Correct derivation for CDF of monotonically decreasing transforms of discrete variables

[ricardoV94]

This is not an issue yet, because transforms of discrete variables aren't supported, but would be after we start allowing them (e.g., in #6836 and after #6360)

import pymc as pm
import numpy as np

p = 0.7
rv = -pm.Bernoulli.dist(p=p)

# A negated Bernoulli has pmf {p if x == -1; 1-p if x == 0; 0 otherwise}
assert pm.logp(rv, -2).eval() == -np.inf  # Correct
assert pm.logp(rv, -1).eval() == np.log(p)  # Correct
assert pm.logp(rv, 0).eval() == np.log(1-p)  # Correct
assert pm.logp(rv, 1).eval() == -np.inf  # Correct

The logic here works correctly for continuous variables, but for discrete variables we need the survival function (logccdf) evaluated at backward_value-1 and not backward_value. Otherwise the following checks would fail:

assert pm.logcdf(rv, -2).eval() == -np.inf  # Correct
assert pm.logcdf(rv, -1).eval() == np.log(p)  # Incorrect
assert pm.logcdf(rv, 0).eval() == 0  # Incorrect
assert pm.logcdf(rv, 1).eval() == 0  # Correct

pymc/pymc/logprob/transforms.py

Lines 466 to 499 in f5c5c9c

	@_logcdf.register(MeasurableTransform)
	def measurable_transform_logcdf(op: MeasurableTransform, value, *inputs, **kwargs):
	"""Compute the log-CDF graph for a `MeasurabeTransform`."""
	other_inputs = list(inputs)
	measurable_input = other_inputs.pop(op.measurable_input_idx)
	backward_value = op.transform_elemwise.backward(value, *other_inputs)
	# Fail if transformation is not injective
	# A TensorVariable is returned in 1-to-1 inversions, and a tuple in 1-to-many
	if isinstance(backward_value, tuple):
	raise NotImplementedError
	logcdf = _logcdf_helper(measurable_input, backward_value)
	logccdf = pt.log1mexp(logcdf)
	if isinstance(op.scalar_op, MONOTONICALLY_INCREASING_OPS):
	pass
	elif isinstance(op.scalar_op, MONOTONICALLY_DECREASING_OPS):
	logcdf = logccdf
	# mul is monotonically increasing for scale > 0, and monotonically decreasing otherwise
	elif isinstance(op.scalar_op, Mul):
	[scale] = other_inputs
	logcdf = pt.switch(pt.ge(scale, 0), logcdf, logccdf)
	# pow is increasing if pow > 0, and decreasing otherwise (even powers are rejected above)!
	# Care must be taken to handle negative values (https://math.stackexchange.com/a/442362/783483)
	elif isinstance(op.scalar_op, Pow):
	if op.transform_elemwise.power < 0:
	logcdf_zero = _logcdf_helper(measurable_input, 0)
	logcdf = pt.switch(
	pt.lt(backward_value, 0),
	pt.log(pt.exp(logcdf_zero) - pt.exp(logcdf)),
	pt.logaddexp(logccdf, logcdf_zero),
	)

Some care must also be taken for the negative odd powers

Likewise, the logic may be wrong for icdf, but I haven't checked:

pymc/pymc/logprob/transforms.py

Lines 510 to 522 in f5c5c9c

	@_icdf.register(MeasurableTransform)
	def measurable_transform_icdf(op: MeasurableTransform, value, *inputs, **kwargs):
	"""Compute the inverse CDF graph for a `MeasurabeTransform`."""
	other_inputs = list(inputs)
	measurable_input = other_inputs.pop(op.measurable_input_idx)
	if isinstance(op.scalar_op, MONOTONICALLY_INCREASING_OPS):
	pass
	elif isinstance(op.scalar_op, MONOTONICALLY_DECREASING_OPS):
	value = 1 - value
	elif isinstance(op.scalar_op, Mul):
	[scale] = other_inputs
	value = pt.switch(pt.lt(scale, 0), 1 - value, value)