Partial preview of the text
Download Stochastic Explorations Using MCMC and more Study notes Mathematics in PDF only on Docsity!
Markov Chain Monte Carlo (MCMC) for Model Exploration Markov Chain Monte Carlo (MCMC) enables exploration of large model spaces in Bayesian regression by sampling models proportional to their posterior probabil ies without enumerating all possibilities. In the kid's cognitive score dataset (kidiq.dta), exact posterior model probabilities are feasible due to a small model space, but MCMC generalizes to larger spaces like the US crime dataset with 2!° — 32, 768 models +4 Models are visualized by size (predictors + intercept on x-axis) vs. posterior probability (y-axis). Simple random sampling with replacement approximates p(M,,,|\data) = 3 1H , (Mj; = M,,,) where J is the indicator function (1 if models match, 0 otherwise), and J is samples drawn +5 . This frequency estimates posterior probability via counting, avoiding direct computation +3 Metropolis-Hastings Algorithm The Metropolis-Hastings algorithm samples models proportional to posterior p(data|M,,,)p(M,,,). Start with initial model 14), at iteration i, propose M/- (+1) randomly (é.g., add/drop/swap predictors via random walk or coin flips for efficiency) +5 P(M)...)| data) iG) » MM) usi OM data) = BF|M;.,, :M™) x O[Mi:.1) : M) using Bayes Factor (BF) and prior odds (0), avoiding full posteriors +6 Compute acceptance ratio R — * if R> 1,accept: Mi) — Mi.) p14 pis. + If.R < 1, accept with probability FR (e.g., generate uniform random u ~ U(0, 1); accept if u < Rviacoin flip analogy) +4 . This ensures lower-probability models appear proportionally. After J samples, estimate p(B; # O\data) = 5 x ,1(X; € M")) 42. Biases from proposal schemes (e.g., neighbor- only moves) are adjusted for convergence to true posteriors p.20 . Example Step-by-Step (Hypothetical Iteration): 1. Current: AZ) = {0,1}, propose Mf... = {80, 81, 82} (add predictor). 2. Compute BF — MéatalM*)/nlaatal) yin priors), O = rs. 3.R— 2.5 > 1— Accept. 4. Or if R = 0.3, draw u = 0.2 < 0.3 + Accept; else retain M@) +27 Zellner's g-Prior and Variants For centered regression y; = 38) + yh 1 Bj(@j4 — Bj) + in ae N(0,c7) 43, Zellner's g- prior is Bla? ~ N (bo, ga7Sz2) where $,. — (X —X)"(X — X) scales prior variance like OLS covariance +3 . Posterior: Blo? data ~N (48 + Tigbo: 40°53) +3 . Shrinks OLS § toward by; as g — ©, recovers OLS +3 . Bayes Factor vs. null (intercept-only): BF([My, : Mo] = (1+ g)7Pe HP. 4 g(1 — R2.))-(-? 5.22 Often set by = 0 for slopes, reference prior p(5, 77) oa 1/o? p3t pat. Variants (resolve paradoxes): « Unit Information: g =n paz. » Zellner-Siow Cauchy: n/g ~ Gamma(1/2, 1/2) (E[n/g]=1) 44, « Hyper-g/n:1/(1+ n/g) ~ Beta(a/2,6/2) +5 . Paradoxes: g + oo yields BF — 0 (Bartlett/Jeffreys-Lindley); fixed g bounds BF despite R? —. 1 (information paradox). g growing with n balances +3 . Kid's Cognitive Score Application Using BAS package (uniform model prior), compare posterior inclusion probabilities (PIP) p(8; # 0) for predictors (IQ, hs, work, age) +3 Cheat Sheet « MCMC Goal: Sample J models « posterior; PIP = frequency of X;. « Metropolis-Hastings: + Propose Md* (add/drop/swap). « R- BF x O;acceptifR > loru< R. g-Prior: Shrinks § to by by g/(1 + g); BF= (1+ g)™P-Y/7(1 + g(1 — BR?) DP, PIP Formula: > I( Xi © Mm.) pB0s".. BF 00 BAS RA: bas.Im(.., method="MCMC", prior="ZS"); diagnostics(); image plots; predict(estimator="BMA"). Convergence: Points on 45° line; cumulative prob levels off. Kid's 1Q: Include IQ (PIP~1), hs (>0.5); drop work /age. Crime: Poi > Po2 (corr=0.993); top models size 8-9. Short Summary MCMC (Metropolis-Hastings) explores large regression model spaces by sampling « posterior via R — BF ~x O acceptance. Zellner's g-prior simplifies Bayesian analysis (shrinks OLS); variants (g=n, Zellner-Siow, hyper-g/n) applied to kid's 1Q (IQ essential) and US crime (Pol key, multicollinearity Pol/Po2) data via BAS R package. Diagnostics confirm convergence; BMA for predictions +6 . 
