# Uniform ergodicity

A first thought here is that some advantage may be gained from the observation that it seems like it should be relatively straight-forwards to identify conditions on the prior+likelihood pairing that would ensure every likelihood-constrained prior admits a random walk MCMC algorithm that is uniformly ergodic. That is, for n-step transition kernel, $P^n(x,\cdot)$, target $\pi^\ast(\cdot) \propto \pi(\cdot)I(L(\cdot)>L^\ast)$, positive constant, $M$, positive constant, $r < 1$, and state space, $E$,
$\mathrm{sup}_{x \in E} ||P^n(x,\cdot)-\pi^\ast(\cdot)|| \le Mr^n$,
where $||\cdot||$ denotes the total variation distance (see Meyn & Tweedie’s excellent, free, online book for a high level of detail).

Now, I say that conditions for the NS case should be relatively straight-forwards to identify based on Corollary 3 from Tierney (1994):
“A Metropolis kernel with $\mu(E^+) < \infty$ and $q$ and $\pi$ bounded and bounded away from 0 on $E^+$ satisfies a minorization condition $M(1,\beta,E,\nu)$ with $\nu$ proportional to the restriction of $\mu$ to $E^+$, and is therefore uniformly ergodic.”
Here $E^+$ is the support of (what in NS is our) $\pi^\ast$, the likelihood-constrained prior; so if $E$ is $\mathcal{R}^k$ then provided the posterior is proper the density in its tails must go to zero and $E^+$ for any $L^\ast >0$ becomes bounded (hence, $\mu(E^+) < \infty$ modulo measurability concerns).

Yet to do is to precise the necessary conditions on $\pi(\cdot)$ and $L(\cdot)$ to make this true, and generally useful. Also the utility of this observation may further depend on any statements about the existence and nature of a corresponding drift condition (possibly implied by uniform ergodicity (that is, I need to research further on this topic).

A note of confusion: The theorem from Tierney (1994) quoted above is directly contradicted by Theorem 3.1 in Mengersen & Tweedie (1996); namely,
“If $Q$ satisfies (23) [23: is a random walk MCMC algorithm, my note] on $\mathcal{R}^k$, then the Metropolis algorithm is not uniformly ergodic for any $\pi$.”
The difference here seems to be that these papers take a very different definition of the Metropolis algorithm when $x$ lies outside of the support of the target density, i.e., $\pi(x)=0$. In both the authors define the kernel to have acceptance 1 when $\pi(x)q(x,y)=0$, but in Tierney (1994) it is assumed that “to avoid some trivial special cases, let $Q(x,E^+)=1$ for $x \not\in E^+$“, whereas in Mengersen & Tweedie (1996) this is not stipulated. To my mind the Tierney (1994) definition is more ‘useful’, so for the remainder I will assume this one.

It is also worth noting that there are quite a few think-o’s in the introduction to Mengersen & Tweedie (1996) that can cause confusion. For instance, the definition of a small set is quoted as being from Meyn & Tweedie, but includes a clause not in their defintion, namely that $\nu$ be concentrated on $C$. [I’m wondering if this might be deliberate though to preclude having to specific the aperiodic+irreducilble clause on Theorem 1.3 (ii) and (iii), which is omitted in Mengernsen & Tweedie but present in Meyn & Tweedie??] Also, the defintion of the total variation norm given after their Equation (9) is missing a $-\mathrm{inf} |\mu(A)|$. And the proof of Lemma 1.2 is quite a bit off: “non-empty” should be $\mu(C) > 0$, the equality in (8) should be an inequality, and the “choose $B \subset C$” should be omitted to focus on A only.

Update:
I’ve just come across Roberts & Tweedie (1996) which gives some conditions for contours of densities in $\mathcal{R}^k$ being smooth enough for geometric ergodicity, so I will read this and comment when I have time.