tfd_dirichlet_multinomial {tfprobability}

Dirichlet-Multinomial compound distribution

Description

The Dirichlet-Multinomial distribution is parameterized by a (batch of) length-K concentration vectors (K > 1) and a total_count number of trials, i.e., the number of trials per draw from the DirichletMultinomial. It is defined over a (batch of) length-K vector counts such that tf$reduce_sum(counts, -1) = total_count. The Dirichlet-Multinomial is identically the Beta-Binomial distribution when K = 2.

Usage

tfd_dirichlet_multinomial(
  total_count,
  concentration,
  validate_args = FALSE,
  allow_nan_stats = TRUE,
  name = "DirichletMultinomial"
)

Arguments

total_count

Non-negative floating point tensor, whose dtype is the same as concentration. The shape is broadcastable to ⁠[N1,..., Nm]⁠ with m >= 0. Defines this as a batch of ⁠N1 x ... x Nm⁠ different Dirichlet multinomial distributions. Its components should be equal to integer values.

concentration

Positive floating point tensor, whose dtype is the same as n with shape broadcastable to ⁠[N1,..., Nm, K]⁠ m >= 0. Defines this as a batch of ⁠N1 x ... x Nm⁠ different K class Dirichlet multinomial distributions.

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-Multinomial is a distribution over K-class counts, i.e., a length-K vector of non-negative integer ⁠counts = n = [n_0, ..., n_{K-1}]⁠.

The probability mass function (pmf) is,

pmf(n; alpha, N) = Beta(alpha + n) / (prod_j n_j!) / Z
Z = Beta(alpha) / N!

where:

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

• total_count = N, N a positive integer,

• ⁠N!⁠ is N factorial, and,

• ⁠Beta(x) = prod_j Gamma(x_j) / Gamma(sum_j x_j)⁠ is the multivariate beta function, and,

• Gamma is the gamma function.

Dirichlet-Multinomial is a compound distribution, i.e., its samples are generated as follows.

1. Choose class probabilities: ⁠probs = [p_0,...,p_{K-1}] ~ Dir(concentration)⁠

2. Draw integers: ⁠counts = [n_0,...,n_{K-1}] ~ Multinomial(total_count, probs)⁠

The last concentration dimension parametrizes a single Dirichlet-Multinomial distribution. When calling distribution functions (e.g., dist$prob(counts)), concentration, total_count and counts are broadcast to the same shape. The last dimension of counts corresponds single Dirichlet-Multinomial distributions. Distribution parameters are automatically broadcast in all functions; see examples for details.

Pitfalls The number of classes, K, must not exceed:

• the largest integer representable by self$dtype, i.e., 2**(mantissa_bits+1) (IEE754),

• the maximum Tensor index, i.e., 2**31-1.

Note: This condition is validated only when validate_args = TRUE.

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(), 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()