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Fix math equation format in tf.contrib.bayesflow.monte_carlo #18089

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merged 2 commits into from
Mar 29, 2018
Merged

Fix math equation format in tf.contrib.bayesflow.monte_carlo #18089

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1829d42
Fix math equation format in contrib\bayesflow
imsheridan Mar 29, 2018
bbe9488
Fix minor pylint error
imsheridan Mar 29, 2018
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39 changes: 20 additions & 19 deletions tensorflow/contrib/bayesflow/python/ops/monte_carlo_impl.py
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Original file line number Diff line number Diff line change
Expand Up @@ -44,14 +44,14 @@ def expectation_importance_sampler(f,
n=None,
seed=None,
name='expectation_importance_sampler'):
r"""Monte Carlo estimate of `E_p[f(Z)] = E_q[f(Z) p(Z) / q(Z)]`.
r"""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
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)]
\\(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
Expand Down Expand Up @@ -95,9 +95,9 @@ def _importance_sampler_positive_f(log_f_z):
log_values = log_f_z + log_p_z - q_log_prob_z
return _logspace_mean(log_values)

# With f_plus(z) = max(0, f(z)), f_minus(z) = max(0, -f(z)),
# E_p[f(Z)] = E_p[f_plus(Z)] - E_p[f_minus(Z)]
# = E_p[f_plus(Z) + 1] - E_p[f_minus(Z) + 1]
# With \\(f_{plus}(z) = max(0, f(z)), f_{minus}(z) = max(0, -f(z))\\),
# \\(E_p[f(Z)] = E_p[f_{plus}(Z)] - E_p[f_{minus}(Z)]\\)
# \\( = E_p[f_{plus}(Z) + 1] - E_p[f_{minus}(Z) + 1]\\)
# Without incurring bias, 1 is added to each to prevent zeros in logspace.
# The logarithm is approximately linear around 1 + epsilon, so this is good
# for small values of 'z' as well.
Expand All @@ -121,13 +121,13 @@ def expectation_importance_sampler_logspace(
name='expectation_importance_sampler_logspace'):
r"""Importance sampling with a positive function, in log-space.

With `p(z) := exp{log_p(z)}`, and `f(z) = exp{log_f(z)}`, this `Op`
returns
With `\\(p(z) := exp^{log_p(z)}\\)`, and `\\(f(z) = exp{log_f(z)}\\)`,
this `Op` returns

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

This integral is done in log-space with max-subtraction to better handle the
Expand Down Expand Up @@ -196,12 +196,12 @@ def _logspace_mean(log_values):

def expectation(f, samples, log_prob=None, use_reparametrization=True,
axis=0, keep_dims=False, name=None):
"""Computes the Monte-Carlo approximation of `E_p[f(X)]`.
"""Computes the Monte-Carlo approximation of `\\(E_p[f(X)]\\)`.

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

```none
E_p[f(X)] approx= m**-1 sum_i^m f(x_j), x_j ~iid p(X)
\\(E_p[f(X)] \approx= m^{-1} sum_i^m f(x_j), x_j\ ~iid\ p(X)\\)
```

where:
Expand All @@ -216,8 +216,8 @@ def expectation(f, samples, log_prob=None, use_reparametrization=True,
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`.
`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.
Expand Down Expand Up @@ -296,7 +296,8 @@ def expectation(f, samples, log_prob=None, use_reparametrization=True,
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.
`\\(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`.
Expand All @@ -316,7 +317,7 @@ def expectation(f, samples, log_prob=None, use_reparametrization=True,

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

Raises:
ValueError: if `f` is not a Python `callable`.
Expand Down