Monte Carlo integration and helpers.

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\\)`

.

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.

`tf.contrib.bayesflow.monte_carlo.expectation`

`tf.contrib.bayesflow.monte_carlo.expectation_importance_sampler`

`tf.contrib.bayesflow.monte_carlo.expectation_importance_sampler_logspace`

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Licensed under the Creative Commons Attribution License 3.0.

Code samples licensed under the Apache 2.0 License.

https://www.tensorflow.org/api_guides/python/contrib.bayesflow.monte_carlo