*Bounty: 50*

Let

- $(E,mathcal E,lambda)$ be a $sigma$-finite measure space;
- $f:Eto[0,infty)^3$ be a bounded Bochner integrable function on $(E,mathcal E,lambda)$ and $p:=alpha_1f_1+alpha_2f_2+alpha_3f_3$ for some $alpha_1,alpha_2,alpha_3ge0$ with $alpha_1+alpha_2+alpha_3=1$ and $$int p:{rm d}lambdain(0,infty)tag1$$
- $I$ be a finite nonempty set
- $r:(Itimes E)times E$ be the density of a Markov kernel with source $(Itimes E,2^Iotimesmathcal E)$ and target $(E,mathcal E)$ with $$E_1:={p>0}subseteq{r((i,x),;cdot;)>0};;;text{for all }iin Itag2$$

Fix $xin E$. I want to find the index $iin I$ minimizing $$sigma_i:=lambda_ileft|f-frac{lambda_if}{lambda_i(E_1)}right|^2=lambda_i(E_1)operatorname E_ileft[left|f-operatorname E_i[f]right|^2right],$$ where $$lambda_i:=frac1{r((i,x),;cdot;)}left.lambdaright|_{E_1}$$ and $operatorname E_i$ is the expectation wrt $$operatorname P_i:=frac{lambda_i}{lambda_i(E_1)}$$ for $iin I$.

I’m not interested in the value $sigma_i$ itself.

Currently I’m estimating each $sigma_i$ using Monte Carlo integration and then compute the minimum. However, this is extremely slow.

I’m not familiar with this kind of problem and hence this might be nonsensical, but note that if $Y_i$ is an $(E,mathcal E)$-valued random variable with $Y_isim r((i,x),;cdot;)lambda$, then begin{equation}begin{split}&sigma_i=operatorname Eleft[1_{E_1}(Y_i)frac{|f(Y_i)|^2}{left|r((i,x),Y_i)right|^2}right]\&;;;;;;;;;;;;-left(operatorname Eleft[1_{E_1}(Y_i)frac1{left|r((i,x),Y_i)right|^2}right]right)^{-1}left|operatorname Eleft[1_{E_1}(Y_i)frac{f(Y_i)}{left|r((i,x),Y_i)right|^2}right]right|^2end{split}tag3end{equation} for all $iin I$. So, we might approximate $sigma_i$ by an independent identically distributed process $left(Y_i^{(n)}right)*{ninmathbb N}$** with $Y_i^{(1)}sim r((i,x),;cdot;)lambda$ via $$A_i^{(n)}-frac1{B_i^{(n)}}left|C_i^{(n)}right|^2xrightarrow{ntoinfty}sigma_i;;;text{almost surely},tag4$$ where begin{align}A_i^{(n)}:=frac1nsum*{i=1}^nfrac{1_{E_1}f}{left|r((i,x),;cdot;)right|^2}left(Y_i^{(n)}right),\B_i^{(n)}:=frac1nsum_{i=1}^nfrac{1_{E_1}}{left|r((i,x),;cdot;)right|^2}left(Y_i^{(n)}right),\C_i^{(n)}:=frac1nsum_{i=1}^nleft|frac{1_{E_1}f}{r((i,x),;cdot;)}right|^2left(Y_i^{(n)}right)end{align} for $ninmathbb N$, for all $iin I$.

Currently I’m using $(4)$ to estimate $sigma_i$ and then compute the minimum among the estimates. But this is extreme slowly and might even be erroneous.

Get this bounty!!!