The posterior distribution is the complete answer to a Bayesian inference problem, but it is usually summarised by a point estimate, an interval or a decision. These summaries are derived from decision theory, in which the consequences of each choice are assessed under the posterior distribution. Let A\mathcal{A} be a set of actions and L:Θ×A→[0,∞)L:\Theta\times\mathcal{A}\rightarrow\left[0,\infty\right) be a loss function, where L(θ,a)L\left(\theta,a\right) is the loss incurred by taking action aa when the true parameter value is θ.\theta. The utility of an action is the negative of its loss, so minimising loss and maximising utility are equivalent.

The posterior expected loss of an action is its expected loss under the posterior distribution,

ρ(a∣x‾):=E[L(θ,a)∣x‾]=∫ΘL(θ,a)π(θ∣x‾)dθ,\rho\left(a\mid\underline{x}\right):=\mathbb{E}\left[L\left(\theta,a\right)\mid\underline{x}\right]=\int_{\Theta}L\left(\theta,a\right)\pi\left(\theta\mid\underline{x}\right)d\theta,

and a Bayes action minimises the posterior expected loss.

a∗(x‾):=arg min⁡a∈Aρ(a∣x‾)a^*\left(\underline{x}\right):=\operatorname*{arg\,min}_{a\in\mathcal{A}}\rho\left(a\mid\underline{x}\right)

A decision rule δ\delta maps each possible data set to an action, and its frequentist risk at θ\theta is its expected loss over repeated sampling with θ\theta fixed, R(θ,δ):=E[L(θ,δ(X‾))∣θ].R\left(\theta,\delta\right):=\mathbb{E}\left[L\left(\theta,\delta\left(\underline{X}\right)\right)\mid\theta\right]. The Bayes risk of a decision rule is its frequentist risk averaged over the prior,

r(π,δ):=∫ΘR(θ,δ)π(θ)dθ,r\left(\pi,\delta\right):=\int_{\Theta}R\left(\theta,\delta\right)\pi\left(\theta\right)d\theta,

and a Bayes rule minimises the Bayes risk. The decision rule that takes the Bayes action for every data set is a Bayes rule, so acting optimally conditional on the observed data is also optimal on average over data sets. Estimating θ\theta is the decision problem with A=Θ,\mathcal{A}=\Theta, and a Bayes action for it is a Bayes estimator.

The Bayes estimator depends on the loss function. The squared error loss L(θ,a)=(θ−a)2L\left(\theta,a\right)=\left(\theta-a\right)^2 gives the posterior mean,

θ^=E[θ∣x‾]=∫Θθ π(θ∣x‾)dθ,\hat{\theta}=\mathbb{E}\left[\theta\mid\underline{x}\right]=\int_{\Theta}\theta\,\pi\left(\theta\mid\underline{x}\right)d\theta,

whose posterior expected loss is the posterior variance var[θ∣x‾].\mathrm{var}\left[\theta\mid\underline{x}\right]. The absolute error loss L(θ,a)=∣θ−a∣L\left(\theta,a\right)=\lvert\theta-a\rvert gives the posterior median, the median of the posterior distribution. The zero-one loss L(θ,a)=1{∣θ−a∣>ε}L\left(\theta,a\right)=\mathbf{1}_{\{\lvert\theta-a\rvert>\varepsilon\}} for small ε\varepsilon gives, in the limit ε→0,\varepsilon\rightarrow 0, the posterior mode or maximum a posteriori (MAP) estimate,

θ^MAP=arg max⁡θ∈Θπ(θ∣x‾)=arg max⁡θ∈Θ(ℓ(θ∣x‾)+log⁡eπ(θ)).\hat{\theta}_{\mathrm{MAP}}=\operatorname*{arg\,max}_{\theta\in\Theta}\pi\left(\theta\mid\underline{x}\right)=\operatorname*{arg\,max}_{\theta\in\Theta}\left(\ell\left(\theta\mid\underline{x}\right)+\log_e\pi\left(\theta\right)\right).

The MAP estimate is the maximum likelihood estimate under a flat prior and in general is a penalised maximum likelihood estimate with penalty −log⁡eπ(θ):-\log_e\pi\left(\theta\right): a normal prior centred at zero penalises θ2\theta^2 and a Laplace prior penalises ∣θ∣,\lvert\theta\rvert, which are the penalties of ridge and lasso regularisation. The posterior median is invariant under monotonic reparametrisation, the posterior mean only under affine reparametrisation and the posterior mode under neither, since the change of variables factor moves the maximum of the density. The posterior standard deviation var[θ∣x‾]\sqrt{\mathrm{var}\left[\theta\mid\underline{x}\right]} is the Bayesian analogue of the standard error.

Let α∈(0,1).\alpha\in\left(0,1\right). A credible set of credible level 1−α1-\alpha is a set C⊆ΘC\subseteq\Theta that has posterior probability 1−α,1-\alpha,

P(θ∈C∣x‾)=∫Cπ(θ∣x‾)dθ=1−α,\mathbb{P}\left(\theta\in C\mid\underline{x}\right)=\int_C\pi\left(\theta\mid\underline{x}\right)d\theta=1-\alpha,

and a credible interval is a credible set that is an interval. Unlike a confidence interval, which has probability 1−α1-\alpha of containing θ\theta over repeated samples before the data are observed, a credible interval contains θ\theta with posterior probability 1−α1-\alpha given the observed data, which is the statement that a confidence interval is often mistakenly taken to make. Credible sets of a given level are not unique, and two choices are standard. The equal-tailed credible interval places probability α2\frac{\alpha}{2} in each tail of the posterior distribution,

[Fθ∣x‾−1(α2),Fθ∣x‾−1(1−α2)],\left[F^{-1}_{\theta\mid\underline{x}}\left(\tfrac{\alpha}{2}\right),F^{-1}_{\theta\mid\underline{x}}\left(1-\tfrac{\alpha}{2}\right)\right],

where Fθ∣x‾−1F^{-1}_{\theta\mid\underline{x}} is the posterior quantile function, and transforms correctly under monotonic reparametrisation. The highest posterior density (HPD) region is the set of parameter values with posterior density above a threshold chosen to give the required level,

CHPD={θ∈Θ:π(θ∣x‾)≥k}C_{\mathrm{HPD}}=\left\{\theta\in\Theta:\pi\left(\theta\mid\underline{x}\right)\geq k\right\} P(θ∈CHPD∣x‾)=1−α\mathbb{P}\left(\theta\in C_{\mathrm{HPD}}\mid\underline{x}\right)=1-\alpha

which is the smallest set of that level, is an interval when the posterior is unimodal, may be a union of intervals when it is multimodal and is not invariant under reparametrisation. The two coincide for a symmetric unimodal posterior. By the Bernstein–von Mises theorem both are approximately θ^±z1−α2(n I(θ^))−12\hat{\theta}\pm z_{1-\frac{\alpha}{2}}\left(n\,I\left(\hat{\theta}\right)\right)^{-\frac{1}{2}} for large n,n, where z1−α2z_{1-\frac{\alpha}{2}} is the (1−α2)\left(1-\frac{\alpha}{2}\right)-quantile of the standard normal distribution.

Let H0:θ∈Θ0H_0:\theta\in\Theta_0 and H1:θ∈Θ1H_1:\theta\in\Theta_1 be hypotheses with Θ0∪Θ1=Θ.\Theta_0\cup\Theta_1=\Theta. The posterior probability of a hypothesis is the posterior probability of its parameter set,

P(H0∣x‾)=∫Θ0π(θ∣x‾)dθ\mathbb{P}\left(H_0\mid\underline{x}\right)=\int_{\Theta_0}\pi\left(\theta\mid\underline{x}\right)d\theta P(H1∣x‾)=1−P(H0∣x‾)\mathbb{P}\left(H_1\mid\underline{x}\right)=1-\mathbb{P}\left(H_0\mid\underline{x}\right)

and the Bayes action under the loss that charges L0L_0 for wrongly rejecting H0H_0 and L1L_1 for wrongly accepting it is to reject H0H_0 if and only if P(H1∣x‾)P(H0∣x‾)>L0L1.\frac{\mathbb{P}\left(H_1\mid\underline{x}\right)}{\mathbb{P}\left(H_0\mid\underline{x}\right)}>\frac{L_0}{L_1}. The hypotheses are treated symmetrically and H0H_0 can be accepted, in contrast to a frequentist test, and the posterior probability of a composite hypothesis is a direct measure of its plausibility rather than a pp-value. A simple hypothesis H0:θ=θ0H_0:\theta=\theta_0 has posterior probability zero under a continuous prior, so testing it requires a prior that places positive probability on θ0,\theta_0, which is treated as a comparison of two models.

The posterior predictive distribution is summarised in the same way. The Bayes prediction of a future observation X~\tilde{X} under squared error loss is the posterior predictive mean E[X~∣x‾]=E[E[X~∣θ]∣x‾],\mathbb{E}\left[\tilde{X}\mid\underline{x}\right]=\mathbb{E}\left[\mathbb{E}\left[\tilde{X}\mid\theta\right]\mid\underline{x}\right], and a predictive interval of level 1−α1-\alpha is an interval with posterior predictive probability 1−α,1-\alpha, which is wider than a credible interval for E[X~∣θ]\mathbb{E}\left[\tilde{X}\mid\theta\right] because by the law of total variance

var[X~∣x‾]=E[var[X~∣θ]∣x‾]+var[E[X~∣θ]∣x‾]\mathrm{var}\left[\tilde{X}\mid\underline{x}\right]=\mathbb{E}\left[\mathrm{var}\left[\tilde{X}\mid\theta\right]\mid\underline{x}\right]+\mathrm{var}\left[\mathbb{E}\left[\tilde{X}\mid\theta\right]\mid\underline{x}\right]

adds the variability of the observation to the uncertainty about the parameter.