Let $w(\theta)=\pi(\theta)/\mu(\theta)$ be the <importance weight>. The two normalized densities give
$$
\log w(\theta)
=-\lambda_2\{Z(\theta)-F(\theta)\}
-\widetilde Z(\lambda)+\widetilde F(\lambda).
$$
Since $|F-Z|<C$, there is a finite constant $M$ such that $w(\theta)\leq M$ everywhere. The <Independence Metropolis–Hastings algorithm> has an accepted transition density satisfying
$$
\mu(y)\min\left\{1,\frac{w(y)}{w(x)}\right\}
\geq\frac1M\mu(y)w(y)
=\frac1M\pi(y).
$$
Consequently the whole state space is a , with the one-step <minorization condition> $K(x,\mathord\cdot)\geq M^{-1}\pi(\mathord\cdot)$. Iterating this <Doeblin condition> gives uniform geometric convergence in <total variation distance>, so the chain is geometrically ergodic.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 216 / 2 / c / Solution

For target density $f$ and proposal density $h$, the importance weight is the density ratio $w(x)=f(x)/h(x)$ wherever the target is supported.


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 Incoming links: Importance weight