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.

Ops