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`.

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`.

tf.contrib.bayesflow.monte_carlo.expectation

Example Use:

Args

Returns

Raises