A Transformed Distribution

A TransformedDistribution models p(y) given a base distribution p(x), and a deterministic, invertible, differentiable transform,Y = g(X). The transform is typically an instance of the Bijector class and the base distribution is typically an instance of the Distribution class.

tfd_transformed_distribution(
  distribution,
  bijector,
  batch_shape = NULL,
  event_shape = NULL,
  kwargs_split_fn = NULL,
  validate_args = FALSE,
  parameters = NULL,
  name = NULL
)

Arguments

distribution	The base distribution instance to transform. Typically an instance of Distribution.
bijector	The object responsible for calculating the transformation. Typically an instance of Bijector.
batch_shape	integer vector Tensor which overrides distribution batch_shape; valid only if distribution.is_scalar_batch().
event_shape	integer vector Tensor which overrides distribution event_shape; valid only if distribution.is_scalar_event().
kwargs_split_fn	Python callable which takes a kwargs dict and returns a tuple of kwargs dicts for each of the distribution and bijector parameters respectively. Default value: _default_kwargs_split_fn (i.e., lambda kwargs: (kwargs.get('distribution_kwargs', {}), kwargs.get('bijector_kwargs', {})))
validate_args	Logical, default FALSE. When TRUE distribution parameters are checked for validity despite possibly degrading runtime performance. When FALSE invalid inputs may silently render incorrect outputs. Default value: FALSE.
parameters	Locals dict captured by subclass constructor, to be used for copy/slice re-instantiation operations.
name	The name for ops managed by the distribution. Default value: bijector.name + distribution.name.

Value

a distribution instance.

Details

A Bijector is expected to implement the following functions:

• forward,

• inverse,

• inverse_log_det_jacobian.

The semantics of these functions are outlined in the Bijector documentation.

We now describe how a TransformedDistribution alters the input/outputs of a Distribution associated with a random variable (rv) X. Write cdf(Y=y) for an absolutely continuous cumulative distribution function of random variable Y; write the probability density function pdf(Y=y) := d^k / (dy_1,...,dy_k) cdf(Y=y) for its derivative wrt to Y evaluated at y. Assume that Y = g(X) where g is a deterministic diffeomorphism, i.e., a non-random, continuous, differentiable, and invertible function. Write the inverse of g as X = g^{-1}(Y) and (J o g)(x) for the Jacobian of g evaluated at x.

A TransformedDistribution implements the following operations:

• sample Mathematically: Y = g(X) Programmatically: bijector.forward(distribution.sample(...))

• log_prob Mathematically: (log o pdf)(Y=y) = (log o pdf o g^{-1})(y) + (log o abs o det o J o g^{-1})(y) Programmatically: (distribution.log_prob(bijector.inverse(y)) + bijector.inverse_log_det_jacobian(y))

• log_cdf Mathematically: (log o cdf)(Y=y) = (log o cdf o g^{-1})(y) Programmatically: distribution.log_cdf(bijector.inverse(x))

• and similarly for: cdf, prob, log_survival_function, survival_function.

See also

For usage examples see e.g. tfd_sample(), tfd_log_prob(), tfd_mean().

Other distributions: tfd_autoregressive(), tfd_batch_reshape(), tfd_bates(), tfd_bernoulli(), tfd_beta_binomial(), tfd_beta(), tfd_binomial(), tfd_categorical(), tfd_cauchy(), tfd_chi2(), tfd_chi(), tfd_cholesky_lkj(), tfd_continuous_bernoulli(), tfd_deterministic(), tfd_dirichlet_multinomial(), tfd_dirichlet(), tfd_empirical(), tfd_exp_gamma(), tfd_exp_inverse_gamma(), tfd_exponential(), tfd_gamma_gamma(), tfd_gamma(), tfd_gaussian_process_regression_model(), tfd_gaussian_process(), tfd_generalized_normal(), tfd_geometric(), tfd_gumbel(), tfd_half_cauchy(), tfd_half_normal(), tfd_hidden_markov_model(), tfd_horseshoe(), tfd_independent(), tfd_inverse_gamma(), tfd_inverse_gaussian(), tfd_johnson_s_u(), tfd_joint_distribution_named_auto_batched(), tfd_joint_distribution_named(), tfd_joint_distribution_sequential_auto_batched(), tfd_joint_distribution_sequential(), tfd_kumaraswamy(), tfd_laplace(), tfd_linear_gaussian_state_space_model(), tfd_lkj(), tfd_log_logistic(), tfd_log_normal(), tfd_logistic(), tfd_mixture_same_family(), tfd_mixture(), tfd_multinomial(), tfd_multivariate_normal_diag_plus_low_rank(), tfd_multivariate_normal_diag(), tfd_multivariate_normal_full_covariance(), tfd_multivariate_normal_linear_operator(), tfd_multivariate_normal_tri_l(), tfd_multivariate_student_t_linear_operator(), tfd_negative_binomial(), tfd_normal(), tfd_one_hot_categorical(), tfd_pareto(), tfd_pixel_cnn(), tfd_poisson_log_normal_quadrature_compound(), tfd_poisson(), tfd_power_spherical(), tfd_probit_bernoulli(), tfd_quantized(), tfd_relaxed_bernoulli(), tfd_relaxed_one_hot_categorical(), tfd_sample_distribution(), tfd_sinh_arcsinh(), tfd_skellam(), tfd_spherical_uniform(), tfd_student_t_process(), tfd_student_t(), tfd_triangular(), tfd_truncated_cauchy(), tfd_truncated_normal(), tfd_uniform(), tfd_variational_gaussian_process(), tfd_vector_diffeomixture(), tfd_vector_exponential_diag(), tfd_vector_exponential_linear_operator(), tfd_vector_laplace_diag(), tfd_vector_laplace_linear_operator(), tfd_vector_sinh_arcsinh_diag(), tfd_von_mises_fisher(), tfd_von_mises(), tfd_weibull(), tfd_wishart_linear_operator(), tfd_wishart_tri_l(), tfd_wishart(), tfd_zipf()