tf.contrib.bayesflow.monte_carlo.expectation

Computes the Monte-Carlo approximation of \(E_p[f(X)]\). (deprecated)

tf.contrib.bayesflow.monte_carlo.expectation(
    f, samples, log_prob=None, use_reparametrization=True, axis=0, keep_dims=False,
    name=None
)

This function computes the Monte-Carlo approximation of an expectation, i.e.,

\(E_p[f(X)] \approx= m^{-1} sum_i^m f(x_j), x_j\ ~iid\ p(X)\)

where:

• x_j = samples[j, ...],
• log(p(samples)) = log_prob(samples) and
• m = prod(shape(samples)[axis]).

Tricks: Reparameterization and Score-Gradient

When p is "reparameterized", i.e., a diffeomorphic transformation of a parameterless distribution (e.g., Normal(Y; m, s) <=> Y = sX + m, X ~ Normal(0,1)), we can swap gradient and expectation, i.e., grad[ Avg{ \(s_i : i=1...n\) } ] = Avg{ grad[\(s_i\)] : i=1...n } where S_n = Avg{\(s_i\)}and\(s_i = f(x_i), x_i ~ p\).

However, if p is not reparameterized, TensorFlow's gradient will be incorrect since the chain-rule stops at samples of non-reparameterized distributions. (The non-differentiated result, approx_expectation, is the same regardless of use_reparametrization.) In this circumstance using the Score-Gradient trick results in an unbiased gradient, i.e.,

grad[ E_p[f(X)] ]
= grad[ int dx p(x) f(x) ]
= int dx grad[ p(x) f(x) ]
= int dx [ p'(x) f(x) + p(x) f'(x) ]
= int dx p(x) [p'(x) / p(x) f(x) + f'(x) ]
= int dx p(x) grad[ f(x) p(x) / stop_grad[p(x)] ]
= E_p[ grad[ f(x) p(x) / stop_grad[p(x)] ] ]

Unless p is not reparametrized, it is usually preferable to use_reparametrization = True.

Example Use:

import tensorflow_probability as tfp
tfd = tfp.distributions

# Monte-Carlo approximation of a reparameterized distribution, e.g., Normal.

num_draws = int(1e5)
p = tfd.Normal(loc=0., scale=1.)
q = tfd.Normal(loc=1., scale=2.)
exact_kl_normal_normal = tfd.kl_divergence(p, q)
# ==> 0.44314718
approx_kl_normal_normal = tfp.monte_carlo.expectation(
    f=lambda x: p.log_prob(x) - q.log_prob(x),
    samples=p.sample(num_draws, seed=42),
    log_prob=p.log_prob,
    use_reparametrization=(p.reparameterization_type
                           == distribution.FULLY_REPARAMETERIZED))
# ==> 0.44632751
# Relative Error: <1%

# Monte-Carlo approximation of non-reparameterized distribution, e.g., Gamma.

num_draws = int(1e5)
p = ds.Gamma(concentration=1., rate=1.)
q = ds.Gamma(concentration=2., rate=3.)
exact_kl_gamma_gamma = tfd.kl_divergence(p, q)
# ==> 0.37999129
approx_kl_gamma_gamma = tfp.monte_carlo.expectation(
    f=lambda x: p.log_prob(x) - q.log_prob(x),
    samples=p.sample(num_draws, seed=42),
    log_prob=p.log_prob,
    use_reparametrization=(p.reparameterization_type
                           == distribution.FULLY_REPARAMETERIZED))
# ==> 0.37696719
# Relative Error: <1%

# For comparing the gradients, see `monte_carlo_test.py`.

Note: The above example is for illustration only. To compute approximate KL-divergence, the following is preferred:

approx_kl_p_q = tfp.vi.monte_carlo_csiszar_f_divergence(
    f=bf.kl_reverse,
    p_log_prob=q.log_prob,
    q=p,
    num_draws=num_draws)

Args
f	Python callable which can return f(samples).
samples	Tensor of samples used to form the Monte-Carlo approximation of \(E_p[f(X)]\). A batch of samples should be indexed by axis dimensions.
log_prob	Python callable which can return log_prob(samples). Must correspond to the natural-logarithm of the pdf/pmf of each sample. Only required/used if use_reparametrization=False. Default value: None.
use_reparametrization	Python bool indicating that the approximation should use the fact that the gradient of samples is unbiased. Whether True or False, this arg only affects the gradient of the resulting approx_expectation. Default value: True.
axis	The dimensions to average. If None, averages all dimensions. Default value: 0 (the left-most dimension).
keep_dims	If True, retains averaged dimensions using size 1. Default value: False.
name	A name_scope for operations created by this function. Default value: None (which implies "expectation").

Returns
approx_expectation	Tensor corresponding to the Monte-Carlo approximation of \(E_p[f(X)]\).

Raises
ValueError	if f is not a Python callable.
ValueError	if use_reparametrization=False and log_prob is not a Python callable.