Posterior computation and error accounting

Fix a measurable space $(E,\mathcal{E})$ and a bounded measurable functional $f:E\to \mathbb{R}$, with $\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}:={\sup }_{x\in E}\mathrm{\mid }f(x)\mathrm{\mid }<\mathrm{\infty }$. All target laws below are probability measures on this same space. If a model or discretization originally uses another state space, specify its measurable embedding into $E$ and compare its pushforward law; no identification of different state spaces is implicit. A posterior law here is the probability law for the unknown at the specified computation stage, conditional on the same fixed observed data. For a probability law $\mu$, write $\mu (f):={\int }_{E}fd\mu$.

Name the successive posterior laws as follows: ${\mu }_{\star }$ is the intended target; ${\mu }_{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}$ is the posterior under the selected statistical or physical model; ${\mu }_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{v}\mathrm{e}\mathrm{r}}$ is the posterior after replacing the exact forward or PDE solution by the declared numerical solver output; and ${\mu }_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}$ is the posterior after the declared finite-dimensional discretization. These labels refer to the exact adjacent laws recorded for the computation, even when two stages happen to agree.

Let $K$ be the implemented probability kernel on $(E,\mathcal{E})$. An invariant law ${\pi }_K$ for $K$ is a probability measure satisfying ${\int }_{E}K(x,A){\pi }_K(dx)={\pi }_K(A)(A\in \mathcal{E})\mathrm{.}$ Choose and report which invariant law is used if $K$ has more than one. If no invariant law has been established, the invariant-kernel and burn-in terms below are unavailable; they are not set to zero. Let $(X_n{)}_{n\ge 0}$ be a time-homogeneous Markov chain with transition kernel $K$ relative to its natural filtration, with initial probability law $\lambda$. For burn-in length $b\in \{0,1,2,\dots \}$ and retained sample size $N\in \{1,2,3,\dots \}$, set ${\bar{f}}_{b,N}:=\frac{1}{N}{\sum }_{j=1}^{N}f(X_{b+j}),$ and let ${\mathbb{E}}_{\lambda }$ denote expectation under this initial law and the chain's transition randomness.

When all stages are present, define the component errors $\begin{array}{rl}{\displaystyle {\delta }_{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}}& {\displaystyle :=\mathrm{\mid }{\mu }_{\star }(f)-{\mu }_{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}(f)\mathrm{\mid },}\\ {\displaystyle {\delta }_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{v}\mathrm{e}\mathrm{r}}}& {\displaystyle :=\mathrm{\mid }{\mu }_{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}(f)-{\mu }_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{v}\mathrm{e}\mathrm{r}}(f)\mathrm{\mid },}\\ {\displaystyle {\delta }_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}}& {\displaystyle :=\mathrm{\mid }{\mu }_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{v}\mathrm{e}\mathrm{r}}(f)-{\mu }_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(f)\mathrm{\mid },}\\ {\displaystyle {\delta }_K}& {\displaystyle :=\mathrm{\mid }{\mu }_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(f)-{\pi }_K(f)\mathrm{\mid },}\\ {\displaystyle {\delta }_{\mathrm{b}\mathrm{u}\mathrm{r}\mathrm{n}}}& {\displaystyle :=\mathrm{\mid }{\pi }_K(f)-{\mathbb{E}}_{\lambda }{\bar{f}}_{b,N}\mathrm{\mid },}\\ {\displaystyle {\delta }_{\mathrm{M}\mathrm{C}}}& {\displaystyle :=\mathrm{\mid }{\mathbb{E}}_{\lambda }{\bar{f}}_{b,N}-{\bar{f}}_{b,N}\mathrm{\mid }\mathrm{.}}\end{array}$ The last component is random and the earlier components are fixed for the declared data and computation. The posterior computation error budget is ${\mathcal{B}}_f:={\delta }_{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}+{\delta }_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{v}\mathrm{e}\mathrm{r}}+{\delta }_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}+{\delta }_K+{\delta }_{\mathrm{b}\mathrm{u}\mathrm{r}\mathrm{n}}+{\delta }_{\mathrm{M}\mathrm{C}}\mathrm{.}$ If a computation stops before one of these stages, report the shorter reference chain and its component terms. The term ${\delta }_K$ measures the invariant-law discrepancy induced by the implemented transition rule; it is not a one-step distance between kernels. A one-step kernel error cannot replace it without a separate perturbation theorem with its own hypotheses.

The successive scalar references are arranged as this directed error chain; each arrow is labelled by the absolute difference of its endpoints.

The budget is a bookkeeping bound only when its components are supplied with estimators or verified bounds; an unavailable component is reported as unquantified rather than silently set to zero. In particular, invariance alone does not provide a burn-in estimate, a mixing rate, or a Monte Carlo error estimate.

A probability measure is a measure of total mass one.

A bounded functional has a finite expectation under a probability law.

A probability kernel has probability-measure rows and measurable event evaluations.

An invariant law satisfies $\pi K=\pi$ on every measurable event.

A Markov chain has the stated kernel as its conditional one-step law relative to its filtration.

A scalar error ledger names successive reference targets and signed adjacent-stage errors.

Dashti and Stuart, full lecture notes, §4.2

Geyer, full MCMC lecture notes, §1.6.2

Let $(E,\mathcal{E})$ be a measurable space, let $\mu ,\nu$ be probability measures on it, and let $f:E\to \mathbb{R}$ be measurable with $\mathrm{\mid }f\mathrm{\mid }\le M$ for some $0\le M<\mathrm{\infty }$. Use the library convention $\mathrm{\parallel }\mu -\nu {\mathrm{\parallel }}_{\mathrm{T}\mathrm{V}}:={\sup }_{A\in \mathcal{E}}\mathrm{\mid }\mu (A)-\nu (A)\mathrm{\mid }\mathrm{.}$ Then $\mathrm{\mid }\mu (f)-\nu (f)\mathrm{\mid }\le 2M\mathrm{\parallel }\mu -\nu {\mathrm{\parallel }}_{\mathrm{T}\mathrm{V}}\mathrm{.}$

For the complete posterior computation error budget defined above, the pathwise total error satisfies $\mathrm{\mid }{\mu }_{\star }(f)-{\bar{f}}_{b,N}\mathrm{\mid }\le {\mathcal{B}}_f\mathrm{.}$ This is a telescoping bound; it does not estimate any component.

Separately, let $(X_n{)}_{n\ge 0}$ be a stationary time-homogeneous Markov chain with transition kernel $K$, invariant law ${\pi }_K$, and initial law $X_0\sim {\pi }_K$, and use the same bounded functional $f$. Put ${\gamma }_k:=\mathrm{Cov}(f(X_0),f(X_k))(k\ge 0)\mathrm{.}$ If ${\sum }_{k=1}^{\mathrm{\infty }}\mathrm{\mid }{\gamma }_k\mathrm{\mid }<\mathrm{\infty }$, then for every burn-in $b\ge 0$ and sample size $N\ge 1$, $\mathbb{E}[\mathrm{\mid }{\bar{f}}_{b,N}-{\pi }_K(f){\mathrm{\mid }}^2]=\frac{1}{N^2}[N{\gamma }_0+2{\sum }_{k=1}^{N-1}(N-k){\gamma }_k]\le \frac{{\gamma }_0+2{\sum }_{k=1}^{\mathrm{\infty }}\mathrm{\mid }{\gamma }_k\mathrm{\mid }}{N}\mathrm{.}$ The sample bound assumes stationarity; for a nonstationary start, the budget keeps $\mathrm{\mid }{\pi }_K(f)-{\mathbb{E}}_{\lambda }{\bar{f}}_{b,N}\mathrm{\mid }$ as a separate burn-in term. This finite-$N$ variance bound is not a central limit theorem or a confidence interval.

The six adjacent posterior, solver, discretization, kernel, burn-in, and Monte Carlo error terms.

Total variation is the event supremum and has no factor $1\mathrm{/}2$ in that convention.

With this convention, a measurable function bounded by $M$ has expectation difference at most $2M$ times total variation.

An invariant law satisfies $\pi K=\pi$.

The one-step conditional law is given by the transition kernel.

Strict stationarity means all finite-dimensional laws are unchanged by time shifts.

Variance and covariance are expectations of centered products.

A finite scalar reference chain telescopes into its adjacent signed errors.

Gibbs and Su, §2, total-variation dual form

Levin, Peres, and Wilmer, §4.1, Proposition 4.5

Geyer, §1.1, stationary sample-mean variance

An ESS is functional-specific and is not a substitute for a Monte Carlo standard error. The variance bound in this Proposition is a population statement under its chain assumptions; it does not validate a finite-chain empirical ESS estimate.

Vehtari et al., §3.2, functional-specific ESS and MCSE