The vectorization of the Exponential distribution on R^k

The vector exponential distribution is defined over a subset of R^k, and parameterized by a (batch of) length-k loc vector and a (batch of) k x k scale matrix: covariance = scale @ scale.T, where @ denotes matrix-multiplication.

tfd_vector_exponential_linear_operator(
  loc = NULL,
  scale = NULL,
  validate_args = FALSE,
  allow_nan_stats = TRUE,
  name = "VectorExponentialLinearOperator"
)

Arguments

loc	Floating point tensor; the means of the distribution(s).
scale	Instance of LinearOperator with same dtype as loc and shape [B1, ..., Bb, k, k].
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.
allow_nan_stats	Logical, default TRUE. When TRUE, statistics (e.g., mean, mode, variance) use the value NaN to indicate the result is undefined. When FALSE, an exception is raised if one or more of the statistic's batch members are undefined.
name	name prefixed to Ops created by this class.

Value

a distribution instance.

Details

Mathematical Details The probability density function (pdf) is

pdf(y; loc, scale) = exp(-||x||_1) / Z,  for y in S(loc, scale),
x = inv(scale) @ (y - loc),
Z = |det(scale)|,

where:

• loc is a vector in R^k,

• scale is a linear operator in R^{k x k}, cov = scale @ scale.T,

• S = {loc + scale @ x : x in R^k, x_1 > 0, ..., x_k > 0}, is an image of the positive half-space,

• ||x||_1 denotes the l1 norm of x, sum_i |x_i|,

• Z denotes the normalization constant.

The VectorExponential distribution is a member of the location-scale family, i.e., it can be constructed as,

X = (X_1, ..., X_k), each X_i ~ Exponential(rate=1)
Y = (Y_1, ...,Y_k) = scale @ X + loc

About VectorExponential and Vector distributions in TensorFlow.

The VectorExponential is a non-standard distribution that has useful properties. The marginals Y_1, ..., Y_k are not Exponential random variables, due to the fact that the sum of Exponential random variables is not Exponential. Instead, Y is a vector whose components are linear combinations of Exponential random variables. Thus, Y lives in the vector space generated by vectors of Exponential distributions. This allows the user to decide the mean and covariance (by setting loc and scale), while preserving some properties of the Exponential distribution. In particular, the tails of Y_i will be (up to polynomial factors) exponentially decaying. To see this last statement, note that the pdf of Y_i is the convolution of the pdf of k independent Exponential random variables. One can then show by induction that distributions with exponential (up to polynomial factors) tails are closed under convolution.

The batch_shape is the broadcast shape between loc and scale arguments. The event_shape is given by last dimension of the matrix implied by scale. The last dimension of loc (if provided) must broadcast with this. Recall that covariance = 2 * scale @ scale.T. Additional leading dimensions (if any) will index batches.

#' @param loc Floating-point Tensor. If this is set to NULL, loc is implicitly 0. When specified, may have shape [B1, ..., Bb, k] where b >= 0 and k is the event size.

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_transformed_distribution(), tfd_triangular(), tfd_truncated_cauchy(), tfd_truncated_normal(), tfd_uniform(), tfd_variational_gaussian_process(), tfd_vector_diffeomixture(), tfd_vector_exponential_diag(), 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()