Probability distributions - torch.distributions

The distributions package contains parameterizable probability distributions and sampling functions. This allows the construction of stochastic computation graphs and stochastic gradient estimators for optimization. This package generally follows the design of the TensorFlow Distributions package.

It is not possible to directly backpropagate through random samples. However, there are two main methods for creating surrogate functions that can be backpropagated through. These are the score function estimator/likelihood ratio estimator/REINFORCE and the pathwise derivative estimator. REINFORCE is commonly seen as the basis for policy gradient methods in reinforcement learning, and the pathwise derivative estimator is commonly seen in the reparameterization trick in variational autoencoders. Whilst the score function only requires the value of samples $f(x)$, the pathwise derivative requires the derivative $f'(x)$. The next sections discuss these two in a reinforcement learning example. For more details see Gradient Estimation Using Stochastic Computation Graphs .

Score function

When the probability density function is differentiable with respect to its parameters, we only need sample() and log_prob() to implement REINFORCE:

$$\Delta\theta = \alpha r \frac{\partial\log p(a|\pi^\theta(s))}{\partial\theta}$$

where $\theta$ are the parameters, $\alpha$ is the learning rate, $r$ is the reward and $p(a|\pi^\theta(s))$ is the probability of taking action $a$ in state $s$ given policy $\pi^\theta$.

In practice we would sample an action from the output of a network, apply this action in an environment, and then use log_prob to construct an equivalent loss function. Note that we use a negative because optimizers use gradient descent, whilst the rule above assumes gradient ascent. With a categorical policy, the code for implementing REINFORCE would be as follows:

probs = policy_network(state)
# Note that this is equivalent to what used to be called multinomial
m = Categorical(probs)
action = m.sample()
next_state, reward = env.step(action)
loss = -m.log_prob(action) * reward
loss.backward()

Pathwise derivative

The other way to implement these stochastic/policy gradients would be to use the reparameterization trick from the rsample() method, where the parameterized random variable can be constructed via a parameterized deterministic function of a parameter-free random variable. The reparameterized sample therefore becomes differentiable. The code for implementing the pathwise derivative would be as follows:

params = policy_network(state)
m = Normal(*params)
# Any distribution with .has_rsample == True could work based on the application
action = m.rsample()
next_state, reward = env.step(action)  # Assuming that reward is differentiable
loss = -reward
loss.backward()

Distribution

class torch.distributions.distribution.Distribution(batch_shape=torch.Size([]), event_shape=torch.Size([]), validate_args=None)

Bases: object

Distribution is the abstract base class for probability distributions.

property arg_constraints: Dict[str, Constraint]

Returns a dictionary from argument names to Constraint objects that should be satisfied by each argument of this distribution. Args that are not tensors need not appear in this dict.

property batch_shape: Size

Returns the shape over which parameters are batched.

cdf(value)

Returns the cumulative density/mass function evaluated at value.

Parameters

value (Tensor) –

Return type

Tensor

entropy()

Returns entropy of distribution, batched over batch_shape.

Returns

Tensor of shape batch_shape.

Return type

Tensor

enumerate_support(expand=True)

Returns tensor containing all values supported by a discrete distribution. The result will enumerate over dimension 0, so the shape of the result will be (cardinality,) + batch_shape + event_shape (where event_shape = () for univariate distributions).

Note that this enumerates over all batched tensors in lock-step [[0, 0], [1, 1], …]. With expand=False, enumeration happens along dim 0, but with the remaining batch dimensions being singleton dimensions, [[0], [1], ...

To iterate over the full Cartesian product use itertools.product(m.enumerate_support()).

Parameters

expand (bool) – whether to expand the support over the batch dims to match the distribution’s batch_shape.

Returns

Tensor iterating over dimension 0.

Return type

Tensor

property event_shape: Size

Returns the shape of a single sample (without batching).

expand(batch_shape, _instance=None)

Returns a new distribution instance (or populates an existing instance provided by a derived class) with batch dimensions expanded to batch_shape. This method calls expand on the distribution’s parameters. As such, this does not allocate new memory for the expanded distribution instance. Additionally, this does not repeat any args checking or parameter broadcasting in __init__.py, when an instance is first created.

Parameters

• batch_shape (torch.Size) – the desired expanded size.

• _instance – new instance provided by subclasses that need to override .expand.

Returns

New distribution instance with batch dimensions expanded to batch_size.

icdf(value)

Returns the inverse cumulative density/mass function evaluated at value.

Parameters

value (Tensor) –

Return type

Tensor

log_prob(value)

Returns the log of the probability density/mass function evaluated at value.

Parameters

value (Tensor) –

Return type

Tensor

property mean: Tensor

Returns the mean of the distribution.

property mode: Tensor

Returns the mode of the distribution.

perplexity()

Returns perplexity of distribution, batched over batch_shape.

Returns

Tensor of shape batch_shape.

Return type

Tensor

rsample(sample_shape=torch.Size([]))

Generates a sample_shape shaped reparameterized sample or sample_shape shaped batch of reparameterized samples if the distribution parameters are batched.

Return type

Tensor

sample(sample_shape=torch.Size([]))

Generates a sample_shape shaped sample or sample_shape shaped batch of samples if the distribution parameters are batched.

Return type

Tensor

sample_n(n)

Generates n samples or n batches of samples if the distribution parameters are batched.

Return type

Tensor

static set_default_validate_args(value)

Sets whether validation is enabled or disabled.

The default behavior mimics Python’s assert statement: validation is on by default, but is disabled if Python is run in optimized mode (via python -O). Validation may be expensive, so you may want to disable it once a model is working.

Parameters

value (bool) – Whether to enable validation.

property stddev: Tensor

Returns the standard deviation of the distribution.

property support: Optional[Any]

Returns a Constraint object representing this distribution’s support.

property variance: Tensor

Returns the variance of the distribution.

ExponentialFamily

class torch.distributions.exp_family.ExponentialFamily(batch_shape=torch.Size([]), event_shape=torch.Size([]), validate_args=None)

Bases: Distribution

ExponentialFamily is the abstract base class for probability distributions belonging to an exponential family, whose probability mass/density function has the form is defined below

$$p_{F}(x; \theta) = \exp(\langle t(x), \theta\rangle - F(\theta) + k(x))$$

where $\theta$ denotes the natural parameters, $t(x)$ denotes the sufficient statistic, $F(\theta)$ is the log normalizer function for a given family and $k(x)$ is the carrier measure.

Note

This class is an intermediary between the Distribution class and distributions which belong to an exponential family mainly to check the correctness of the .entropy() and analytic KL divergence methods. We use this class to compute the entropy and KL divergence using the AD framework and Bregman divergences (courtesy of: Frank Nielsen and Richard Nock, Entropies and Cross-entropies of Exponential Families).

entropy()

Method to compute the entropy using Bregman divergence of the log normalizer.

Bernoulli

class torch.distributions.bernoulli.Bernoulli(probs=None, logits=None, validate_args=None)

Bases: ExponentialFamily

Creates a Bernoulli distribution parameterized by probs or logits (but not both).

Samples are binary (0 or 1). They take the value 1 with probability p and 0 with probability 1 - p.

Example:

>>> m = Bernoulli(torch.tensor([0.3]))
>>> m.sample()  # 30% chance 1; 70% chance 0
tensor([ 0.])

Parameters

• probs (Number, Tensor) – the probability of sampling 1

• logits (Number, Tensor) – the log-odds of sampling 1

arg_constraints = {'logits': Real(), 'probs': Interval(lower_bound=0.0, upper_bound=1.0)}

entropy()

enumerate_support(expand=True)

expand(batch_shape, _instance=None)

has_enumerate_support = True

log_prob(value)

property logits

property mean

property mode

property param_shape

property probs

sample(sample_shape=torch.Size([]))

support = Boolean()

property variance

Beta

class torch.distributions.beta.Beta(concentration1, concentration0, validate_args=None)

Bases: ExponentialFamily

Beta distribution parameterized by concentration1 and concentration0.

Example:

>>> m = Beta(torch.tensor([0.5]), torch.tensor([0.5]))
>>> m.sample()  # Beta distributed with concentration concentration1 and concentration0
tensor([ 0.1046])

Parameters

• concentration1 (float or Tensor) – 1st concentration parameter of the distribution (often referred to as alpha)

• concentration0 (float or Tensor) – 2nd concentration parameter of the distribution (often referred to as beta)

arg_constraints = {'concentration0': GreaterThan(lower_bound=0.0), 'concentration1': GreaterThan(lower_bound=0.0)}

property concentration0

property concentration1

entropy()

expand(batch_shape, _instance=None)

has_rsample = True

log_prob(value)

property mean

property mode

rsample(sample_shape=())

support = Interval(lower_bound=0.0, upper_bound=1.0)

property variance

Binomial

class torch.distributions.binomial.Binomial(total_count=1, probs=None, logits=None, validate_args=None)

Bases: Distribution

Creates a Binomial distribution parameterized by total_count and either probs or logits (but not both). total_count must be broadcastable with probs/logits.

Example:

>>> m = Binomial(100, torch.tensor([0 , .2, .8, 1]))
>>> x = m.sample()
tensor([   0.,   22.,   71.,  100.])

>>> m = Binomial(torch.tensor([[5.], [10.]]), torch.tensor([0.5, 0.8]))
>>> x = m.sample()
tensor([[ 4.,  5.],
        [ 7.,  6.]])

Parameters

• total_count (int or Tensor) – number of Bernoulli trials

• probs (Tensor) – Event probabilities

• logits (Tensor) – Event log-odds

arg_constraints = {'logits': Real(), 'probs': Interval(lower_bound=0.0, upper_bound=1.0), 'total_count': IntegerGreaterThan(lower_bound=0)}

entropy()

enumerate_support(expand=True)

expand(batch_shape, _instance=None)

has_enumerate_support = True

log_prob(value)

property logits

property mean

property mode

property param_shape

property probs

sample(sample_shape=torch.Size([]))

property support

property variance

Categorical

class torch.distributions.categorical.Categorical(probs=None, logits=None, validate_args=None)

Bases: Distribution

Creates a categorical distribution parameterized by either probs or logits (but not both).

Note

It is equivalent to the distribution that torch.multinomial() samples from.

Samples are integers from $\{0, \ldots, K-1\}$ where K is probs.size(-1).

If probs is 1-dimensional with length-K, each element is the relative probability of sampling the class at that index.

If probs is N-dimensional, the first N-1 dimensions are treated as a batch of relative probability vectors.

Note

The probs argument must be non-negative, finite and have a non-zero sum, and it will be normalized to sum to 1 along the last dimension. probs will return this normalized value. The logits argument will be interpreted as unnormalized log probabilities and can therefore be any real number. It will likewise be normalized so that the resulting probabilities sum to 1 along the last dimension. logits will return this normalized value.

See also: torch.multinomial()

Example:

>>> m = Categorical(torch.tensor([ 0.25, 0.25, 0.25, 0.25 ]))
>>> m.sample()  # equal probability of 0, 1, 2, 3
tensor(3)

Parameters

• probs (Tensor) – event probabilities

• logits (Tensor) – event log probabilities (unnormalized)

arg_constraints = {'logits': IndependentConstraint(Real(), 1), 'probs': Simplex()}

entropy()

enumerate_support(expand=True)

expand(batch_shape, _instance=None)

has_enumerate_support = True

log_prob(value)

property logits

property mean

property mode

property param_shape

property probs

sample(sample_shape=torch.Size([]))

property support

property variance

Cauchy

class torch.distributions.cauchy.Cauchy(loc, scale, validate_args=None)

Bases: Distribution

Samples from a Cauchy (Lorentz) distribution. The distribution of the ratio of independent normally distributed random variables with means 0 follows a Cauchy distribution.

Example:

>>> m = Cauchy(torch.tensor([0.0]), torch.tensor([1.0]))
>>> m.sample()  # sample from a Cauchy distribution with loc=0 and scale=1
tensor([ 2.3214])

Parameters

• loc (float or Tensor) – mode or median of the distribution.

• scale (float or Tensor) – half width at half maximum.

arg_constraints = {'loc': Real(), 'scale': GreaterThan(lower_bound=0.0)}

cdf(value)

entropy()

expand(batch_shape, _instance=None)

has_rsample = True

icdf(value)

log_prob(value)

property mean

property mode

rsample(sample_shape=torch.Size([]))

support = Real()

property variance

Chi2

class torch.distributions.chi2.Chi2(df, validate_args=None)

Bases: Gamma

Creates a Chi-squared distribution parameterized by shape parameter df. This is exactly equivalent to Gamma(alpha=0.5*df, beta=0.5)

Example:

>>> m = Chi2(torch.tensor([1.0]))
>>> m.sample()  # Chi2 distributed with shape df=1
tensor([ 0.1046])

Parameters

df (float or Tensor) – shape parameter of the distribution

arg_constraints = {'df': GreaterThan(lower_bound=0.0)}

property df

expand(batch_shape, _instance=None)

ContinuousBernoulli

class torch.distributions.continuous_bernoulli.ContinuousBernoulli(probs=None, logits=None, lims=(0.499, 0.501), validate_args=None)

Bases: ExponentialFamily

Creates a continuous Bernoulli distribution parameterized by probs or logits (but not both).

The distribution is supported in [0, 1] and parameterized by ‘probs’ (in (0,1)) or ‘logits’ (real-valued). Note that, unlike the Bernoulli, ‘probs’ does not correspond to a probability and ‘logits’ does not correspond to log-odds, but the same names are used due to the similarity with the Bernoulli. See [1] for more details.

Example:

>>> m = ContinuousBernoulli(torch.tensor([0.3]))
>>> m.sample()
tensor([ 0.2538])

Parameters

• probs (Number, Tensor) – (0,1) valued parameters

• logits (Number, Tensor) – real valued parameters whose sigmoid matches ‘probs’

[1] The continuous Bernoulli: fixing a pervasive error in variational autoencoders, Loaiza-Ganem G and Cunningham JP, NeurIPS 2019. https://arxiv.org/abs/1907.06845

arg_constraints = {'logits': Real(), 'probs': Interval(lower_bound=0.0, upper_bound=1.0)}

cdf(value)

entropy()

expand(batch_shape, _instance=None)

has_rsample = True

icdf(value)

log_prob(value)

property logits

property mean

property param_shape

property probs

rsample(sample_shape=torch.Size([]))

sample(sample_shape=torch.Size([]))

property stddev

support = Interval(lower_bound=0.0, upper_bound=1.0)

property variance

Dirichlet

class torch.distributions.dirichlet.Dirichlet(concentration, validate_args=None)

Bases: ExponentialFamily

Creates a Dirichlet distribution parameterized by concentration concentration.

Example:

>>> m = Dirichlet(torch.tensor([0.5, 0.5]))
>>> m.sample()  # Dirichlet distributed with concentration [0.5, 0.5]
tensor([ 0.1046,  0.8954])

Parameters

concentration (Tensor) – concentration parameter of the distribution (often referred to as alpha)

arg_constraints = {'concentration': IndependentConstraint(GreaterThan(lower_bound=0.0), 1)}

entropy()

expand(batch_shape, _instance=None)

has_rsample = True

log_prob(value)

property mean

property mode

rsample(sample_shape=())

support = Simplex()

property variance

Exponential

class torch.distributions.exponential.Exponential(rate, validate_args=None)

Bases: ExponentialFamily

Creates a Exponential distribution parameterized by rate.

Example:

>>> m = Exponential(torch.tensor([1.0]))
>>> m.sample()  # Exponential distributed with rate=1
tensor([ 0.1046])

Parameters

rate (float or Tensor) – rate = 1 / scale of the distribution

arg_constraints = {'rate': GreaterThan(lower_bound=0.0)}

cdf(value)

entropy()

expand(batch_shape, _instance=None)

has_rsample = True

icdf(value)

log_prob(value)

property mean

property mode

rsample(sample_shape=torch.Size([]))

property stddev

support = GreaterThanEq(lower_bound=0.0)

property variance

FisherSnedecor

class torch.distributions.fishersnedecor.FisherSnedecor(df1, df2, validate_args=None)

Bases: Distribution

Creates a Fisher-Snedecor distribution parameterized by df1 and df2.

Example:

>>> m = FisherSnedecor(torch.tensor([1.0]), torch.tensor([2.0]))
>>> m.sample()  # Fisher-Snedecor-distributed with df1=1 and df2=2
tensor([ 0.2453])

Parameters

• df1 (float or Tensor) – degrees of freedom parameter 1

• df2 (float or Tensor) – degrees of freedom parameter 2

arg_constraints = {'df1': GreaterThan(lower_bound=0.0), 'df2': GreaterThan(lower_bound=0.0)}

expand(batch_shape, _instance=None)

has_rsample = True

log_prob(value)

property mean

property mode

rsample(sample_shape=torch.Size([]))

support = GreaterThan(lower_bound=0.0)

property variance

Gamma

class torch.distributions.gamma.Gamma(concentration, rate, validate_args=None)

Bases: ExponentialFamily

Creates a Gamma distribution parameterized by shape concentration and rate.

Example:

>>> m = Gamma(torch.tensor([1.0]), torch.tensor([1.0]))
>>> m.sample()  # Gamma distributed with concentration=1 and rate=1
tensor([ 0.1046])

Parameters

• concentration (float or Tensor) – shape parameter of the distribution (often referred to as alpha)

• rate (float or Tensor) – rate = 1 / scale of the distribution (often referred to as beta)

arg_constraints = {'concentration': GreaterThan(lower_bound=0.0), 'rate': GreaterThan(lower_bound=0.0)}

cdf(value)

entropy()

expand(batch_shape, _instance=None)

has_rsample = True

log_prob(value)

property mean

property mode

rsample(sample_shape=torch.Size([]))

support = GreaterThanEq(lower_bound=0.0)

property variance

Geometric

class torch.distributions.geometric.Geometric(probs=None, logits=None, validate_args=None)

Bases: Distribution

Creates a Geometric distribution parameterized by probs, where probs is the probability of success of Bernoulli trials.

$$P(X=k) = (1-p)^{k} p, k = 0, 1, ...$$

Note

torch.distributions.geometric.Geometric() $(k+1)$-th trial is the first success hence draws samples in $\{0, 1, \ldots\}$, whereas torch.Tensor.geometric_() k-th trial is the first success hence draws samples in $\{1, 2, \ldots\}$.

Example:

>>> m = Geometric(torch.tensor([0.3]))
>>> m.sample()  # underlying Bernoulli has 30% chance 1; 70% chance 0
tensor([ 2.])

Parameters

• probs (Number, Tensor) – the probability of sampling 1. Must be in range (0, 1]

• logits (Number, Tensor) – the log-odds of sampling 1.

arg_constraints = {'logits': Real(), 'probs': Interval(lower_bound=0.0, upper_bound=1.0)}

entropy()

expand(batch_shape, _instance=None)

log_prob(value)

property logits

property mean

property mode

property probs

sample(sample_shape=torch.Size([]))

support = IntegerGreaterThan(lower_bound=0)

property variance

Gumbel

class torch.distributions.gumbel.Gumbel(loc, scale, validate_args=None)

Bases: TransformedDistribution

Samples from a Gumbel Distribution.

Examples:

>>> m = Gumbel(torch.tensor([1.0]), torch.tensor([2.0]))
>>> m.sample()  # sample from Gumbel distribution with loc=1, scale=2
tensor([ 1.0124])

Parameters

• loc (float or Tensor) – Location parameter of the distribution

• scale (float or Tensor) – Scale parameter of the distribution

arg_constraints: Dict[str, Constraint] = {'loc': Real(), 'scale': GreaterThan(lower_bound=0.0)}

entropy()

expand(batch_shape, _instance=None)

log_prob(value)

property mean

property mode

property stddev

support = Real()

property variance

HalfCauchy

class torch.distributions.half_cauchy.HalfCauchy(scale, validate_args=None)

Bases: TransformedDistribution

Creates a half-Cauchy distribution parameterized by scale where:

X ~ Cauchy(0, scale)
Y = |X| ~ HalfCauchy(scale)

Example:

>>> m = HalfCauchy(torch.tensor([1.0]))
>>> m.sample()  # half-cauchy distributed with scale=1
tensor([ 2.3214])

Parameters

scale (float or Tensor) – scale of the full Cauchy distribution

arg_constraints: Dict[str, Constraint] = {'scale': GreaterThan(lower_bound=0.0)}

cdf(value)

entropy()

expand(batch_shape, _instance=None)

has_rsample = True

icdf(prob)

log_prob(value)

property mean

property mode

property scale

support = GreaterThanEq(lower_bound=0.0)

property variance

HalfNormal

class torch.distributions.half_normal.HalfNormal(scale, validate_args=None)

Bases: TransformedDistribution

Creates a half-normal distribution parameterized by scale where:

X ~ Normal(0, scale)
Y = |X| ~ HalfNormal(scale)

Example:

>>> m = HalfNormal(torch.tensor([1.0]))
>>> m.sample()  # half-normal distributed with scale=1
tensor([ 0.1046])

Parameters

scale (float or Tensor) – scale of the full Normal distribution

arg_constraints: Dict[str, Constraint] = {'scale': GreaterThan(lower_bound=0.0)}

cdf(value)

entropy()

expand(batch_shape, _instance=None)

has_rsample = True

icdf(prob)

log_prob(value)

property mean

property mode

property scale

support = GreaterThanEq(lower_bound=0.0)

property variance

Independent

class torch.distributions.independent.Independent(base_distribution, reinterpreted_batch_ndims, validate_args=None)

Bases: Distribution

Reinterprets some of the batch dims of a distribution as event dims.

This is mainly useful for changing the shape of the result of log_prob(). For example to create a diagonal Normal distribution with the same shape as a Multivariate Normal distribution (so they are interchangeable), you can:

>>> from torch.distributions.multivariate_normal import MultivariateNormal
>>> from torch.distributions.normal import Normal
>>> loc = torch.zeros(3)
>>> scale = torch.ones(3)
>>> mvn = MultivariateNormal(loc, scale_tril=torch.diag(scale))
>>> [mvn.batch_shape, mvn.event_shape]
[torch.Size([]), torch.Size([3])]
>>> normal = Normal(loc, scale)
>>> [normal.batch_shape, normal.event_shape]
[torch.Size([3]), torch.Size([])]
>>> diagn = Independent(normal, 1)
>>> [diagn.batch_shape, diagn.event_shape]
[torch.Size([]), torch.Size([3])]

Parameters

• base_distribution (torch.distributions.distribution.Distribution) – a base distribution

• reinterpreted_batch_ndims (int) – the number of batch dims to reinterpret as event dims

arg_constraints: Dict[str, Constraint] = {}

entropy()

enumerate_support(expand=True)

expand(batch_shape, _instance=None)

property has_enumerate_support

property has_rsample

log_prob(value)

property mean

property mode

rsample(sample_shape=torch.Size([]))

sample(sample_shape=torch.Size([]))

property support

property variance

InverseGamma

class torch.distributions.inverse_gamma.InverseGamma(concentration, rate, validate_args=None)

Bases: TransformedDistribution

Creates an inverse gamma distribution parameterized by concentration and rate where:

X ~ Gamma(concentration, rate)
Y = 1 / X ~ InverseGamma(concentration, rate)

Example:

>>> m = InverseGamma(torch.tensor([2.0]), torch.tensor([3.0]))
>>> m.sample()
tensor([ 1.2953])

Parameters

• concentration (float or Tensor) – shape parameter of the distribution (often referred to as alpha)

• rate (float or Tensor) – rate = 1 / scale of the distribution (often referred to as beta)

arg_constraints: Dict[str, Constraint] = {'concentration': GreaterThan(lower_bound=0.0), 'rate': GreaterThan(lower_bound=0.0)}

property concentration

entropy()

expand(batch_shape, _instance=None)

has_rsample = True

property mean

property mode

property rate

support = GreaterThan(lower_bound=0.0)

property variance

Kumaraswamy

class torch.distributions.kumaraswamy.Kumaraswamy(concentration1, concentration0, validate_args=None)

Bases: TransformedDistribution

Samples from a Kumaraswamy distribution.

Example:

>>> m = Kumaraswamy(torch.tensor([1.0]), torch.tensor([1.0]))
>>> m.sample()  # sample from a Kumaraswamy distribution with concentration alpha=1 and beta=1
tensor([ 0.1729])

Parameters

• concentration1 (float or Tensor) – 1st concentration parameter of the distribution (often referred to as alpha)

• concentration0 (float or Tensor) – 2nd concentration parameter of the distribution (often referred to as beta)

arg_constraints: Dict[str, Constraint] = {'concentration0': GreaterThan(lower_bound=0.0), 'concentration1': GreaterThan(lower_bound=0.0)}

entropy()

expand(batch_shape, _instance=None)

has_rsample = True

property mean

property mode

support = Interval(lower_bound=0.0, upper_bound=1.0)

property variance

LKJCholesky

class torch.distributions.lkj_cholesky.LKJCholesky(dim, concentration=1.0, validate_args=None)

Bases: Distribution

LKJ distribution for lower Cholesky factor of correlation matrices. The distribution is controlled by concentration parameter $\eta$ to make the probability of the correlation matrix $M$ generated from a Cholesky factor proportional to $\det(M)^{\eta - 1}$. Because of that, when concentration == 1, we have a uniform distribution over Cholesky factors of correlation matrices:

L ~ LKJCholesky(dim, concentration)
X = L @ L' ~ LKJCorr(dim, concentration)

Note that this distribution samples the Cholesky factor of correlation matrices and not the correlation matrices themselves and thereby differs slightly from the derivations in [1] for the LKJCorr distribution. For sampling, this uses the Onion method from [1] Section 3.

Example:

>>> l = LKJCholesky(3, 0.5)
>>> l.sample()  # l @ l.T is a sample of a correlation 3x3 matrix
tensor([[ 1.0000,  0.0000,  0.0000],
        [ 0.3516,  0.9361,  0.0000],
        [-0.1899,  0.4748,  0.8593]])

Parameters

• dimension (dim) – dimension of the matrices

• concentration (float or Tensor) – concentration/shape parameter of the distribution (often referred to as eta)

References

[1] Generating random correlation matrices based on vines and extended onion method (2009), Daniel Lewandowski, Dorota Kurowicka, Harry Joe. Journal of Multivariate Analysis. 100. 10.1016/j.jmva.2009.04.008

arg_constraints = {'concentration': GreaterThan(lower_bound=0.0)}

expand(batch_shape, _instance=None)

log_prob(value)

sample(sample_shape=torch.Size([]))

support = CorrCholesky()

Laplace

class torch.distributions.laplace.Laplace(loc, scale, validate_args=None)

Bases: Distribution

Creates a Laplace distribution parameterized by loc and scale.

Example:

>>> m = Laplace(torch.tensor([0.0]), torch.tensor([1.0]))
>>> m.sample()  # Laplace distributed with loc=0, scale=1
tensor([ 0.1046])

Parameters

• loc (float or Tensor) – mean of the distribution

• scale (float or Tensor) – scale of the distribution

arg_constraints = {'loc': Real(), 'scale': GreaterThan(lower_bound=0.0)}

cdf(value)

entropy()

expand(batch_shape, _instance=None)

has_rsample = True

icdf(value)

log_prob(value)

property mean

property mode

rsample(sample_shape=torch.Size([]))

property stddev

support = Real()

property variance

LogNormal

class torch.distributions.log_normal.LogNormal(loc, scale, validate_args=None)

Bases: TransformedDistribution

Creates a log-normal distribution parameterized by loc and scale where:

X ~ Normal(loc, scale)
Y = exp(X) ~ LogNormal(loc, scale)

Example:

>>> m = LogNormal(torch.tensor([0.0]), torch.tensor([1.0]))
>>> m.sample()  # log-normal distributed with mean=0 and stddev=1
tensor([ 0.1046])

Parameters

• loc (float or Tensor) – mean of log of distribution

• scale (float or Tensor) – standard deviation of log of the distribution

arg_constraints: Dict[str, Constraint] = {'loc': Real(), 'scale': GreaterThan(lower_bound=0.0)}

entropy()

expand(batch_shape, _instance=None)

has_rsample = True

property loc

property mean

property mode

property scale

support = GreaterThan(lower_bound=0.0)

property variance

LowRankMultivariateNormal

class torch.distributions.lowrank_multivariate_normal.LowRankMultivariateNormal(loc, cov_factor, cov_diag, validate_args=None)

Bases: Distribution

Creates a multivariate normal distribution with covariance matrix having a low-rank form parameterized by cov_factor and cov_diag:

covariance_matrix = cov_factor @ cov_factor.T + cov_diag

Example

>>> m = LowRankMultivariateNormal(torch.zeros(2), torch.tensor([[1.], [0.]]), torch.ones(2))
>>> m.sample()  # normally distributed with mean=`[0,0]`, cov_factor=`[[1],[0]]`, cov_diag=`[1,1]`
tensor([-0.2102, -0.5429])

Parameters

• loc (Tensor) – mean of the distribution with shape batch_shape + event_shape

• cov_factor (Tensor) – factor part of low-rank form of covariance matrix with shape batch_shape + event_shape + (rank,)

• cov_diag (Tensor) – diagonal part of low-rank form of covariance matrix with shape batch_shape + event_shape

Note

The computation for determinant and inverse of covariance matrix is avoided when cov_factor.shape[1] << cov_factor.shape[0] thanks to Woodbury matrix identity and matrix determinant lemma. Thanks to these formulas, we just need to compute the determinant and inverse of the small size “capacitance” matrix:

capacitance = I + cov_factor.T @ inv(cov_diag) @ cov_factor

arg_constraints = {'cov_diag': IndependentConstraint(GreaterThan(lower_bound=0.0), 1), 'cov_factor': IndependentConstraint(Real(), 2), 'loc': IndependentConstraint(Real(), 1)}

property covariance_matrix

entropy()

expand(batch_shape, _instance=None)

has_rsample = True

log_prob(value)

property mean

property mode

property precision_matrix

rsample(sample_shape=torch.Size([]))

property scale_tril

support = IndependentConstraint(Real(), 1)

property variance

MixtureSameFamily

class torch.distributions.mixture_same_family.MixtureSameFamily(mixture_distribution, component_distribution, validate_args=None)

Bases: Distribution

The MixtureSameFamily distribution implements a (batch of) mixture distribution where all component are from different parameterizations of the same distribution type. It is parameterized by a Categorical “selecting distribution” (over k component) and a component distribution, i.e., a Distribution with a rightmost batch shape (equal to [k]) which indexes each (batch of) component.

Examples:

>>> # Construct Gaussian Mixture Model in 1D consisting of 5 equally
>>> # weighted normal distributions
>>> mix = D.Categorical(torch.ones(5,))
>>> comp = D.Normal(torch.randn(5,), torch.rand(5,))
>>> gmm = MixtureSameFamily(mix, comp)

>>> # Construct Gaussian Mixture Model in 2D consisting of 5 equally
>>> # weighted bivariate normal distributions
>>> mix = D.Categorical(torch.ones(5,))
>>> comp = D.Independent(D.Normal(
...          torch.randn(5,2), torch.rand(5,2)), 1)
>>> gmm = MixtureSameFamily(mix, comp)

>>> # Construct a batch of 3 Gaussian Mixture Models in 2D each
>>> # consisting of 5 random weighted bivariate normal distributions
>>> mix = D.Categorical(torch.rand(3,5))
>>> comp = D.Independent(D.Normal(
...         torch.randn(3,5,2), torch.rand(3,5,2)), 1)
>>> gmm = MixtureSameFamily(mix, comp)

Parameters

• mixture_distribution – torch.distributions.Categorical-like instance. Manages the probability of selecting component. The number of categories must match the rightmost batch dimension of the component_distribution. Must have either scalar batch_shape or batch_shape matching component_distribution.batch_shape[:-1]

• component_distribution – torch.distributions.Distribution-like instance. Right-most batch dimension indexes component.

arg_constraints: Dict[str, Constraint] = {}

cdf(x)

property component_distribution

expand(batch_shape, _instance=None)

has_rsample = False

log_prob(x)

property mean

property mixture_distribution

sample(sample_shape=torch.Size([]))

property support

property variance

Multinomial

class torch.distributions.multinomial.Multinomial(total_count=1, probs=None, logits=None, validate_args=None)

Bases: Distribution

Creates a Multinomial distribution parameterized by total_count and either probs or logits (but not both). The innermost dimension of probs indexes over categories. All other dimensions index over batches.

Note that total_count need not be specified if only log_prob() is called (see example below)

Note

The probs argument must be non-negative, finite and have a non-zero sum, and it will be normalized to sum to 1 along the last dimension. probs will return this normalized value. The logits argument will be interpreted as unnormalized log probabilities and can therefore be any real number. It will likewise be normalized so that the resulting probabilities sum to 1 along the last dimension. logits will return this normalized value.

• sample() requires a single shared total_count for all parameters and samples.

• log_prob() allows different total_count for each parameter and sample.

Example:

>>> m = Multinomial(100, torch.tensor([ 1., 1., 1., 1.]))
>>> x = m.sample()  # equal probability of 0, 1, 2, 3
tensor([ 21.,  24.,  30.,  25.])

>>> Multinomial(probs=torch.tensor([1., 1., 1., 1.])).log_prob(x)
tensor([-4.1338])

Parameters

• total_count (int) – number of trials

• probs (Tensor) – event probabilities

• logits (Tensor) – event log probabilities (unnormalized)

arg_constraints = {'logits': IndependentConstraint(Real(), 1), 'probs': Simplex()}

entropy()

expand(batch_shape, _instance=None)

log_prob(value)

property logits

property mean

property param_shape

property probs

sample(sample_shape=torch.Size([]))

property support

total_count: int

property variance

MultivariateNormal

class torch.distributions.multivariate_normal.MultivariateNormal(loc, covariance_matrix=None, precision_matrix=None, scale_tril=None, validate_args=None)

Bases: Distribution

Creates a multivariate normal (also called Gaussian) distribution parameterized by a mean vector and a covariance matrix.

The multivariate normal distribution can be parameterized either in terms of a positive definite covariance matrix $\mathbf{\Sigma}$ or a positive definite precision matrix $\mathbf{\Sigma}^{-1}$ or a lower-triangular matrix $\mathbf{L}$ with positive-valued diagonal entries, such that $\mathbf{\Sigma} = \mathbf{L}\mathbf{L}^\top$. This triangular matrix can be obtained via e.g. Cholesky decomposition of the covariance.

Example

>>> m = MultivariateNormal(torch.zeros(2), torch.eye(2))
>>> m.sample()  # normally distributed with mean=`[0,0]` and covariance_matrix=`I`
tensor([-0.2102, -0.5429])

Parameters

• loc (Tensor) – mean of the distribution

• covariance_matrix (Tensor) – positive-definite covariance matrix

• precision_matrix (Tensor) – positive-definite precision matrix

• scale_tril (Tensor) – lower-triangular factor of covariance, with positive-valued diagonal

Note

Only one of covariance_matrix or precision_matrix or scale_tril can be specified.

Using scale_tril will be more efficient: all computations internally are based on scale_tril. If covariance_matrix or precision_matrix is passed instead, it is only used to compute the corresponding lower triangular matrices using a Cholesky decomposition.

arg_constraints = {'covariance_matrix': PositiveDefinite(), 'loc': IndependentConstraint(Real(), 1), 'precision_matrix': PositiveDefinite(), 'scale_tril': LowerCholesky()}

property covariance_matrix

entropy()

expand(batch_shape, _instance=None)

has_rsample = True

log_prob(value)

property mean

property mode

property precision_matrix

rsample(sample_shape=torch.Size([]))

property scale_tril

support = IndependentConstraint(Real(), 1)

property variance

NegativeBinomial

class torch.distributions.negative_binomial.NegativeBinomial(total_count, probs=None, logits=None, validate_args=None)

Bases: Distribution

Creates a Negative Binomial distribution, i.e. distribution of the number of successful independent and identical Bernoulli trials before total_count failures are achieved. The probability of success of each Bernoulli trial is probs.

Parameters

• total_count (float or Tensor) – non-negative number of negative Bernoulli trials to stop, although the distribution is still valid for real valued count

• probs (Tensor) – Event probabilities of success in the half open interval [0, 1)

• logits (Tensor) – Event log-odds for probabilities of success

arg_constraints = {'logits': Real(), 'probs': HalfOpenInterval(lower_bound=0.0, upper_bound=1.0), 'total_count': GreaterThanEq(lower_bound=0)}

expand(batch_shape, _instance=None)

log_prob(value)

property logits

property mean

property mode

property param_shape

property probs

sample(sample_shape=torch.Size([]))

support = IntegerGreaterThan(lower_bound=0)

property variance

Normal

class torch.distributions.normal.Normal(loc, scale, validate_args=None)

Bases: ExponentialFamily

Creates a normal (also called Gaussian) distribution parameterized by loc and scale.

Example:

>>> m = Normal(torch.tensor([0.0]), torch.tensor([1.0]))
>>> m.sample()  # normally distributed with loc=0 and scale=1
tensor([ 0.1046])

Parameters

• loc (float or Tensor) – mean of the distribution (often referred to as mu)

• scale (float or Tensor) – standard deviation of the distribution (often referred to as sigma)

arg_constraints = {'loc': Real(), 'scale': GreaterThan(lower_bound=0.0)}

cdf(value)

entropy()

expand(batch_shape, _instance=None)

has_rsample = True

icdf(value)

log_prob(value)

property mean

property mode

rsample(sample_shape=torch.Size([]))

sample(sample_shape=torch.Size([]))

property stddev

support = Real()

property variance

OneHotCategorical

class torch.distributions.one_hot_categorical.OneHotCategorical(probs=None, logits=None, validate_args=None)

Bases: Distribution

Creates a one-hot categorical distribution parameterized by probs or logits.

Samples are one-hot coded vectors of size probs.size(-1).

Note

The probs argument must be non-negative, finite and have a non-zero sum, and it will be normalized to sum to 1 along the last dimension. probs will return this normalized value. The logits argument will be interpreted as unnormalized log probabilities and can therefore be any real number. It will likewise be normalized so that the resulting probabilities sum to 1 along the last dimension. logits will return this normalized value.

See also: torch.distributions.Categorical() for specifications of probs and logits.

Example:

>>> m = OneHotCategorical(torch.tensor([ 0.25, 0.25, 0.25, 0.25 ]))
>>> m.sample()  # equal probability of 0, 1, 2, 3
tensor([ 0.,  0.,  0.,  1.])

Parameters

• probs (Tensor) – event probabilities

• logits (Tensor) – event log probabilities (unnormalized)

arg_constraints = {'logits': IndependentConstraint(Real(), 1), 'probs': Simplex()}

entropy()

enumerate_support(expand=True)

expand(batch_shape, _instance=None)

has_enumerate_support = True

log_prob(value)

property logits

property mean

property mode

property param_shape

property probs

sample(sample_shape=torch.Size([]))

support = OneHot()

property variance

Pareto

class torch.distributions.pareto.Pareto(scale, alpha, validate_args=None)

Bases: TransformedDistribution

Samples from a Pareto Type 1 distribution.

Example:

>>> m = Pareto(torch.tensor([1.0]), torch.tensor([1.0]))
>>> m.sample()  # sample from a Pareto distribution with scale=1 and alpha=1
tensor([ 1.5623])

Parameters

• scale (float or Tensor) – Scale parameter of the distribution

• alpha (float or Tensor) – Shape parameter of the distribution

arg_constraints: Dict[str, Constraint] = {'alpha': GreaterThan(lower_bound=0.0), 'scale': GreaterThan(lower_bound=0.0)}

entropy()

expand(batch_shape, _instance=None)

property mean

property mode

property support

property variance

Poisson

class torch.distributions.poisson.Poisson(rate, validate_args=None)

Bases: ExponentialFamily

Creates a Poisson distribution parameterized by rate, the rate parameter.

Samples are nonnegative integers, with a pmf given by

$$\mathrm{rate}^k \frac{e^{-\mathrm{rate}}}{k!}$$

Example:

>>> m = Poisson(torch.tensor([4]))
>>> m.sample()
tensor([ 3.])

Parameters

rate (Number, Tensor) – the rate parameter

arg_constraints = {'rate': GreaterThanEq(lower_bound=0.0)}

expand(batch_shape, _instance=None)

log_prob(value)

property mean

property mode

sample(sample_shape=torch.Size([]))

support = IntegerGreaterThan(lower_bound=0)

property variance

RelaxedBernoulli

class torch.distributions.relaxed_bernoulli.RelaxedBernoulli(temperature, probs=None, logits=None, validate_args=None)

Bases: TransformedDistribution

Creates a RelaxedBernoulli distribution, parametrized by temperature, and either probs or logits (but not both). This is a relaxed version of the Bernoulli distribution, so the values are in (0, 1), and has reparametrizable samples.

Example:

>>> m = RelaxedBernoulli(torch.tensor([2.2]),
...                      torch.tensor([0.1, 0.2, 0.3, 0.99]))
>>> m.sample()
tensor([ 0.2951,  0.3442,  0.8918,  0.9021])

Parameters

• temperature (Tensor) – relaxation temperature

• probs (Number, Tensor) – the probability of sampling 1

• logits (Number, Tensor) – the log-odds of sampling 1

arg_constraints: Dict[str, Constraint] = {'logits': Real(), 'probs': Interval(lower_bound=0.0, upper_bound=1.0)}

expand(batch_shape, _instance=None)

has_rsample = True

property logits

property probs

support = Interval(lower_bound=0.0, upper_bound=1.0)

property temperature

LogitRelaxedBernoulli

class torch.distributions.relaxed_bernoulli.LogitRelaxedBernoulli(temperature, probs=None, logits=None, validate_args=None)

Bases: Distribution

Creates a LogitRelaxedBernoulli distribution parameterized by probs or logits (but not both), which is the logit of a RelaxedBernoulli distribution.

Samples are logits of values in (0, 1). See [1] for more details.

Parameters

• temperature (Tensor) – relaxation temperature

• probs (Number, Tensor) – the probability of sampling 1

• logits (Number, Tensor) – the log-odds of sampling 1

[1] The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables (Maddison et al., 2017)

[2] Categorical Reparametrization with Gumbel-Softmax (Jang et al., 2017)

arg_constraints = {'logits': Real(), 'probs': Interval(lower_bound=0.0, upper_bound=1.0)}

expand(batch_shape, _instance=None)

log_prob(value)

property logits

property param_shape

property probs

rsample(sample_shape=torch.Size([]))

support = Real()

RelaxedOneHotCategorical

class torch.distributions.relaxed_categorical.RelaxedOneHotCategorical(temperature, probs=None, logits=None, validate_args=None)

Bases: TransformedDistribution

Creates a RelaxedOneHotCategorical distribution parametrized by temperature, and either probs or logits. This is a relaxed version of the OneHotCategorical distribution, so its samples are on simplex, and are reparametrizable.

Example:

>>> m = RelaxedOneHotCategorical(torch.tensor([2.2]),
...                              torch.tensor([0.1, 0.2, 0.3, 0.4]))
>>> m.sample()
tensor([ 0.1294,  0.2324,  0.3859,  0.2523])

Parameters

• temperature (Tensor) – relaxation temperature

• probs (Tensor) – event probabilities

• logits (Tensor) – unnormalized log probability for each event

arg_constraints: Dict[str, Constraint] = {'logits': IndependentConstraint(Real(), 1), 'probs': Simplex()}

expand(batch_shape, _instance=None)

has_rsample = True

property logits

property probs

support = Simplex()

property temperature

StudentT

class torch.distributions.studentT.StudentT(df, loc=0.0, scale=1.0, validate_args=None)

Bases: Distribution

Creates a Student’s t-distribution parameterized by degree of freedom df, mean loc and scale scale.

Example:

>>> m = StudentT(torch.tensor([2.0]))
>>> m.sample()  # Student's t-distributed with degrees of freedom=2
tensor([ 0.1046])

Parameters

• df (float or Tensor) – degrees of freedom

• loc (float or Tensor) – mean of the distribution

• scale (float or Tensor) – scale of the distribution

arg_constraints = {'df': GreaterThan(lower_bound=0.0), 'loc': Real(), 'scale': GreaterThan(lower_bound=0.0)}

entropy()

expand(batch_shape, _instance=None)

has_rsample = True

log_prob(value)

property mean

property mode

rsample(sample_shape=torch.Size([]))

support = Real()

property variance

TransformedDistribution

class torch.distributions.transformed_distribution.TransformedDistribution(base_distribution, transforms, validate_args=None)

Bases: Distribution

Extension of the Distribution class, which applies a sequence of Transforms to a base distribution. Let f be the composition of transforms applied:

X ~ BaseDistribution
Y = f(X) ~ TransformedDistribution(BaseDistribution, f)
log p(Y) = log p(X) + log |det (dX/dY)|

Note that the .event_shape of a TransformedDistribution is the maximum shape of its base distribution and its transforms, since transforms can introduce correlations among events.

An example for the usage of TransformedDistribution would be:

# Building a Logistic Distribution
# X ~ Uniform(0, 1)
# f = a + b * logit(X)
# Y ~ f(X) ~ Logistic(a, b)
base_distribution = Uniform(0, 1)
transforms = [SigmoidTransform().inv, AffineTransform(loc=a, scale=b)]
logistic = TransformedDistribution(base_distribution, transforms)

For more examples, please look at the implementations of Gumbel, HalfCauchy, HalfNormal, LogNormal, Pareto, Weibull, RelaxedBernoulli and RelaxedOneHotCategorical

arg_constraints: Dict[str, Constraint] = {}

cdf(value)

Computes the cumulative distribution function by inverting the transform(s) and computing the score of the base distribution.

expand(batch_shape, _instance=None)

property has_rsample

icdf(value)

Computes the inverse cumulative distribution function using transform(s) and computing the score of the base distribution.

log_prob(value)

Scores the sample by inverting the transform(s) and computing the score using the score of the base distribution and the log abs det jacobian.

rsample(sample_shape=torch.Size([]))

Generates a sample_shape shaped reparameterized sample or sample_shape shaped batch of reparameterized samples if the distribution parameters are batched. Samples first from base distribution and applies transform() for every transform in the list.

sample(sample_shape=torch.Size([]))

Generates a sample_shape shaped sample or sample_shape shaped batch of samples if the distribution parameters are batched. Samples first from base distribution and applies transform() for every transform in the list.

property support

Uniform

class torch.distributions.uniform.Uniform(low, high, validate_args=None)

Bases: Distribution

Generates uniformly distributed random samples from the half-open interval [low, high).

Example:

>>> m = Uniform(torch.tensor([0.0]), torch.tensor([5.0]))
>>> m.sample()  # uniformly distributed in the range [0.0, 5.0)
tensor([ 2.3418])

Parameters

• low (float or Tensor) – lower range (inclusive).

• high (float or Tensor) – upper range (exclusive).

arg_constraints = {'high': Dependent(), 'low': Dependent()}

cdf(value)

entropy()

expand(batch_shape, _instance=None)

has_rsample = True

icdf(value)

log_prob(value)

property mean

property mode

rsample(sample_shape=torch.Size([]))

property stddev

property support

property variance

VonMises

class torch.distributions.von_mises.VonMises(loc, concentration, validate_args=None)

Bases: Distribution

A circular von Mises distribution.

This implementation uses polar coordinates. The loc and value args can be any real number (to facilitate unconstrained optimization), but are interpreted as angles modulo 2 pi.

Example::

>>> m = VonMises(torch.tensor([1.0]), torch.tensor([1.0]))
>>> m.sample()  # von Mises distributed with loc=1 and concentration=1
tensor([1.9777])

Parameters

• loc (torch.Tensor) – an angle in radians.

• concentration (torch.Tensor) – concentration parameter

arg_constraints = {'concentration': GreaterThan(lower_bound=0.0), 'loc': Real()}

expand(batch_shape)

has_rsample = False

log_prob(value)

property mean

The provided mean is the circular one.

property mode

sample(sample_shape=torch.Size([]))

The sampling algorithm for the von Mises distribution is based on the following paper: D.J. Best and N.I. Fisher, “Efficient simulation of the von Mises distribution.” Applied Statistics (1979): 152-157.

Sampling is always done in double precision internally to avoid a hang in _rejection_sample() for small values of the concentration, which starts to happen for single precision around 1e-4 (see issue #88443).

support = Real()

property variance

The provided variance is the circular one.

Weibull

class torch.distributions.weibull.Weibull(scale, concentration, validate_args=None)

Bases: TransformedDistribution

Samples from a two-parameter Weibull distribution.

Example

>>> m = Weibull(torch.tensor([1.0]), torch.tensor([1.0]))
>>> m.sample()  # sample from a Weibull distribution with scale=1, concentration=1
tensor([ 0.4784])

Parameters

• scale (float or Tensor) – Scale parameter of distribution (lambda).

• concentration (float or Tensor) – Concentration parameter of distribution (k/shape).

arg_constraints: Dict[str, Constraint] = {'concentration': GreaterThan(lower_bound=0.0), 'scale': GreaterThan(lower_bound=0.0)}

entropy()

expand(batch_shape, _instance=None)

property mean

property mode

support = GreaterThan(lower_bound=0.0)

property variance

Wishart

class torch.distributions.wishart.Wishart(df, covariance_matrix=None, precision_matrix=None, scale_tril=None, validate_args=None)

Bases: ExponentialFamily

Creates a Wishart distribution parameterized by a symmetric positive definite matrix $\Sigma$, or its Cholesky decomposition $\mathbf{\Sigma} = \mathbf{L}\mathbf{L}^\top$

Example

>>> m = Wishart(torch.Tensor([2]), covariance_matrix=torch.eye(2))
>>> m.sample()  # Wishart distributed with mean=`df * I` and
>>>             # variance(x_ij)=`df` for i != j and variance(x_ij)=`2 * df` for i == j

Parameters

• df (float or Tensor) – real-valued parameter larger than the (dimension of Square matrix) - 1

• covariance_matrix (Tensor) – positive-definite covariance matrix

• precision_matrix (Tensor) – positive-definite precision matrix

• scale_tril (Tensor) – lower-triangular factor of covariance, with positive-valued diagonal

Note

Only one of covariance_matrix or precision_matrix or scale_tril can be specified. Using scale_tril will be more efficient: all computations internally are based on scale_tril. If covariance_matrix or precision_matrix is passed instead, it is only used to compute the corresponding lower triangular matrices using a Cholesky decomposition. ‘torch.distributions.LKJCholesky’ is a restricted Wishart distribution.[1]

References

[1] Wang, Z., Wu, Y. and Chu, H., 2018. On equivalence of the LKJ distribution and the restricted Wishart distribution. [2] Sawyer, S., 2007. Wishart Distributions and Inverse-Wishart Sampling. [3] Anderson, T. W., 2003. An Introduction to Multivariate Statistical Analysis (3rd ed.). [4] Odell, P. L. & Feiveson, A. H., 1966. A Numerical Procedure to Generate a SampleCovariance Matrix. JASA, 61(313):199-203. [5] Ku, Y.-C. & Bloomfield, P., 2010. Generating Random Wishart Matrices with Fractional Degrees of Freedom in OX.

arg_constraints = {'covariance_matrix': PositiveDefinite(), 'df': GreaterThan(lower_bound=0), 'precision_matrix': PositiveDefinite(), 'scale_tril': LowerCholesky()}

property covariance_matrix

entropy()

expand(batch_shape, _instance=None)

has_rsample = True

log_prob(value)

property mean

property mode

property precision_matrix

rsample(sample_shape=torch.Size([]), max_try_correction=None)

Warning

In some cases, sampling algorithm based on Bartlett decomposition may return singular matrix samples. Several tries to correct singular samples are performed by default, but it may end up returning singular matrix samples. Singular samples may return -inf values in .log_prob(). In those cases, the user should validate the samples and either fix the value of df or adjust max_try_correction value for argument in .rsample accordingly.

property scale_tril

support = PositiveDefinite()

property variance

KL Divergence

torch.distributions.kl.kl_divergence(p, q)

Compute Kullback-Leibler divergence $KL(p \| q)$ between two distributions.

$$KL(p \| q) = \int p(x) \log\frac {p(x)} {q(x)} \,dx$$

Parameters

• p (Distribution) – A Distribution object.

• q (Distribution) – A Distribution object.

Returns

A batch of KL divergences of shape batch_shape.

Return type

Tensor

Raises

NotImplementedError – If the distribution types have not been registered via register_kl().

KL divergence is currently implemented for the following distribution pairs:

• Bernoulli and Bernoulli

• Bernoulli and Poisson

• Beta and Beta

• Beta and ContinuousBernoulli

• Beta and Exponential

• Beta and Gamma

• Beta and Normal

• Beta and Pareto

• Beta and Uniform

• Binomial and Binomial

• Categorical and Categorical

• Cauchy and Cauchy

• ContinuousBernoulli and ContinuousBernoulli

• ContinuousBernoulli and Exponential

• ContinuousBernoulli and Normal

• ContinuousBernoulli and Pareto

• ContinuousBernoulli and Uniform

• Dirichlet and Dirichlet

• Exponential and Beta

• Exponential and ContinuousBernoulli

• Exponential and Exponential

• Exponential and Gamma

• Exponential and Gumbel

• Exponential and Normal

• Exponential and Pareto

• Exponential and Uniform

• ExponentialFamily and ExponentialFamily

• Gamma and Beta

• Gamma and ContinuousBernoulli

• Gamma and Exponential

• Gamma and Gamma

• Gamma and Gumbel

• Gamma and Normal

• Gamma and Pareto

• Gamma and Uniform

• Geometric and Geometric

• Gumbel and Beta

• Gumbel and ContinuousBernoulli

• Gumbel and Exponential

• Gumbel and Gamma

• Gumbel and Gumbel

• Gumbel and Normal

• Gumbel and Pareto

• Gumbel and Uniform

• HalfNormal and HalfNormal

• Independent and Independent

• Laplace and Beta

• Laplace and ContinuousBernoulli

• Laplace and Exponential

• Laplace and Gamma

• Laplace and Laplace

• Laplace and Normal

• Laplace and Pareto

• Laplace and Uniform

• LowRankMultivariateNormal and LowRankMultivariateNormal

• LowRankMultivariateNormal and MultivariateNormal

• MultivariateNormal and LowRankMultivariateNormal

• MultivariateNormal and MultivariateNormal

• Normal and Beta

• Normal and ContinuousBernoulli

• Normal and Exponential

• Normal and Gamma

• Normal and Gumbel

• Normal and Laplace

• Normal and Normal

• Normal and Pareto

• Normal and Uniform

• OneHotCategorical and OneHotCategorical

• Pareto and Beta

• Pareto and ContinuousBernoulli

• Pareto and Exponential

• Pareto and Gamma

• Pareto and Normal

• Pareto and Pareto

• Pareto and Uniform

• Poisson and Bernoulli

• Poisson and Binomial

• Poisson and Poisson

• TransformedDistribution and TransformedDistribution

• Uniform and Beta

• Uniform and ContinuousBernoulli

• Uniform and Exponential

• Uniform and Gamma

• Uniform and Gumbel

• Uniform and Normal

• Uniform and Pareto

• Uniform and Uniform

torch.distributions.kl.register_kl(type_p, type_q)

Decorator to register a pairwise function with kl_divergence(). Usage:

@register_kl(Normal, Normal)
def kl_normal_normal(p, q):
    # insert implementation here

Lookup returns the most specific (type,type) match ordered by subclass. If the match is ambiguous, a RuntimeWarning is raised. For example to resolve the ambiguous situation:

@register_kl(BaseP, DerivedQ)
def kl_version1(p, q): ...
@register_kl(DerivedP, BaseQ)
def kl_version2(p, q): ...

you should register a third most-specific implementation, e.g.:

register_kl(DerivedP, DerivedQ)(kl_version1)  # Break the tie.

Parameters

• type_p (type) – A subclass of Distribution.

• type_q (type) – A subclass of Distribution.

Transforms

class torch.distributions.transforms.AbsTransform(cache_size=0)

Transform via the mapping $y = |x|$.

class torch.distributions.transforms.AffineTransform(loc, scale, event_dim=0, cache_size=0)

Transform via the pointwise affine mapping $y = \text{loc} + \text{scale} \times x$.

Parameters

• loc (Tensor or float) – Location parameter.

• scale (Tensor or float) – Scale parameter.

• event_dim (int) – Optional size of event_shape. This should be zero for univariate random variables, 1 for distributions over vectors, 2 for distributions over matrices, etc.

class torch.distributions.transforms.CatTransform(tseq, dim=0, lengths=None, cache_size=0)

Transform functor that applies a sequence of transforms tseq component-wise to each submatrix at dim, of length lengths[dim], in a way compatible with torch.cat().

Example:

x0 = torch.cat([torch.range(1, 10), torch.range(1, 10)], dim=0)
x = torch.cat([x0, x0], dim=0)
t0 = CatTransform([ExpTransform(), identity_transform], dim=0, lengths=[10, 10])
t = CatTransform([t0, t0], dim=0, lengths=[20, 20])
y = t(x)

class torch.distributions.transforms.ComposeTransform(parts, cache_size=0)

Composes multiple transforms in a chain. The transforms being composed are responsible for caching.

Parameters

• parts (list of Transform) – A list of transforms to compose.

• cache_size (int) – Size of cache. If zero, no caching is done. If one, the latest single value is cached. Only 0 and 1 are supported.

class torch.distributions.transforms.CorrCholeskyTransform(cache_size=0)

Transforms an uncontrained real vector $x$ with length $D*(D-1)/2$ into the Cholesky factor of a D-dimension correlation matrix. This Cholesky factor is a lower triangular matrix with positive diagonals and unit Euclidean norm for each row. The transform is processed as follows:

1. First we convert x into a lower triangular matrix in row order.

2. For each row $X_i$ of the lower triangular part, we apply a signed version of class StickBreakingTransform to transform $X_i$ into a unit Euclidean length vector using the following steps: - Scales into the interval $(-1, 1)$ domain: $r_i = \tanh(X_i)$. - Transforms into an unsigned domain: $z_i = r_i^2$. - Applies $s_i = StickBreakingTransform(z_i)$. - Transforms back into signed domain: $y_i = sign(r_i) * \sqrt{s_i}$.

class torch.distributions.transforms.CumulativeDistributionTransform(distribution, cache_size=0)

Transform via the cumulative distribution function of a probability distribution.

Parameters

distribution (Distribution) – Distribution whose cumulative distribution function to use for the transformation.

Example:

# Construct a Gaussian copula from a multivariate normal.
base_dist = MultivariateNormal(
    loc=torch.zeros(2),
    scale_tril=LKJCholesky(2).sample(),
)
transform = CumulativeDistributionTransform(Normal(0, 1))
copula = TransformedDistribution(base_dist, [transform])

class torch.distributions.transforms.ExpTransform(cache_size=0)

Transform via the mapping $y = \exp(x)$.

class torch.distributions.transforms.IndependentTransform(base_transform, reinterpreted_batch_ndims, cache_size=0)

Wrapper around another transform to treat reinterpreted_batch_ndims-many extra of the right most dimensions as dependent. This has no effect on the forward or backward transforms, but does sum out reinterpreted_batch_ndims-many of the rightmost dimensions in log_abs_det_jacobian().

Parameters

• base_transform (Transform) – A base transform.

• reinterpreted_batch_ndims (int) – The number of extra rightmost dimensions to treat as dependent.

class torch.distributions.transforms.LowerCholeskyTransform(cache_size=0)

Transform from unconstrained matrices to lower-triangular matrices with nonnegative diagonal entries.

This is useful for parameterizing positive definite matrices in terms of their Cholesky factorization.

class torch.distributions.transforms.PositiveDefiniteTransform(cache_size=0)

Transform from unconstrained matrices to positive-definite matrices.

class torch.distributions.transforms.PowerTransform(exponent, cache_size=0)

Transform via the mapping $y = x^{\text{exponent}}$.

class torch.distributions.transforms.ReshapeTransform(in_shape, out_shape, cache_size=0)

Unit Jacobian transform to reshape the rightmost part of a tensor.

Note that in_shape and out_shape must have the same number of elements, just as for torch.Tensor.reshape().

Parameters

• in_shape (torch.Size) – The input event shape.

• out_shape (torch.Size) – The output event shape.

class torch.distributions.transforms.SigmoidTransform(cache_size=0)

Transform via the mapping $y = \frac{1}{1 + \exp(-x)}$ and $x = \text{logit}(y)$.

class torch.distributions.transforms.SoftplusTransform(cache_size=0)

Transform via the mapping $\text{Softplus}(x) = \log(1 + \exp(x))$. The implementation reverts to the linear function when $x > 20$.

class torch.distributions.transforms.TanhTransform(cache_size=0)

Transform via the mapping $y = \tanh(x)$.

It is equivalent to ` ComposeTransform([AffineTransform(0., 2.), SigmoidTransform(), AffineTransform(-1., 2.)]) ` However this might not be numerically stable, thus it is recommended to use TanhTransform instead.

Note that one should use cache_size=1 when it comes to NaN/Inf values.

class torch.distributions.transforms.SoftmaxTransform(cache_size=0)

Transform from unconstrained space to the simplex via $y = \exp(x)$ then normalizing.

This is not bijective and cannot be used for HMC. However this acts mostly coordinate-wise (except for the final normalization), and thus is appropriate for coordinate-wise optimization algorithms.

class torch.distributions.transforms.StackTransform(tseq, dim=0, cache_size=0)

Transform functor that applies a sequence of transforms tseq component-wise to each submatrix at dim in a way compatible with torch.stack().

Example:

x = torch.stack([torch.range(1, 10), torch.range(1, 10)], dim=1)
t = StackTransform([ExpTransform(), identity_transform], dim=1)
y = t(x)

class torch.distributions.transforms.StickBreakingTransform(cache_size=0)

Transform from unconstrained space to the simplex of one additional dimension via a stick-breaking process.

This transform arises as an iterated sigmoid transform in a stick-breaking construction of the Dirichlet distribution: the first logit is transformed via sigmoid to the first probability and the probability of everything else, and then the process recurses.

This is bijective and appropriate for use in HMC; however it mixes coordinates together and is less appropriate for optimization.

class torch.distributions.transforms.Transform(cache_size=0)

Abstract class for invertable transformations with computable log det jacobians. They are primarily used in torch.distributions.TransformedDistribution.

Caching is useful for transforms whose inverses are either expensive or numerically unstable. Note that care must be taken with memoized values since the autograd graph may be reversed. For example while the following works with or without caching:

y = t(x)
t.log_abs_det_jacobian(x, y).backward()  # x will receive gradients.

However the following will error when caching due to dependency reversal:

y = t(x)
z = t.inv(y)
grad(z.sum(), [y])  # error because z is x

Derived classes should implement one or both of _call() or _inverse(). Derived classes that set bijective=True should also implement log_abs_det_jacobian().

Parameters

cache_size (int) – Size of cache. If zero, no caching is done. If one, the latest single value is cached. Only 0 and 1 are supported.

Variables

• domain (Constraint) – The constraint representing valid inputs to this transform.

• codomain (Constraint) – The constraint representing valid outputs to this transform which are inputs to the inverse transform.

• bijective (bool) – Whether this transform is bijective. A transform t is bijective iff t.inv(t(x)) == x and t(t.inv(y)) == y for every x in the domain and y in the codomain. Transforms that are not bijective should at least maintain the weaker pseudoinverse properties t(t.inv(t(x)) == t(x) and t.inv(t(t.inv(y))) == t.inv(y).

• sign (int or Tensor) – For bijective univariate transforms, this should be +1 or -1 depending on whether transform is monotone increasing or decreasing.

property inv

Returns the inverse Transform of this transform. This should satisfy t.inv.inv is t.

property sign

Returns the sign of the determinant of the Jacobian, if applicable. In general this only makes sense for bijective transforms.

log_abs_det_jacobian(x, y)

Computes the log det jacobian log |dy/dx| given input and output.

forward_shape(shape)

Infers the shape of the forward computation, given the input shape. Defaults to preserving shape.

inverse_shape(shape)

Infers the shapes of the inverse computation, given the output shape. Defaults to preserving shape.

Constraints

The following constraints are implemented:

• constraints.boolean

• constraints.cat

• constraints.corr_cholesky

• constraints.dependent

• constraints.greater_than(lower_bound)

• constraints.greater_than_eq(lower_bound)

• constraints.independent(constraint, reinterpreted_batch_ndims)

• constraints.integer_interval(lower_bound, upper_bound)

• constraints.interval(lower_bound, upper_bound)

• constraints.less_than(upper_bound)

• constraints.lower_cholesky

• constraints.lower_triangular

• constraints.multinomial

• constraints.nonnegative

• constraints.nonnegative_integer

• constraints.one_hot

• constraints.positive_integer

• constraints.positive

• constraints.positive_semidefinite

• constraints.positive_definite

• constraints.real_vector

• constraints.real

• constraints.simplex

• constraints.symmetric

• constraints.stack

• constraints.square

• constraints.symmetric

• constraints.unit_interval

class torch.distributions.constraints.Constraint

Abstract base class for constraints.

A constraint object represents a region over which a variable is valid, e.g. within which a variable can be optimized.

Variables

• is_discrete (bool) – Whether constrained space is discrete. Defaults to False.

• event_dim (int) – Number of rightmost dimensions that together define an event. The check() method will remove this many dimensions when computing validity.

check(value)

Returns a byte tensor of sample_shape + batch_shape indicating whether each event in value satisfies this constraint.

torch.distributions.constraints.cat

alias of _Cat

torch.distributions.constraints.dependent_property

alias of _DependentProperty

torch.distributions.constraints.greater_than

alias of _GreaterThan

torch.distributions.constraints.greater_than_eq

alias of _GreaterThanEq

torch.distributions.constraints.independent

alias of _IndependentConstraint

torch.distributions.constraints.integer_interval

alias of _IntegerInterval

torch.distributions.constraints.interval

alias of _Interval

torch.distributions.constraints.half_open_interval

alias of _HalfOpenInterval

torch.distributions.constraints.less_than

alias of _LessThan

torch.distributions.constraints.multinomial

alias of _Multinomial

torch.distributions.constraints.stack

alias of _Stack

Constraint Registry

PyTorch provides two global ConstraintRegistry objects that link Constraint objects to Transform objects. These objects both input constraints and return transforms, but they have different guarantees on bijectivity.

1. biject_to(constraint) looks up a bijective Transform from constraints.real to the given constraint. The returned transform is guaranteed to have .bijective = True and should implement .log_abs_det_jacobian().

2. transform_to(constraint) looks up a not-necessarily bijective Transform from constraints.real to the given constraint. The returned transform is not guaranteed to implement .log_abs_det_jacobian().

The transform_to() registry is useful for performing unconstrained optimization on constrained parameters of probability distributions, which are indicated by each distribution’s .arg_constraints dict. These transforms often overparameterize a space in order to avoid rotation; they are thus more suitable for coordinate-wise optimization algorithms like Adam:

loc = torch.zeros(100, requires_grad=True)
unconstrained = torch.zeros(100, requires_grad=True)
scale = transform_to(Normal.arg_constraints['scale'])(unconstrained)
loss = -Normal(loc, scale).log_prob(data).sum()

The biject_to() registry is useful for Hamiltonian Monte Carlo, where samples from a probability distribution with constrained .support are propagated in an unconstrained space, and algorithms are typically rotation invariant.:

dist = Exponential(rate)
unconstrained = torch.zeros(100, requires_grad=True)
sample = biject_to(dist.support)(unconstrained)
potential_energy = -dist.log_prob(sample).sum()

Note

An example where transform_to and biject_to differ is constraints.simplex: transform_to(constraints.simplex) returns a SoftmaxTransform that simply exponentiates and normalizes its inputs; this is a cheap and mostly coordinate-wise operation appropriate for algorithms like SVI. In contrast, biject_to(constraints.simplex) returns a StickBreakingTransform that bijects its input down to a one-fewer-dimensional space; this a more expensive less numerically stable transform but is needed for algorithms like HMC.

The biject_to and transform_to objects can be extended by user-defined constraints and transforms using their .register() method either as a function on singleton constraints:

transform_to.register(my_constraint, my_transform)

or as a decorator on parameterized constraints:

@transform_to.register(MyConstraintClass)
def my_factory(constraint):
    assert isinstance(constraint, MyConstraintClass)
    return MyTransform(constraint.param1, constraint.param2)

You can create your own registry by creating a new ConstraintRegistry object.

class torch.distributions.constraint_registry.ConstraintRegistry

Registry to link constraints to transforms.

register(constraint, factory=None)

Registers a Constraint subclass in this registry. Usage:

@my_registry.register(MyConstraintClass)
def construct_transform(constraint):
    assert isinstance(constraint, MyConstraint)
    return MyTransform(constraint.arg_constraints)

Parameters

• constraint (subclass of Constraint) – A subclass of Constraint, or a singleton object of the desired class.

• factory (Callable) – A callable that inputs a constraint object and returns a Transform object.