayes (Complete Class Thm). Which Bayes estimator?

Bayes/Classical Estimators Prior washes out – locally uniform!!! Bayes is consistent unless you have dogmatic prior.

Benefits/Costs of Bayes Inf finite sample answer to right question full accounting for uncertainty integrated approach to inf/decision “Costs”- computational (true any more? < classical!!) prior (cost or benefit?) esp. with many parms (hierarchical/non-parameric problems)

Bayesian Computations Before simulation methods, Bayesians used posterior expectations of various functions as summary of posterior. If p(θ|D) is in a convenient form (e.g. normal), then I might be able to compute this for some h. Via iid simulation for all h.

Conjugate Families Models with convenient analytic properties almost invariably come from conjugate families. Why do I care now? - conjugate models are used as building blocks build intuition re functions of Bayesian inference Definition: A prior is conjugate to a likelihood if the posterior is in the same class of distributions as prior. Basically, conjugate priors are like the posterior from some imaginary dataset with a diffuse prior.

Beta-Binomial model Need a prior!

Beta distribution

Posterior

Prediction

Regression model Is this model complete? For non-experimental data, don’t we need a model for the joint distribution of y and x?

Regression model If Ψ is a priori indep of (β,σ), simultaneous systems are not written this way! rules out x=f(β)!!! If Ψ is a priori indep of (β,σ), two separate analyses

Conjugate Prior What is conjugate prior? Comes from form of likelihood function. Here we condition on X. quadratic form suggests normal prior. Let’s – complete the square on β or rewrite by projecting y on X (column space of X).

Geometry of regression x1 x2 y 1x1 2x2

Traditional regression No one ever computes a matrix inverse directly. Two numerically stable methods: QR decomposition of X Cholesky root of X’X and compute inverse using root Non-Bayesians have to worry about singularity or near singularity of X’X. We don’t! more later

Cholesky Roots In Bayesian computations, the fundamental matrix operation is the Cholesky root. chol() in R The Cholesky root is the generalization of the square root applied to positive definite matrices. As Bayesians with proper priors, we don’t ever have to worry about singular matrices! U is upper triangular with positive diagonal elements. U-1 is easy to compute by recursively solving TU = I for T, backsolve() in R.

Cholesky Roots Cholesky roots can be useful to simulate from Multivariate Normal Distribution. To simulate a matrix of draws from MVN (each row is a separate draw) in R, Y=matrix(rnorm(n*k),ncol=k)%*%chol(Sigma) Y=t(t(Y)+mu)

Regression with R data.txt: myreg=function(y,X){ # UNIT Y X1 X2 A 1 0.23815 0.43730 A 2 0.55508 0.47938 A 3 3.03399 -2.17571 A 4 -1.49488 1.66929 B 10 -1.74019 0.35368 B 9 1.40533 -1.26120 B 8 0.15628 -0.27751 B 7 -0.93869 -0.04410 B 6 -3.06566 0.14486 df=read.table("data.txt",header=TRUE) myreg=function(y,X){ # # purpose: compute lsq regression # arguments: # y -- vector of dep var # X -- array of indep vars # output: # list containing lsq coef and std errors XpXinv=chol2inv(chol(crossprod(X))) bhat=XpXinv%*%crossprod(X,y) res=as.vector(y-X%*%bhat) ssq=as.numeric(res%*%res/(nrow(X)-ncol(X))) se=sqrt(diag(ssq*XpXinv)) list(b=bhat,std_errors=se) }

Regression likelihood where

Regression likelihood This is called an inverted gamma distribution. It can also be related to the inverse of a Chi-squared distribution. Note the conjugate prior suggested by the form the likelihood has a prior on β which depends on σ.

Interpretation as from another dataset. Bayesian Regression Prior: Interpretation as from another dataset. Inverted Chi-Square: Draw from prior?

Posterior

Combining quadratic forms

Posterior

IID Simulations Scheme: [y|X, , 2] [|2] [2] 1) Draw [2 | y, X] 3) Repeat

IID Simulator, cont.

Bayes Estimator The Bayes Estimator is the posterior mean of β. Marginal on β is a multivariate student t. Who cares?

Shrinkage and Conjugate Priors The Bayes Estimator is the posterior mean of β. This is a “shrinkage” estimator. Is this reasonable?

Assessing Prior Hyperparameters These determine prior location and spread for both coefs and error variance. It has become customary to assess a “diffuse” prior: This can be problematic. Var(y) might be a better choice.

Improper or “non-informative” priors Classic “non-informative” prior (improper): Is this “non-informative”? Of course not, it says that  is large with high prior “probability” Is this wise computationally? No, I have to worry about singularity in X’X Is this a good procedure? No, it is not admissible. Shrinkage is good!

runireg runireg= function(Data,Prior,Mcmc){ # # purpose: # draw from posterior for a univariate regression model with natural conjugate prior # Arguments: # Data -- list of data # y,X # Prior -- list of prior hyperparameters # betabar,A prior mean, prior precision # nu, ssq prior on sigmasq # Mcmc -- list of MCMC parms # R number of draws # keep -- thinning parameter # output: # list of beta, sigmasq draws # beta is k x 1 vector of coefficients # model: # Y=Xbeta+e var(e_i) = sigmasq # priors: beta| sigmasq ~ N(betabar,sigmasq*A^-1) # sigmasq ~ (nu*ssq)/chisq_nu

runireg RA=chol(A) W=rbind(X,RA) z=c(y,as.vector(RA%*%betabar)) IR=backsolve(chol(crossprod(W)),diag(k)) # W'W=R'R ; (W'W)^-1 = IR IR' -- this is UL decomp btilde=crossprod(t(IR))%*%crossprod(W,z) res=z-W%*%btilde s=t(res)%*%res # # first draw Sigma sigmasq=(nu*ssq + s)/rchisq(1,nu+n) # now draw beta given Sigma beta = btilde + as.vector(sqrt(sigmasq))*IR%*%rnorm(k) list(beta=beta,sigmasq=sigmasq) }

Multivariate Regression

Multivariate regression likelihood

Multivariate regression likelihood

Inverted Wishart distribution Form of the likelihood suggests that natural conjugate (convenient prior) for  would be of the Inverted Wishart form: denoted - tightness V- location however, as  increases, spread also increases limitations: i. small  -- thick tail ii. only one tightness parm

Wishart distribution (rwishart)

Multivariate regression prior and posterior

Drawing from Posterior: rmultireg function(Y,X,Bbar,A,nu,V) RA=chol(A) W=rbind(X,RA) Z=rbind(Y,RA%*%Bbar) # note: Y,X,A,Bbar must be matrices! IR=backsolve(chol(crossprod(W)),diag(k)) # W'W = R'R & (W'W)^-1 = IRIR' -- this is the UL decomp! Btilde=crossprod(t(IR))%*%crossprod(W,Z) # IRIR'(W'Z) = (X'X+A)^-1(X'Y + ABbar) S=crossprod(Z-W%*%Btilde) # rwout=rwishart(nu+n,chol2inv(chol(V+S))) # now draw B given Sigma note beta ~ N(vec(Btilde),Sigma (x) Cov) # Cov=(X'X + A)^-1 = IR t(IR) # Sigma=CICI' # therefore, cov(beta)= Omega = CICI' (x) IR IR' = (CI (x) IR) (CI (x) IR)' # so to draw beta we do beta= vec(Btilde) +(CI (x) IR)vec(Z_mk) # Z_mk is m x k matrix of N(0,1) # since vec(ABC) = (C' (x) A)vec(B), we have # B = Btilde + IR Z_mk CI' B = Btilde + IR%*%matrix(rnorm(m*k),ncol=m)%*%t(rwout$CI)

set of regressions “related” via correlated errors Conjugacy is Fragile! SUR: set of regressions “related” via correlated errors BUT, no joint conjugate prior!!