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:

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

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

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

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