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# tf.contrib.bayesflow.monte_carlo.expectation_importance_sampler

Monte Carlo estimate of $$E_p[f(Z)] = E_q[f(Z) p(Z) / q(Z)]$$.

With $$p(z) := exp^{log_p(z)}$$, this Op returns

$$n^{-1} sum_{i=1}^n [ f(z_i) p(z_i) / q(z_i) ], z_i ~ q,$$ $$\approx E_q[ f(Z) p(Z) / q(Z) ]$$ $$= E_p[f(Z)]$$

This integral is done in log-space with max-subtraction to better handle the often extreme values that f(z) p(z) / q(z) can take on.

If f >= 0, it is up to 2x more efficient to exponentiate the result of expectation_importance_sampler_logspace applied to Log[f].

User supplies either Tensor of samples z, or number of samples to draw n

f Callable mapping samples from sampling_dist_q to Tensors with shape broadcastable to q.batch_shape. For example, f works "just like" q.log_prob.
log_p Callable mapping samples from sampling_dist_q to Tensors with shape broadcastable to q.batch_shape. For example, log_p works "just like" sampling_dist_q.log_prob.
sampling_dist_q The sampling distribution. tfp.distributions.Distribution. float64 dtype recommended. log_p and q should be supported on the same set.
z Tensor of samples from q, produced by q.sample for some n.
n Integer Tensor. Number of samples to generate if z is not provided.
seed Python integer to seed the random number generator.
name A name to give this Op.

The importance sampling estimate. Tensor with shape equal to batch shape of q, and dtype = q.dtype.

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