# Source code for pyro.ops.special

# Copyright Contributors to the Pyro project.

import functools
import math
import operator

import torch

@staticmethod
def forward(ctx, x):
ctx.save_for_backward(x)
return x.log()

@staticmethod
x, = ctx.saved_tensors

[docs]def safe_log(x):
"""
Like :func:torch.log but avoids infinite gradients at log(0)
by clamping them to at most 1 / finfo.eps.
"""
return _SafeLog.apply(x)

[docs]def log_beta(x, y, tol=0.):
"""
Computes log Beta function.

When tol >= 0.02 this uses a shifted Stirling's approximation to the
log Beta function. The approximation adapts Stirling's approximation of the
log Gamma function::

lgamma(z) ≈ (z - 1/2) * log(z) - z + log(2 * pi) / 2

to approximate the log Beta function::

log_beta(x, y) ≈ ((x-1/2) * log(x) + (y-1/2) * log(y)
- (x+y-1/2) * log(x+y) + log(2*pi)/2)

The approximation additionally improves accuracy near zero by iteratively
shifting the log Gamma approximation using the recursion::

lgamma(x) = lgamma(x + 1) - log(x)

If this recursion is applied n times, then absolute error is bounded by
error < 0.082 / n < tol, thus we choose n based on the user
provided tol.

:param torch.Tensor x: A positive tensor.
:param torch.Tensor y: A positive tensor.
:param float tol: Bound on maximum absolute error. Defaults to 0.1. For
very small tol, this function simply defers to :func:log_beta.
:rtype: torch.Tensor
"""
assert isinstance(tol, (float, int)) and tol >= 0
if tol < 0.02:
# At small tolerance it is cheaper to defer to torch.lgamma().
return x.lgamma() + y.lgamma() - (x + y).lgamma()

# This bound holds for arbitrary x,y. We could do better with large x,y.
shift = int(math.ceil(0.082 / tol))

xy = x + y
factors = []
for _ in range(shift):
factors.append(xy / (x * y))
x = x + 1
y = y + 1
xy = xy + 1

log_factor = functools.reduce(operator.mul, factors).log()

return (log_factor + (x - 0.5) * x.log() + (y - 0.5) * y.log()
- (xy - 0.5) * xy.log() + (math.log(2 * math.pi) / 2 - shift))

def log_binomial(n, k, tol=0.):
"""
Computes log binomial coefficient.

When tol >= 0.02 this uses a shifted Stirling's approximation to the
log Beta function via :func:log_beta.

:param torch.Tensor n: A nonnegative integer tensor.
:param torch.Tensor k: An integer tensor ranging in [0, n].
:rtype: torch.Tensor
"""
assert isinstance(tol, (float, int)) and tol >= 0
n_plus_1 = n + 1
if tol < 0.02:
# At small tolerance it is cheaper to defer to torch.lgamma().
return n_plus_1.lgamma() - (k + 1).lgamma() - (n_plus_1 - k).lgamma()

return -n_plus_1.log() - log_beta(k + 1, n_plus_1 - k, tol=tol)