class RejectionStandardGamma(Rejector):
"""
Naive Marsaglia & Tsang rejection sampler for standard Gamma distibution.
This assumes `concentration >= 1` and does not boost `concentration` or augment shape.
"""
def __init__(self, concentration):
if concentration.data.min() < 1:
raise NotImplementedError('concentration < 1 is not supported')
self.concentration = concentration
self._standard_gamma = Gamma(
concentration,
concentration.new([1.]).squeeze().expand_as(concentration))
# The following are Marsaglia & Tsang's variable names.
self._d = self.concentration - 1.0 / 3.0
self._c = 1.0 / torch.sqrt(9.0 * self._d)
# Compute log scale using Gamma.log_prob().
x = self._d.detach() # just an arbitrary x.
log_scale = self.propose_log_prob(x) + self.log_prob_accept(
x) - self.log_prob(x)
super().__init__(self.propose,
self.log_prob_accept,
log_scale,
batch_shape=concentration.shape,
event_shape=())
def expand(self, batch_shape, _instance=None):
new = self._get_checked_instance(RejectionStandardGamma, _instance)
batch_shape = torch.Size(batch_shape)
new.concentration = self.concentration.expand(batch_shape)
new._standard_gamma = self._standard_gamma.expand(batch_shape)
new._d = self._d.expand(batch_shape)
new._c = self._c.expand(batch_shape)
# Compute log scale using Gamma.log_prob().
x = new._d.detach() # just an arbitrary x.
log_scale = new.propose_log_prob(x) + new.log_prob_accept(
x) - new.log_prob(x)
super(RejectionStandardGamma, new).__init__(new.propose,
new.log_prob_accept,
log_scale,
batch_shape=batch_shape,
event_shape=())
new._validate_args = self._validate_args
return new
@weakmethod
def propose(self, sample_shape=torch.Size()):
# Marsaglia & Tsang's x == Naesseth's epsilon`
x = torch.randn(sample_shape + self.concentration.shape,
dtype=self.concentration.dtype,
device=self.concentration.device)
y = 1.0 + self._c * x
v = y * y * y
return (self._d * v).clamp_(1e-30, 1e30)
def propose_log_prob(self, value):
v = value / self._d
result = -self._d.log()
y = v.pow(1 / 3)
result -= torch.log(3 * y**2)
x = (y - 1) / self._c
result -= self._c.log()
result += Normal(torch.zeros_like(self.concentration),
torch.ones_like(self.concentration)).log_prob(x)
return result
@weakmethod
def log_prob_accept(self, value):
v = value / self._d
y = torch.pow(v, 1.0 / 3.0)
x = (y - 1.0) / self._c
log_prob_accept = 0.5 * x * x + self._d * (1.0 - v + torch.log(v))
log_prob_accept[y <= 0] = -float('inf')
return log_prob_accept
def log_prob(self, x):
return self._standard_gamma.log_prob(x)
class GammaPoisson(TorchDistribution):
r"""
Compound distribution comprising of a gamma-poisson pair, also referred to as
a gamma-poisson mixture. The ``rate`` parameter for the
:class:`~pyro.distributions.Poisson` distribution is unknown and randomly
drawn from a :class:`~pyro.distributions.Gamma` distribution.
.. note:: This can be treated as an alternate parametrization of the
:class:`~pyro.distributions.NegativeBinomial` (``total_count``, ``probs``)
distribution, with `concentration = total_count` and `rate = (1 - probs) / probs`.
:param float or torch.Tensor concentration: shape parameter (alpha) of the Gamma
distribution.
:param float or torch.Tensor rate: rate parameter (beta) for the Gamma
distribution.
"""
arg_constraints = {
'concentration': constraints.positive,
'rate': constraints.positive
}
support = Poisson.support
def __init__(self, concentration, rate, validate_args=None):
concentration, rate = broadcast_all(concentration, rate)
self._gamma = Gamma(concentration, rate)
super(GammaPoisson, self).__init__(self._gamma._batch_shape,
validate_args=validate_args)
@property
def concentration(self):
return self._gamma.concentration
@property
def rate(self):
return self._gamma.rate
def expand(self, batch_shape, _instance=None):
new = self._get_checked_instance(GammaPoisson, _instance)
batch_shape = torch.Size(batch_shape)
new._gamma = self._gamma.expand(batch_shape)
super(GammaPoisson, new).__init__(batch_shape, validate_args=False)
new._validate_args = self._validate_args
return new
def sample(self, sample_shape=()):
rate = self._gamma.sample(sample_shape)
return Poisson(rate).sample()
def log_prob(self, value):
if self._validate_args:
self._validate_sample(value)
post_value = self.concentration + value
return -_log_beta(self.concentration, value + 1) - post_value.log() + \
self.concentration * self.rate.log() - post_value * (1 + self.rate).log()
@property
def mean(self):
return self.concentration / self.rate
@property
def variance(self):
return self.concentration / self.rate.pow(2) * (1 + self.rate)