The vectorization of the Laplace distribution on R^k

Source: R/distributions.R

The vector laplace distribution is defined over R^k, and parameterized by a (batch of) length-k loc vector (the means) and a (batch of) k x k scale matrix: covariance = 2 * scale @ scale.T, where @ denotes matrix-multiplication.

tfd_vector_laplace_diag(
  loc = NULL,
  scale_diag = NULL,
  scale_identity_multiplier = NULL,
  validate_args = FALSE,
  allow_nan_stats = TRUE,
  name = "VectorLaplaceDiag"
)

Arguments

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.
scale_diag

Non-zero, floating-point Tensor representing a diagonal matrix added to scale. May have shape [B1, ..., Bb, k], b >= 0, and characterizes b-batches of k x k diagonal matrices added to scale. When both scale_identity_multiplier and scale_diag are NULL then scale is the Identity.
scale_identity_multiplier

Non-zero, floating-point Tensor representing a scaled-identity-matrix added to scale. May have shape [B1, ..., Bb], b >= 0, and characterizes b-batches of scaled k x k identity matrices added to scale. When both scale_identity_multiplier and scale_diag are NULL then scale is the Identity.
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(x; loc, scale) = exp(-||y||_1) / Z
y = inv(scale) @ (x - loc)
Z = 2**k |det(scale)|

where:

  • loc is a vector in R^k,

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

  • Z denotes the normalization constant, and,

  • ||y||_1 denotes the l1 norm of y, `sum_i |y_i|.

A (non-batch) scale matrix is:

scale = diag(scale_diag + scale_identity_multiplier * ones(k))

where:

  • scale_diag.shape = [k], and,

  • scale_identity_multiplier.shape = []. Additional leading dimensions (if any) will index batches. If both scale_diag and scale_identity_multiplier are NULL, then scale is the Identity matrix.

About VectorLaplace and Vector distributions in TensorFlow

The VectorLaplace is a non-standard distribution that has useful properties. The marginals Y_1, ..., Y_k are not Laplace random variables, due to the fact that the sum of Laplace random variables is not Laplace. Instead, Y is a vector whose components are linear combinations of Laplace random variables. Thus, Y lives in the vector space generated by vectors of Laplace distributions. This allows the user to decide the mean and covariance (by setting loc and scale), while preserving some properties of the Laplace 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 Laplace random variables. One can then show by induction that distributions with exponential (up to polynomial factors) tails are closed under convolution.

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_exponential_linear_operator(), 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()