# BayesFlow Monte Carlo (contrib)

Monte Carlo integration and helpers.

## Background

Monte Carlo integration refers to the practice of estimating an expectation with a sample mean. For example, given random variable Z in \$$R^k\$$ with density p, the expectation of function f can be approximated like:

<div> $$E_p[f(Z)] = \int f(z) p(z) dz$$ </div>
<div> $$~ S_n := n^{-1} \sum_{i=1}^n f(z_i), z_i\ iid\ samples\ from\ p.$$</div>


If \$$E_p[|f(Z)|] < infinity\$$, then \$$S_n\$$ --> \$$E_p[f(Z)]\$$ by the strong law of large numbers. If \$$E_p[f(Z)^2] < infinity\$$, then \$$S_n\$$ is asymptotically normal with variance \$$Var[f(Z)] / n\$$.

Practitioners of Bayesian statistics often find themselves wanting to estimate \$$E_p[f(Z)]\$$ when the distribution p is known only up to a constant. For example, the joint distribution p(z, x) may be known, but the evidence \$$p(x) = \int p(z, x) dz\$$ may be intractable. In that case, a parameterized distribution family \$$q_\lambda(z)\$$ may be chosen, and the optimal \$$\lambda\$$ is the one minimizing the KL divergence between \$$q_\lambda(z)\$$ and \$$p(z | x)\$$. We only know p(z, x), but that is sufficient to find \$$\lambda\$$.

## Log-space evaluation and subtracting the maximum

Care must be taken when the random variable lives in a high dimensional space. For example, the naive importance sample estimate \$$E_q[f(Z) p(Z) / q(Z)]\$$ involves the ratio of two terms \$$p(Z) / q(Z)\$$, each of which must have tails dropping off faster than \$$O(|z|^{-(k + 1)})\$$ in order to have finite integral. This ratio would often be zero or infinity up to numerical precision.

For that reason, we write

<div> $$Log E_q[ f(Z) p(Z) / q(Z) ]$$ </div>
<div> $$= Log E_q[ \exp\{Log[f(Z)] + Log[p(Z)] - Log[q(Z)] - C\} ] + C,$$  where </div>
<div> $$C := Max[ Log[f(Z)] + Log[p(Z)] - Log[q(Z)] ].$$ </div>


The maximum value of the exponentiated term will be 0.0, and the expectation can be evaluated in a stable manner.