Independent finite bridge-ratio estimator

Input a finite nonempty state space $\mathsf{X}$, an integer $K\ge 1$, positive finite weights $w_0,\dots ,w_K:\mathsf{X}\to (0,\mathrm{\infty })$, positive integers $n_0,\dots ,n_{K-1}$, and, on a common probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$, supplied random variables $X_{k,j}:\mathrm{\Omega }\to (\mathsf{X},\mathcal{P}(\mathsf{X}))$ for $0\le k<K$ and $1\le j\le n_k$. For each $k$, set $Z_k={\sum }_{x\in \mathsf{X}}{w}_k(x)$ and ${\pi }_k(x)=w_k(x)\mathrm{/}{Z}_k$, a probability mass function on $\mathsf{X}$ and thus a probability measure on $\mathcal{P}(\mathsf{X})$. Require $\mathbb{P}(X_{k,j}=x)={\pi }_k(x)$ for every $x\in \mathsf{X}$ and require the entire finite family $(X_{k,j}{)}_{k,j}$ to be mutually independent. These exact draws are inputs, not outputs of this procedure; approximate Gibbs draws, correlated trajectories, and adaptively chosen bridges are outside its contract.

For each $k=0,\dots ,K-1$, compute ${\bar{R}}_k=n_k^{-1}{\sum }_{j=1}^{n_k}{w}_{k+1}(X_{k,j})\mathrm{/}{w}_k(X_{k,j})$. Return $\widehat{R}={\prod }_{k=0}^{K-1}{\bar{R}}_k$. The procedure evaluates only raw-weight ratios; it neither computes nor requires any $Z_k$ as a numeric input.

Finite partition-ratio estimator and overlap variance