The Bayesian Inference Framework | Prior, Data & Posterior Distribution

Markov Chain Monte Carlo (MCMC) algorithms solve Bayesian computation challenges by sampling from probability distributions without exhaustive grid searches. These algorithms walk through parameter space in clever ways, revisiting each combination proportionally to its posterior probability. Common variants include Metropolis-Hastings and Gibbs sampling. MCMC enables Bayesian inference even when parameter spaces are vast and multi-dimensional, making complex models computationally tractable. The development of efficient MCMC algorithms was crucial for Bayesian methods becoming widely practical.

Markov Chain Monte Carlo (MCMC) methods enable sampling from complex posterior distributions when direct sampling is impossible. A Markov chain is a sequence of states where each state depends only on the previous state, with key properties including irreducibility, aperiodicity, and detailed balance. The Metropolis-Hastings algorithm implements this by decomposing the transition kernel into a proposal distribution G(x'|x) and an acceptance probability A(x'|x) = min(1, P(x')/P(x) × G(x|x')/G(x'|x)). Detailed balance ensures the stationary distribution equals the target distribution. Gibbs sampling is a specialized MCMC method where each step samples one parameter from its conditional distribution given all other parameters. For BART, this means cycling through each tree and sampling a new tree from its conditional distribution given all other trees, which depends only on residuals from other trees.

This video explains how Markov Chain Monte Carlo (MCMC) methods extend from univariate to multivariate settings. The Metropolis-Hastings algorithm generalizes by proposing new locations from a multi-dimensional proposal distribution (such as uniform rectangles or multivariate Gaussians) and accepting/rejecting them based on the target density ratio. However, in high dimensions, simultaneous proposal leads to exponentially decreasing acceptance probabilities (approximately 2^(-D) for D dimensions), making the algorithm inefficient. The Gibbs sampler addresses this by factoring the multivariate target distribution into conditional distributions and updating one coordinate at a time, effectively reducing each step to a univariate sampling problem, which works much better in high-dimensional spaces.

This section presents two fundamental MCMC algorithms for sampling from complex distributions. Metropolis-Hastings constructs a Markov chain with desired stationary distribution π by proposing moves to adjacent nodes and accepting them with probability min(1, π_j/π_i), ensuring detailed balance. Gibbs sampling operates on multi-dimensional spaces by updating one coordinate at a time, sampling from conditional distributions P(x_k' | other coordinates). Both methods reduce complex multi-dimensional sampling to simpler operations, with convergence depending on the underlying graph structure and dimensionality.

MCMC (Markov Chain Monte Carlo) methods provide a way to sample from complex distributions by constructing a Markov chain whose stationary distribution is the target distribution. A Markov chain is a sequence where the next state depends only on the current state. The chain has a stationary distribution π such that if we start from π, the distribution remains π after any number of transitions. For MCMC, we want the chain to converge to our target distribution P. The detailed balance condition states that P(x)·Q(x'|x) = P(x')·Q(x|x') for all x, x', ensuring probability flow is balanced in both directions. The Metropolis-Hastings algorithm constructs a Markov chain that satisfies detailed balance with respect to any target distribution P. The algorithm: sample a candidate x' from a proposal distribution Q(x'|x), compute the acceptance ratio α = min(1, [P(x')·Q(x|x')]/[P(x)·Q(x'|x)]), accept x' with probability α, otherwise stay at x. The key insight is that this algorithm works for any P, even when we only know P up to a proportionality constant. Gibbs sampling is a special case of Metropolis-Hastings where the proposal distribution is deterministic. For a distribution P(x₁,...,xₙ), Gibbs sampling: sample x₁ from P(x₁|x₂,...,xₙ), sample x₂ from P(x₂|x₁,...,xₙ), ..., sample xₙ from P(xₙ|x₁,...,xₙ₋₁), repeat. This works because the conditional distributions are easy to sample from.

Credible intervals are the Bayesian equivalent of confidence intervals, representing regions containing the true parameter with specified probability. For a skewed posterior distribution, multiple 95% credible intervals exist depending on boundary placement. The Highest Density Interval (HDI) is preferred - it contains 95% of the distribution and has no point outside with higher density than inside. With large samples, posteriors approach normality, allowing simple symmetric credible intervals constructed like frequentist confidence intervals. Unlike confidence intervals, credible intervals directly state the probability that the parameter falls within the interval, representing the statistician's degree of belief.

A credible interval is a Bayesian method for summarizing uncertainty about parameter estimates. It represents an interval that contains a specified percentage (e.g., 90%) of the probability mass in the posterior distribution. For the salmon subscription example, the 90% credible interval for the sign-up rate was found to be between 30% and 54%, meaning there is a 90% probability that the true sign-up rate falls within this range.

Maximum A Posteriori (MAP) estimation is a Bayesian parameter estimation technique that combines likelihood functions with prior distributions to compute posterior distributions using Bayes' theorem. The MAP estimate finds the parameter value maximizing the posterior distribution. Unlike maximum likelihood estimation, MAP incorporates prior beliefs about parameter values through the prior distribution p(m). For continuous parameters, common prior choices include beta distribution for [0,1] bounded parameters, gamma distribution for semi-bounded [0,∞) parameters, and normal distribution for unbounded parameters. The beta distribution is characterized by shape parameters α and β with mean μ = α/(α+β) and variance σ² = αβ/[(α+β)²(α+β+1)], while the gamma distribution uses shape k and scale θ with mean μ = kθ and variance σ² = kθ². Calibration formulas allow solving for these parameters given desired mean and variance. A key strength of MAP estimation is its iterative nature: the posterior distribution from one round becomes the prior for the next round when new data is collected, allowing successive refinements of both parameter estimates and their confidence levels.

Bayesian parameter estimates are derived from the posterior distribution. Point estimates include posterior mean, median, or mode. Credible intervals (e.g., 95%) contain specified posterior probability mass. These intervals represent uncertainty about parameter values given observed data, providing a complete picture of estimation uncertainty.

Bayesian point estimation uses the posterior distribution. The posterior mean (Bayesian estimate) is E[θ|x] = a'/(a'+b'), representing the expected value of the posterior. For Beta(4,11), this is 4/15. The Maximum A Posteriori (MAP) estimate is the posterior mode: (a'-1)/(a'+b'-2). For Beta(4,11), MAP = 3/13. MAP incorporates prior information and is useful when prior knowledge is strong. Both estimates differ from the frequentist maximum likelihood estimate.

Bayesian hypothesis testing provides an alternative to traditional frequentist p-value testing by using Bayes factors (BF), which are ratios of the probability of data under the null hypothesis to the probability under the alternative hypothesis. Unlike p-values that only assess the null model, Bayes factors directly compare both models' fit to the data, allowing researchers to quantify evidence for either hypothesis. A Bayes factor greater than 1 indicates the data are more likely under the null, while less than 1 indicates stronger support for the alternative. Bayes factors can be interpreted as relative predictive adequacy (how much more likely the data are under one model versus another) or as an updating factor that multiplies prior odds to produce posterior odds. For example, a BF₀₁ of 9 means the data are nine times more likely under H₀ than H₁, and with equal prior odds of 1:1, this updates to posterior odds of 9:1, corresponding to a 90% posterior probability for H₀. Jeffreys' guidelines suggest interpreting Bayes factors as anecdotal (1-3), moderate (3-10), strong (10-30), or very strong (>30).

Bayesian statistics provides an alternative to frequentist hypothesis testing by quantifying evidence as relative probabilities between competing hypotheses using Bayes factors, which indicate how many times more likely the data are under one hypothesis compared to another, allowing researchers to quantify evidence both for and against hypotheses rather than only rejecting null hypotheses as in traditional p-value approaches.

Bayes factor compares evidence for competing hypotheses by dividing the probability of data under one hypothesis by the probability under another. Unlike frequentist p-values that only test against the null, Bayes factors quantify evidence for both hypotheses simultaneously. Values above 1 indicate more evidence for the null, while values below 1 indicate more evidence for the alternative. Interpretation follows conventions: 1-3 (anecdotal), 3-10 (moderate), 10+ (strong) evidence for the hypothesis in the numerator. A BF of 8.44 means data are 8.44 times more likely under the null than the alternative, not that the null is 8.44 times more probable.

Bayesian hypothesis testing uses Bayes factors to compare competing models by calculating the ratio of their marginal likelihoods (evidence). This measures how many times more probable the data are under one hypothesis versus another. Unlike p-values, Bayes factors provide symmetric evidence measures without arbitrary thresholds. Values near 1 indicate no discrimination between hypotheses; values around 3 suggest substantial evidence; higher values indicate stronger support. This approach naturally incorporates theoretical predictions and provides continuous evidence measures suitable for scientific communication.

Bayesian hypothesis testing compares multiple competing hypotheses (e.g., difference = 0, difference = 1, difference = 2) by calculating the posterior probability of each given the data. Bayes factors quantify relative evidence: a Bayes factor of 12:1 means data is 12 times more likely under the alternative hypothesis than the null. This provides more intuitive interpretation than p-values, which only measure evidence against the null. The approach directly answers 'What is the probability that the difference equals zero?' rather than providing binary reject/fail-to-reject decisions.