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 ) f(x) f(x), the pathwise derivative requires the derivative f′ ( x ) f'(x) 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:
Δ θ = α r ∂ log p ( a ∣ πθ ( s ) ) ∂ θ \Delta\theta = \alpha r \frac{\partial\log p(a|\pi^\theta(s))}{\partial\theta} Δθ=αr∂θ∂logp(a∣πθ(s))
where θ \theta θ are the parameters, α \alpha α is the learning rate, r r r is the reward and p ( a ∣ πθ ( s ) ) p(a|\pi^\theta(s)) p(a∣πθ(s)) is the probability of taking action a a a in state s s s given policy πθ \pi^\theta πθ.
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¶
Bases: object
Distribution is the abstract base class for probability distributions.
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.
Returns the shape over which parameters are batched.
Returns the cumulative density/mass function evaluated at value.
Returns entropy of distribution, batched over batch_shape.
Tensor of shape batch_shape.
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()).
Returns the shape of a single sample (without batching).
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.
batch_shape (torch.Size) – the desired expanded size.
_instance – new instance provided by subclasses that need to override .expand.
New distribution instance with batch dimensions expanded to batch_size.
Returns the inverse cumulative density/mass function evaluated at value.
Returns the log of the probability density/mass function evaluated at value.
Returns the mean of the distribution.
Returns the mode of the distribution.
Returns perplexity of distribution, batched over batch_shape.
Tensor of shape batch_shape.
Generates a sample_shape shaped reparameterized sample or sample_shape shaped batch of reparameterized samples if the distribution parameters are batched.
Generates a sample_shape shaped sample or sample_shape shaped batch of samples if the distribution parameters are batched.
Generates n samples or n batches of samples if the distribution parameters are batched.
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.
value (bool) – Whether to enable validation.
Returns the standard deviation of the distribution.
Returns a Constraint object representing this distribution’s support.
Returns the variance of the distribution.
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 ; θ ) = exp ( ⟨ t ( x ) , θ ⟩ − F ( θ ) + k ( x ) ) p_{F}(x; \theta) = \exp(\langle t(x), \theta\rangle - F(\theta) + k(x)) pF(x;θ)=exp(⟨t(x),θ⟩−F(θ)+k(x))
where θ \theta θ denotes the natural parameters, t ( x ) t(x) t(x) denotes the sufficient statistic, F ( θ ) F(\theta) F(θ) is the log normalizer function for a given family and k ( x ) k(x) 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).
Method to compute the entropy using Bregman divergence of the log normalizer.
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.])
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])
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.]])
_DependentProperty
Bases: Distribution
Creates a categorical distribution parameterized by either probs or logits (but not both).
Samples are integers from { 0 , … , K − 1 } \{0, \ldots, K-1\} {0,…,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)
_DependentProperty
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])
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])
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])
[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
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])
concentration (Tensor) – concentration parameter of the distribution (often referred to as alpha)
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])
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])
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 , . . . P(X=k) = (1-p)^{k} p, k = 0, 1, ... P(X=k)=(1−p)kp,k=0,1,...
Note
torch.distributions.geometric.Geometric() ( k + 1 ) (k+1) (k+1)-th trial is the first success hence draws samples in { 0 , 1 , … } \{0, 1, \ldots\} {0,1,…}, whereas torch.Tensor.geometric_() k-th trial is the first success hence draws samples in { 1 , 2 , … } \{1, 2, \ldots\} {1,2,…}.
Example:
>>> m = Geometric(torch.tensor([0.3])) >>> m.sample() # underlying Bernoulli has 30% chance 1; 70% chance 0 tensor([ 2.])
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])]
base_distribution (torch.distributions.distribution.Distribution) – a base distribution
reinterpreted_batch_ndims (int) – the number of batch dims to reinterpret as event dims
_DependentProperty
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])
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])
Bases: Distribution
LKJ distribution for lower Cholesky factor of correlation matrices. The distribution is controlled by concentration parameter η \eta η to make the probability of the correlation matrix M M M generated from a Cholesky factor proportional to det ( M ) η − 1 \det(M)^{\eta - 1} det(M)η−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]])
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
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], [0.0]]), torch.ones(2) ... ) >>> m.sample() # normally distributed with mean=`[0,0]`, cov_factor=`[[1],[0]]`, cov_diag=`[1,1]` tensor([-0.2102, -0.5429])
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
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)
mixture_distribution (Categorical) – 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 (Distribution) – torch.distributions.Distribution-like instance. Right-most batch dimension indexes component.
_DependentProperty
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])
_DependentProperty
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 Σ \mathbf{\Sigma} Σ or a positive definite precision matrix Σ − 1 \mathbf{\Sigma}^{-1} Σ−1 or a lower-triangular matrix L \mathbf{L} L with positive-valued diagonal entries, such that Σ = L L⊤ \mathbf{\Sigma} = \mathbf{L}\mathbf{L}^\top Σ=LL⊤. 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])
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.
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.])
Bases: ExponentialFamily
Creates a Poisson distribution parameterized by rate, the rate parameter.
Samples are nonnegative integers, with a pmf given by
r a t e k e − r a t e k ! \mathrm{rate}^k \frac{e^{-\mathrm{rate}}}{k!} ratekk!e−rate
Example:
>>> m = Poisson(torch.tensor([4])) >>> m.sample() tensor([ 3.])
rate (Number, Tensor) – the rate parameter
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])
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.
[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)
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])
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])
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
Computes the cumulative distribution function by inverting the transform(s) and computing the score of the base distribution.
Computes the inverse cumulative distribution function using transform(s) and computing the score of the base distribution.
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.
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.
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.
_DependentProperty
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.
>>> m = VonMises(torch.tensor([1.0]), torch.tensor([1.0])) >>> m.sample() # von Mises distributed with loc=1 and concentration=1 tensor([1.9777])
loc (torch.Tensor) – an angle in radians.
concentration (torch.Tensor) – concentration parameter
The provided mean is the circular one.
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).
The provided variance is the circular one.
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])
Bases: ExponentialFamily
Creates a Wishart distribution parameterized by a symmetric positive definite matrix Σ \Sigma Σ, or its Cholesky decomposition Σ = L L⊤ \mathbf{\Sigma} = \mathbf{L}\mathbf{L}^\top Σ=LL⊤
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
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
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.
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.
Compute Kullback-Leibler divergence K L ( p ∥ q ) KL(p \| q) KL(p∥q) between two distributions.
K L ( p ∥ q ) = ∫ p ( x ) log p ( x ) q ( x ) d x KL(p \| q) = \int p(x) \log\frac {p(x)} {q(x)} \,dx KL(p∥q)=∫p(x)logq(x)p(x)dx
p (Distribution) – A Distribution object.
q (Distribution) – A Distribution object.
A batch of KL divergences of shape batch_shape.
NotImplementedError – If the distribution types have not been registered via register_kl().
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
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.
Transform via the mapping y = ∣ x ∣ y = |x| y=∣x∣.
Transform via the pointwise affine mapping y = loc + scale × x y = \text{loc} + \text{scale} \times x y=loc+scale×x.
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)
Composes multiple transforms in a chain. The transforms being composed are responsible for caching.
Transforms an uncontrained real vector x x x with length D ∗ ( D − 1 ) / 2 D*(D-1)/2 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:
First we convert x into a lower triangular matrix in row order.
For each row X i X_i Xi of the lower triangular part, we apply a signed version of class
StickBreakingTransformto transform X i X_i Xi into a unit Euclidean length vector using the following steps: - Scales into the interval ( − 1 , 1 ) (-1, 1) (−1,1) domain: r i = tanh ( X i ) r_i = \tanh(X_i) ri=tanh(Xi). - Transforms into an unsigned domain: z i = r i 2 z_i = r_i^2 zi=ri2. - Applies s i = S t i c k B r e a k i n g T r a n s f o r m ( z i ) s_i = StickBreakingTransform(z_i) si=StickBreakingTransform(zi). - Transforms back into signed domain: y i = s i g n ( r i ) ∗ s i y_i = sign(r_i) * \sqrt{s_i} yi=sign(ri)∗si .
Transform via the cumulative distribution function of a probability distribution.
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])
Transform via the mapping y = exp ( x ) y = \exp(x) y=exp(x).
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().
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.
Transform from unconstrained matrices to positive-definite matrices.
Transform via the mapping y = xexponent y = x^{\text{exponent}} y=xexponent.
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().
in_shape (torch.Size) – The input event shape.
out_shape (torch.Size) – The output event shape.
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. (Default 0.)
Transform via the mapping y = 1 1 + exp ( − x ) y = \frac{1}{1 + \exp(-x)} y=1+exp(−x)1 and x = logit ( y ) x = \text{logit}(y) x=logit(y).
Transform via the mapping Softplus ( x ) = log ( 1 + exp ( x ) ) \text{Softplus}(x) = \log(1 + \exp(x)) Softplus(x)=log(1+exp(x)). The implementation reverts to the linear function when x > 20 x > 20 x>20.
Transform via the mapping y = tanh ( x ) y = \tanh(x) y=tanh(x).
It is equivalent to
ComposeTransform(
[
AffineTransform(0.0, 2.0),
SigmoidTransform(),
AffineTransform(-1.0, 2.0),
]
)
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.
Transform from unconstrained space to the simplex via y = exp ( x ) y = \exp(x) 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.
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)
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.
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().
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.
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.
Returns the inverse Transform of this transform. This should satisfy t.inv.inv is t.
Returns the sign of the determinant of the Jacobian, if applicable. In general this only makes sense for bijective transforms.
Computes the log det jacobian log |dy/dx| given input and output.
Infers the shape of the forward computation, given the input shape. Defaults to preserving shape.
Infers the shapes of the inverse computation, given the output shape. Defaults to preserving shape.
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.
Returns a byte tensor of sample_shape + batch_shape indicating whether each event in value satisfies this constraint.
alias of _Cat
alias of _DependentProperty
alias of _GreaterThan
alias of _GreaterThanEq
alias of _IndependentConstraint
alias of _IntegerInterval
alias of _Interval
alias of _HalfOpenInterval
Checks if constraint is a _Dependent object.
constraint – A Constraint object.
True if constraint can be refined to the type _Dependent, False otherwise.
bool
Examples
>>> import torch >>> from torch.distributions import Bernoulli >>> from torch.distributions.constraints import is_dependent
>>> dist = Bernoulli(probs=torch.tensor([0.6], requires_grad=True)) >>> constraint1 = dist.arg_constraints["probs"] >>> constraint2 = dist.arg_constraints["logits"]
>>> for constraint in [constraint1, constraint2]: >>> if is_dependent(constraint): >>> continue
alias of _LessThan
alias of _Multinomial
alias of _Stack
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.
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().
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.
Registry to link constraints to transforms.
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)
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.
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