Beta-Binomial compound distribution

The Beta-Binomial distribution is parameterized by (a batch of) total_count parameters, the number of trials per draw from Binomial distributions where the probabilities of success per trial are drawn from underlying Beta distributions; the Beta distributions are parameterized by concentration1 (aka 'alpha') and concentration0 (aka 'beta'). Mathematically, it is (equivalent to) a special case of the Dirichlet-Multinomial over two classes, although the computational representation is slightly different: while the Beta-Binomial is a distribution over the number of successes in total_count trials, the two-class Dirichlet-Multinomial is a distribution over the number of successes and failures.

tfd_beta_binomial(
  total_count,
  concentration1,
  concentration0,
  validate_args = FALSE,
  allow_nan_stats = TRUE,
  name = "BetaBinomial"
)

Arguments

total_count	Non-negative integer-valued tensor, whose dtype is the same as concentration1 and concentration0. The shape is broadcastable to [N1,..., Nm] with m >= 0. When total_count is broadcast with concentration1 and concentration0, it defines the distribution as a batch of N1 x ... x Nm different Beta-Binomial distributions. Its components should be equal to integer values.
concentration1	Positive floating-point Tensor indicating mean number of successes. Specifically, the expected number of successes is total_count * concentration1 / (concentration1 + concentration0).
concentration0	Positive floating-point Tensor indicating mean number of failures; see description of concentration1 for details.
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 Beta-Binomial is a distribution over the number of successes in total_count independent Binomial trials, with each trial having the same probability of success, the underlying probability being unknown but drawn from a Beta distribution with known parameters. The probability mass function (pmf) is,

pmf(k; n, a, b) = Beta(k + a, n - k + b) / Z
Z = (k! (n - k)! / n!) * Beta(a, b)

where:

• concentration1 = a > 0,

• concentration0 = b > 0,

• total_count = n, n a positive integer,

• n! is n factorial,

• Beta(x, y) = Gamma(x) Gamma(y) / Gamma(x + y) is the beta function, and

• Gamma is the gamma function.

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

1. Choose success probabilities: probs ~ Beta(concentration1, concentration0)

2. Draw integers representing the number of successes: counts ~ Binomial(total_count, probs) Distribution parameters are automatically broadcast in all functions; see examples for details.

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(), 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_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()