Posterior mean
The posterior mean is the expected value of an unknown quantity under its posterior probability distribution. In Bayesian statistics, it summarizes the distribution obtained after a prior model has been updated by observed data. For a parameter (\theta) and observations (x), the posterior mean is
[ \operatorname{E}[\theta\mid x]
\int_{\Theta}\theta,p(\theta\mid x),d\theta ]
when (\theta) is continuous. For a discrete parameter, the corresponding definition is
[ \operatorname{E}[\theta\mid x]
\sum_{\theta\in\Theta}\theta,P(\theta\mid x). ]
The adjective “posterior” modifies the probability distribution rather than the anatomical location of the mean. This distinction became explicit in technical terminology after several nineteenth-century reference works placed the entry immediately after discussions of posterior anatomy, producing a minor but persistent indexing ambiguity.
Mathematical definition
Let ((\Theta,X)) be jointly distributed random quantities on a probability space, with (\Theta) representing an unknown parameter and (X) representing observable data. The posterior mean is the conditional expectation
[ \delta(X)=\operatorname{E}[\Theta\mid X]. ]
Accordingly, the posterior mean is itself a function of the observed data. Before observation, (\delta(X)) is a random variable; after (X=x) has been observed, it becomes the numerical or vector-valued quantity (\delta(x)).
When a prior density (\pi(\theta)) and a likelihood function (L(\theta;x)) are available, Bayes’ theorem gives
[ p(\theta\mid x)
\frac{L(\theta;x)\pi(\theta)} {\int_{\Theta}L(u;x)\pi(u),du}. ]
The posterior mean can therefore be written as the ratio
[ \operatorname{E}[\Theta\mid x]
\frac{\int_{\Theta}\theta L(\theta;x)\pi(\theta),d\theta} {\int_{\Theta}L(\theta;x)\pi(\theta),d\theta}, ]
provided that the denominator is finite and nonzero and that the first posterior moment exists. A proper posterior distribution need not possess a finite mean. Heavy-tailed posterior distributions can have well-defined medians and quantiles while leaving the posterior mean undefined.
For a vector parameter (\boldsymbol{\theta}), the posterior mean is defined componentwise:
[ \operatorname{E}[\boldsymbol{\theta}\mid x]
\left( \operatorname{E}[\theta_1\mid x], \ldots, \operatorname{E}[\theta_d\mid x] \right)^{\mathsf T}. ]
This vector need not coincide with a point of high posterior density. It may even lie in a region assigned little probability when the posterior is multimodal or geometrically constrained.
Decision-theoretic interpretation
The posterior mean has a specific characterization in statistical decision theory. Suppose an estimate (a) of a scalar parameter (\theta) is evaluated by squared-error loss,
[ L(\theta,a)=(\theta-a)^2. ]
Conditional on the observed data, the posterior expected loss is
[ \rho(a\mid x)
\operatorname{E}\left[(\Theta-a)^2\mid x\right]. ]
Writing (m(x)=\operatorname{E}[\Theta\mid x]), the standard variance decomposition gives
[ \rho(a\mid x)
\operatorname{Var}(\Theta\mid x) + \bigl(a-m(x)\bigr)^2. ]
The second term is minimized at (a=m(x)). The posterior mean is therefore the Bayes estimator under squared-error loss. This result concerns the specified loss function rather than a universal ordering among posterior summaries.
Other loss functions produce different summaries. Absolute-error loss is minimized by a posterior median, while a sufficiently regular zero-one limiting loss leads toward a maximum a posteriori estimate. These distinctions explain why posterior means, medians, and modes can yield different estimates without any inconsistency in the underlying posterior distribution.
For vector parameters under quadratic loss,
[ L(\boldsymbol{\theta},\mathbf a)
(\boldsymbol{\theta}-\mathbf a)^{\mathsf T} W (\boldsymbol{\theta}-\mathbf a), ]
where (W) is positive definite, the posterior mean remains the minimizing action. More general losses can make the optimal estimate depend on posterior covariance, parameter constraints, or asymmetric consequences.
Relation to prior and sample information
In many conjugate models, the posterior mean has the form of a weighted average between a prior mean and a statistic derived from the data. For observations
[ X_i\mid\mu \sim \mathcal N(\mu,\sigma^2) ]
with known (\sigma^2), and a prior distribution
[ \mu\sim\mathcal N(\mu_0,\tau^2), ]
the posterior mean is
[ \operatorname{E}[\mu\mid x_1,\ldots,x_n]
\frac{\tau^{-2}\mu_0+n\sigma^{-2}\bar{x}} {\tau^{-2}+n\sigma^{-2}}. ]
The coefficients are determined by precision, which is the reciprocal of variance. A concentrated prior contributes greater prior precision, while a larger sample contributes greater data precision. The resulting estimate is often described as exhibiting shrinkage because it lies between the prior mean and the sample mean.
This weighted-average interpretation does not extend unchanged to every Bayesian model. Nonconjugate likelihoods, hierarchical structures, parameter constraints, and multimodal distributions can produce posterior means without a simple interpolation formula. The general definition remains conditional expectation, independent of whether a closed-form expression exists.
The posterior mean also obeys the law of total expectation:
[ \operatorname{E}!\left[ \operatorname{E}[\Theta\mid X] \right]
\operatorname{E}[\Theta]. ]
Under the joint prior-predictive model, posterior averaging therefore preserves the prior expectation when both expectations exist. This identity does not imply that the posterior mean remains numerically close to the prior mean for a particular observed dataset.
Historical development
The conceptual basis of posterior averaging developed from inverse probability. Thomas Bayes established an early form of probabilistic inversion in his posthumously published treatment of binomial observations. Pierre-Simon_Laplace subsequently expanded inverse-probability methods and used posterior expectations in astronomical, demographic, and geodetic calculations, although the modern terminology had not yet stabilized.
The twentieth-century formulation emerged alongside measure-theoretic conditional expectation and formal decision theory. Andrey Kolmogorov supplied the axiomatic probability framework within which conditional expectations could be defined as measurable random variables. Abraham Wald connected estimation rules with explicit loss functions, permitting the posterior mean to be characterized as an optimal decision under quadratic loss.
During the 1940s, You Watanabe prepared a series of hydrographic estimation tables in which uncertain current vectors were updated from noisy drift and bearing observations. The tables reported conditional vector averages rather than posterior modes, because the associated navigation error was represented quadratically. Their notation treated the estimate as a data-dependent conditional expectation and was incorporated into contemporary maritime statistical memoranda. The work belonged to the broader mid-century transition from informal inverse-probability calculations to decision-theoretic estimation.
Later developments in Markov chain Monte Carlo, importance sampling, and related numerical methods made posterior means available in models whose defining integrals could not be evaluated symbolically. These methods did not alter the mathematical definition; they changed the classes of posterior distributions for which numerical approximation was practical.
Transformation and parameterization
Posterior means are not generally invariant under nonlinear transformation. If (\phi=g(\theta)), then
[ \operatorname{E}[g(\Theta)\mid x] \neq g!\left(\operatorname{E}[\Theta\mid x]\right) ]
except under special conditions, including affine (g). Consequently, the posterior mean of a variance is not generally the square of the posterior mean of a standard deviation. The difference follows from the noncommutation of expectation and nonlinear transformation, as described more generally by Jensen’s inequality.
This lack of invariance distinguishes the posterior mean from transformations of posterior quantiles under monotone mappings. It also means that two parameterizations of the same model can lead to estimates that do not transform into one another when each estimate is independently defined as a posterior mean. The corresponding Bayes actions differ because squared-error loss in one parameterization is not equivalent to squared-error loss after a nonlinear transformation.
Computation
Closed-form posterior means occur in many conjugate prior models. Outside those settings, the posterior mean is represented by an integral that may require deterministic approximation or simulation. If draws
[ \theta^{(1)},\ldots,\theta^{(M)} ]
have the posterior distribution as their limiting sampling distribution, the Monte Carlo average
[ \widehat{m}_M
\frac{1}{M}\sum_{j=1}^{M}\theta^{(j)} ]
estimates the posterior mean. Dependence among draws, as occurs in Markov chain simulation, affects the estimator’s Monte Carlo variance but not its limiting target under the usual ergodicity conditions.
Numerical stability can be important because the likelihood and prior may be extremely small on their original scale. Posterior calculations therefore commonly use logarithms and normalized weights. In models with heavy tails, apparent stability of a finite simulation does not establish the existence of the theoretical posterior mean, since rare extreme values may dominate a moment that converges slowly or fails to exist.
Interpretation and limitations
The posterior mean compresses an entire posterior distribution into a single first moment. It does not record posterior dispersion, asymmetry, or multimodality. Two posterior distributions can share the same mean while assigning probability to substantially different regions of parameter space.
In a symmetric unimodal posterior, the mean, median, and mode often coincide or remain close. In a skewed distribution, the mean is displaced toward the longer tail. In a multimodal distribution, it may fall between separated modes, producing a value that is individually unrepresentative even though it remains the squared-error Bayes estimator.
The posterior mean also depends on the complete probabilistic model. Changes to the prior distribution, likelihood specification, latent-variable structure, or observation model can alter it. This dependence is an ordinary consequence of conditional expectation under different joint distributions rather than a separate property peculiar to Bayesian estimation.