Ideal and finite-simulation synthetic likelihood

Let $\mathsf{E}=((\mathcal{X},\mathcal{A}),\mathrm{\Theta },(P_{\theta }{)}_{\theta \in \mathrm{\Theta }})$ be a statistical experiment and let $S:\mathcal{X}\to {\mathbb{R}}^d$ be a statistic, meaning a measurable map to its declared Borel target as in Statistics and estimators, where $d\ge 1$ is finite. Expectations are Lebesgue integrals under $P_{\theta }$ as in Expectation of a nonnegative or integrable random variable, and variances and covariances use centered second moments as in Moments, variance, and covariance on a probability space. For every $\theta \in \mathrm{\Theta }$, assume ${\mathbb{E}}_{\theta }\text{\hspace{0.17em}⁣}\left[\mathrm{\parallel }S{\mathrm{\parallel }}_2^2\right]<\mathrm{\infty },\mathrm{\parallel }x{\mathrm{\parallel }}_2^2:={\sum }_{j=1}^d{x}_j^2\mathrm{.}$ The binding mean and summary covariance are the pointwise-defined quantities $b_j(\theta ):={\mathbb{E}}_{\theta }[S_j],{\mathrm{\Sigma }}_{jk}(\theta ):={\mathbb{E}}_{\theta }\text{\hspace{0.17em}⁣}[(S_j-b_j(\theta ))(S_k-b_k(\theta ))](1\le j,k\le d)\mathrm{.}$ Each covariance entry is finite since $2\mathrm{\mid }uv\mathrm{\mid }\le u^2+v^2$ and the coordinates are square-integrable. The matrix is symmetric and positive semidefinite because for $a\in {\mathbb{R}}^d$, $a^T\mathrm{\Sigma }(\theta )a={\mathbb{E}}_{\theta }\text{\hspace{0.17em}⁣}[(a^T(S-b(\theta )){)}^2]\ge 0.$ No measurability or continuity of $\theta \mapsto b(\theta )$ or $\theta \mapsto \mathrm{\Sigma }(\theta )$ follows from these pointwise definitions.

Assume AC (The Axiom of Choice). If $\mathrm{\Sigma }(\theta )$ is positive definite, the ideal Gaussian synthetic likelihood for observed data $y\in \mathcal{X}$ is the Gaussian density evaluated at its summary, as supplied by Multivariate normal law, including singular covariance and A positive-definite multivariate normal law has its Gaussian density, $g_{\mathrm{S}\mathrm{L}}(y\mid \theta ):={\phi }_d\text{\hspace{0.17em}⁣}(S(y);b(\theta ),\mathrm{\Sigma }(\theta ))=(2\pi {)}^{-d\mathrm{/}2}\det (\mathrm{\Sigma }(\theta ){)}^{-1\mathrm{/}2}\exp \text{\hspace{0.17em}⁣}[-\frac{1}{2}(S(y)-b(\theta ){)}^T\mathrm{\Sigma }(\theta {)}^{-1}(S(y)-b(\theta ))]\mathrm{.}$ Here ${\phi }_d(\cdot ;m,\mathrm{\Sigma })$ is the Lebesgue density of $N_d(m,\mathrm{\Sigma })$. If $\mathrm{\Sigma }(\theta )$ is singular, this positive-definite-covariance density is not defined by this item.

Let $Q_{\theta }:=P_{\theta }\circ S^{-1}$ be the pushforward law of the statistic under $P_{\theta }$ as in Law or distribution of a random element. For $m\ge 2$, let $S_1,\dots ,S_m$ be IID simulated summaries with common law $Q_{\theta }$, where IID means a common law and mutual independence as in Identical distribution and IID families. Set ${\bar{S}}_m:=m^{-1}{\sum }_{i=1}^m{S}_i,{\widehat{\mathrm{\Sigma }}}_m:=(m-1{)}^{-1}{\sum }_{i=1}^m(S_i-{\bar{S}}_m)(S_i-{\bar{S}}_m{)}^T\mathrm{.}$ The finite-simulation plug-in synthetic likelihood is the random quantity ${\widehat{g}}_{\mathrm{S}\mathrm{L},m}(y\mid \theta ):={\phi }_d(S(y);{\bar{S}}_m,{\widehat{\mathrm{\Sigma }}}_m)\mathrm{.}$ when ${\widehat{\mathrm{\Sigma }}}_m$ is positive definite, and is unavailable otherwise. The finite plug-in quantity is a function of simulated summaries; it is not defined as an unbiased estimator of the ideal Gaussian density.

The exact experiment’s likelihood density, when the experiment is dominated as in Dominated statistical experiments and given a pointwise density as in Likelihood densities, and this Gaussian summary approximation are different objects. The latter density-selection interface assumes AC and does not assert a common exceptional null set across an arbitrary parameter set; the present definition adds no parameter-indexed measurability. This definition makes no claim that the actual summary law is Gaussian, that $g_{\mathrm{S}\mathrm{L}}$ equals an exact likelihood, or that it yields posterior calibration, consistency, a Bernstein–von Mises theorem, or coverage. No optimizer or optimizer convergence property is included.

If a positive-definite covariance matrix varies differentiably in a real matrix direction $H$, Jacobi’s formula gives $D\det (\mathrm{\Sigma })[H]=\det (\mathrm{\Sigma })\mathrm{tr}({\mathrm{\Sigma }}^{-1}H)$ by The determinant differential is $D\det (A)[H]=\mathrm{tr}(\mathrm{adj}(A)H)$ at every matrix, and Jacobi's formula holds on the invertible locus. This differentiates the determinant factor only; this definition assumes no differentiability of the binding mean or covariance.

A positive-definite matrix has the unique Cholesky factor used in numerical evaluation by A matrix admits a Cholesky factorisation with positive diagonal exactly when it is Hermitian positive definite, and that factor is unique. This factorization does not ensure that a finite-simulation covariance is positive definite.