Let $d\in\mathbb N$. I'm searching for a Markov kernel $\kappa$ on $\left([0,1)^d,{\mathcal B([0,1))}^{\otimes d}\right)$ suitable for the following application:
Given $x\in[0,1)^d$, I want to sample $y\in[0,1)^d$ with probability $\beta\in[0,1]$ from the uniform distribution $\mathcal U_{[0,\:1)^d}$ on $[0,1)^d$ and with probability $1-\beta$ from $\kappa(x,\;\cdot\;)$.
The idea is that with probability $\beta$ a "large step" transformation is performed and with probability $1-\beta$ a "small step" transformation is performed. This corresponds to sampling from the Markov kernel $$Q(x,B):=\beta\mathcal U_{[0,\:1)^d}(B)+(1-\beta)\kappa(x,B)\;\;\;\text{for }(x,B)\in[0,1)^d\times{\mathcal B([0,1))}^{\otimes d}.$$
Currently, I'm considering the wrapped normal distribution kernel $\mathcal W_{\sigma^2}$ with mean $0$ and variance $\sigma^2$, i.e. $$\mathcal W(x,\;\cdot\;):=\mathcal N(x,\;\cdot\;)\circ\iota^{-1}\;\;\;\text{for }x\in[0,1),$$ where $$\mathcal N(x,B):=\int_B\varphi_{\sigma^2}(y-x)\:{\rm d}y\;\;\;\text{for }(x,B)\in\mathbb R\times\mathcal B(\mathbb R),$$ $\varphi_{\sigma^2}$ denotes the density of the normal distribution with mean $0$ and variance $\sigma^2$ and $$\iota:\mathbb R\to[0,\infty)\;,\;\;\;x\mapsto x-\lfloor x\rfloor.$$ A typical choice is $\beta=0.3$ and $\sigma=0.01$. Now I choose $$\kappa(x,B):=\left(\bigotimes_{i=1}^d\mathcal W_{\sigma^2}(x_i,\;\cdot\;)\right)(B)\;\;\;\text{for }(x,B)\in[0,1)^d\times{\mathcal B([0,1))}^{\otimes d}.$$
This choice for $\kappa$ captures the idea of a "small step" (or a "local perturbation") of the current sample $x$ pretty well.
Assume that we have chosen a $\kappa$ such that $Q$ has a density $u$ with respect to the trace of the $d$-dimensional Lebesgue measure $\lambda^{\otimes d}$ on $[0,1)^d$. In my application, I have a finite set $I$ and a consider a varying number of dimensions $d_i$ and $\kappa_i$'s with density $u_i$, $i\in I$. Moreover, I have bounded measurable functions $f_i:[0,1)^{d_i}\to[0,\infty)$ and need to find the index $i\in I$ minimizing $$\int\lambda^{\otimes d_i}({\rm d}y)\frac{f_i(y)}{u_i\left(x^{(i)},y\right)}\tag1$$ for given fixed $x^{(i)}\in[0,1)^{d_i}$. (cf. my other question: Compute which of a finite number of integrals is minimal (not interested in the actual value of the integral)).
I need to solve this problem over and over again (inside a Metropolis-Hastings update) for different $x^{(i)}$'s. With the given choice of $\kappa$ this problem seems to be hard to solve and clearly depends on the $x^{(i)}$. However, if the $u_i$ would be such that the integral $\tag1$ does not depend on $x^{(i)}$, I could estimate $(1)$ and find the minimum in a precomputation step.
So, I'm searching for a different choice of $\kappa$ for which $(1)$ does not depend on $x^{(i)}$. $\kappa$ should still capture the idea of a "local, small step" and allow me to control the "size" of the step (as $\sigma$ does in the wrapped normal distribution).
Any suggestions?
