A full-rank local Hessian does not capture both posterior modes

Given: Assume Countable Choice. Let $\mathrm{\Theta }=[-2,2]$ with its Borel sigma-algebra, let $n\ge 1$ be an integer, and set $h(\theta )=-({\theta }^2-1{)}^2\mathrm{.}$ Give the data space $\mathcal{X}=\{0,1,2\}$ its power-set sigma-algebra and counting measure, and define the $n$th experiment by $p_{n,\theta }(0)=\frac{1}{2}{e}^{nh(\theta )},p_{n,\theta }(1)=\frac{1}{4}+\frac{\theta }{32},p_{n,\theta }(2)=1-p_{n,\theta }(0)-p_{n,\theta }(1)\mathrm{.}$ Use the uniform prior ${\mathrm{\Pi }}_0(A)=\lambda (A\cap \mathrm{\Theta })\mathrm{/}4$ on Borel $A$, observe $x=0$ in the $n$th experiment, and use the likelihood version ${\mathrm{\ell }}_{n,0}(\theta ):=p_{n,\theta }(0)$. Let ${\mathrm{\Pi }}_n$ be the exact target and let ${\bar{\mathrm{\Pi }}}_n$ be its image on $\mathbb{R}$ under the inclusion $\mathrm{\Theta }\hookrightarrow \mathbb{R}$. Define the one-mode Laplace proxy $Q_n$ on $\mathbb{R}$ by the density $q_n(t)=\sqrt{4n\mathrm{/}\pi }{e}^{-4n(t-1{)}^2}\mathrm{.}$

Countable Choice means every countable family of nonempty sets has a choice function. The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

A fixed measurable likelihood version and finite positive evidence define ${\mathrm{\Pi }}_x(A)=Z_x^{-1}{\int }_A{\mathrm{\ell }}_{x}d{\mathrm{\Pi }}_0$; the likelihood version here is the density relative to the stated counting measure. Exact inferential targets and approximation chains Dominated statistical experiments A measurable function between measurable spaces

In the parametric model, observational equivalence means equality of sampling laws, and the identity estimand is identifiable when equivalent parameters agree. Statistical experiments Parametric statistical models Observational equivalence Estimands on a statistical model Identifiability of an estimand

The cited Laplace asymptotic theorem assumes a unique interior global maximizer in addition to a positive-definite negative Hessian and local and global domination conditions. Laplace asymptotic under a unique nondegenerate interior maximum

Under Countable Choice, the one-dimensional Gaussian integral gives ${\int }_{\mathbb{R}}{e}^{-u^2\mathrm{/}2}du=\sqrt{2\pi }$, and the positive-definite Gaussian integral lemma includes its normalization. Gaussian integral for a positive-definite quadratic form

Under Countable Choice, an affine $C^1$ diffeomorphism changes a nonnegative Lebesgue integral by its absolute Jacobian. A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions

For increasing measurable sets, measure of the union is the supremum of their measures. Continuity from below for measures

Total variation is the supremum of event-probability differences. Total variation distance for probability laws

Borel subsets of Euclidean space are Lebesgue measurable, bounded boxes have their usual volume, and the Borel sigma-algebra on $\mathrm{\Theta }$ is the trace of the one on $\mathbb{R}$. Assuming countable choice, every Borel subset of ${\mathbb{R}}^n$ is Lebesgue measurable A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included The Borel sigma-algebra of a subspace is the trace of the ambient Borel sigma-algebra

The real numbers form a complete ordered field, so their natural numbers are unbounded; $\mathrm{\mid }z\mathrm{\mid }\ge 0$ and $\mathrm{\mid }z{\mathrm{\mid }}^2=z^2$, and squaring is strictly increasing on nonnegative numbers. Uniqueness of nonnegative square roots therefore makes the square root order-preserving. The Cauchy-sequence reals have the least-upper-bound property The reals form a totally ordered field Every complete ordered field is Archimedean Basic properties of the absolute value Squaring is monotone on the nonnegatives Square roots exist: a unique $\sqrt{a}\ge 0$ with $(\sqrt{a}{)}^2=a$; the positives are $\{x^2:x\mathrm{\ne }0\}$