Bayesian MCMC Reference Cheat Sheet

Key Facts

• Bayes’ theorem updates prior beliefs using data: $p(\theta \mid y) = \frac{p(y \mid \theta)p(\theta)}{p(y)}$.
• The posterior normalizing constant is the marginal likelihood $p(y) = \int p(y \mid \theta)p(\theta)\,d\theta$.
• MCMC estimates posterior expectations with sample averages: $E[g(\theta) \mid y] \approx \frac{1}{S}\sum_{s=1}^{S} g(\theta^{(s)})$.
• A Markov chain has transition rule $p(\theta^{(s+1)} \mid \theta^{(s)}, \theta^{(s-1)}, \ldots) = p(\theta^{(s+1)} \mid \theta^{(s)})$.
• In Metropolis-Hastings, a proposal $\theta^{*} \sim q(\theta^{*} \mid \theta)$ is accepted with probability $\alpha = \min\left(1, \frac{p(\theta^{*} \mid y)q(\theta \mid \theta^{*})}{p(\theta \mid y)q(\theta^{*} \mid \theta)}\right)$.
• For a symmetric proposal $q(\theta^{*} \mid \theta) = q(\theta \mid \theta^{*})$, the Metropolis-Hastings acceptance probability reduces to $\alpha = \min\left(1, \frac{p(\theta^{*} \mid y)}{p(\theta \mid y)}\right)$.
• Gibbs sampling updates each parameter from its full conditional distribution, such as $\theta_j^{(s+1)} \sim p(\theta_j \mid \theta_{-j}^{(s)}, y)$.
• Effective sample size adjusts for autocorrelation: $ESS \approx \frac{S}{1 + 2\sum_{k=1}^{\infty}\rho_k}$.

Vocabulary

Posterior distribution

The distribution $p(\theta \mid y)$ describing uncertainty about parameters $\theta$ after observing data $y$.

Prior distribution

The distribution $p(\theta)$ representing beliefs about parameters before observing the current data.

Likelihood

The function $p(y \mid \theta)$ that measures how plausible the observed data are for different parameter values.

Markov chain

A sequence of random states where the next state depends only on the current state, not the full past history.

Burn-in

The early part of an MCMC run often discarded because the chain may not yet represent the target posterior distribution.

Autocorrelation

Dependence between MCMC draws at different lags, often measured by correlations such as $\rho_k$.