tfd_dirichlet {tfprobability}

Dirichlet distribution

Description

The Dirichlet distribution is defined over the (k-1)-simplex using a positive, length-k vector concentration (k > 1). The Dirichlet is identically the Beta distribution when k = 2.

Usage

tfd_dirichlet(
  concentration,
  validate_args = FALSE,
  allow_nan_stats = TRUE,
  name = "Dirichlet"
)

Arguments

concentration

Positive floating-point Tensor indicating mean number of class occurrences; aka "alpha". Implies self$dtype, and self$batch_shape, self$event_shape, i.e., if ⁠concentration$shape = [N1, N2, ..., Nm, k]⁠ then ⁠batch_shape = [N1, N2, ..., Nm]⁠ and ⁠event_shape = [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.

Details

Mathematical Details

The Dirichlet is a distribution over the open (k-1)-simplex, i.e.,

S^{k-1} = { (x_0, ..., x_{k-1}) in R^k : sum_j x_j = 1 and all_j x_j > 0 }.

The probability density function (pdf) is,

pdf(x; alpha) = prod_j x_j**(alpha_j - 1) / Z
Z = prod_j Gamma(alpha_j) / Gamma(sum_j alpha_j)

where:

• ⁠x in S^{k-1}⁠, i.e., the (k-1)-simplex,

• ⁠concentration = alpha = [alpha_0, ..., alpha_{k-1}]⁠, alpha_j > 0,

• Z is the normalization constant aka the multivariate beta function, and,

• Gamma is the gamma function.

The concentration represents mean total counts of class occurrence, i.e.,

concentration = alpha = mean * total_concentration

where mean in S^{k-1} and total_concentration is a positive real number representing a mean total count. Distribution parameters are automatically broadcast in all functions; see examples for details. Warning: Some components of the samples can be zero due to finite precision. This happens more often when some of the concentrations are very small. Make sure to round the samples to np$finfo(dtype)$tiny before computing the density. Samples of this distribution are reparameterized (pathwise differentiable). The derivatives are computed using the approach described in the paper Michael Figurnov, Shakir Mohamed, Andriy Mnih. Implicit Reparameterization Gradients, 2018

Value

a distribution instance.

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_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_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()