Let $\Theta$ be a more or less arbitrary space carrying a probability distribution $\pi(d \theta)$ on it. Suppose that for every $\theta \in \Theta$ one can simulate a random variable $X_{\theta}$ with mean $\mu_{\theta} \in \mathbb{R}$. How would you study the distribution $\tilde{\pi}$ on $\Theta$ defined through the change of probability $$\frac{d \tilde{\pi}}{d \pi}(\theta) \propto V(\mu_{\theta})$$ where $V:\mathbb{R} \to (0; +\infty)$ is given function. Ideally, I would like to simulate from $\tilde{\pi}$, which might be a little bit too ambitious. Can one construct a Markov chain with $\tilde{\pi}$ as invariant distribution, or study $\tilde{\pi}$ trough importance sampling? I have not been able to find any reference for these kind of problems.
If it were possible to simulate a Bernoulli random variable with success probability $V(\mu_{\theta})$ one could use a simple accept-reject approach. In this problem, one can only simulate from $X_{\theta}$.
