PosteriorMean

Posterior mean refers to the expected value of a parameter θ under its posterior distribution after observing data. In Bayesian inference, the posterior distribution p(θ|D) is proportional to the product of the likelihood p(D|θ) and the prior p(θ): p(θ|D) ∝ p(D|θ) p(θ). The posterior mean is defined as E[θ|D] = ∫ θ p(θ|D) dθ.

As a point estimator, the posterior mean is the Bayes estimator of θ under squared error loss, since it minimizes the posterior expected squared error. It naturally combines prior beliefs with observed data, leading to shrinkage toward the prior when data are limited or noisy, and toward the data as information increases.

Computation of the posterior mean depends on the model. In conjugate models, closed-form expressions often exist. For example, in a normal model with known variance σ^2 and a normal prior θ ~ N(μ0, τ0^2), the posterior is θ|D ~ N(μn, τn^2) with τn^2 = 1/(n/σ^2 + 1/τ0^2) and μn = τn^2 (n x̄/σ^2 + μ0/τ0^2); the posterior mean is μn. In a Beta-Binomial model with θ ~ Beta(α, β) and k successes in n trials, θ|D ~ Beta(α+k, β+n−k), giving a posterior mean (α+k)/(α+β+n). For non-conjugate cases, numerical methods such as quadrature, Markov chain Monte Carlo, or variational inference are used.

The posterior mean differs from the maximum a posteriori estimate, which is the posterior mode. It also informs Bayesian predictive distributions when integrating over the posterior rather than conditioning on a single point estimate.