tfd_gamma_gamma {tfprobability}

Gamma-Gamma distribution

Description

Gamma-Gamma is a compound distribution defined over positive real numbers using parameters concentration, mixing_concentration and mixing_rate.

Usage

tfd_gamma_gamma(
  concentration,
  mixing_concentration,
  mixing_rate,
  validate_args = FALSE,
  allow_nan_stats = TRUE,
  name = "GammaGamma"
)

Arguments

concentration

Floating point tensor, the concentration params of the distribution(s). Must contain only positive values.

mixing_concentration

Floating point tensor, the concentration params of the mixing Gamma distribution(s). Must contain only positive values.

mixing_rate

Floating point tensor, the rate params of the mixing Gamma distribution(s). Must contain only positive values.

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

This distribution is also referred to as the beta of the second kind (B2), and can be useful for transaction value modeling, as in Fader and Hardi, 2013.

Mathematical Details

It is derived from the following Gamma-Gamma hierarchical model by integrating out the random variable beta.

beta ~ Gamma(alpha0, beta0)
X | beta ~ Gamma(alpha, beta)

where

• concentration = alpha

• mixing_concentration = alpha0

• mixing_rate = beta0

The probability density function (pdf) is

x**(alpha - 1)
pdf(x; alpha, alpha0, beta0) =  Z * (x + beta0)**(alpha + alpha0)

where the normalizing constant Z = Beta(alpha, alpha0) * beta0**(-alpha0). Samples of this distribution are reparameterized as samples of the Gamma distribution are reparameterized using the technique described in (Figurnov et al., 2018).

@section References:

• Peter S. Fader, Bruce G. S. Hardi. The Gamma-Gamma Model of Monetary Value. Technical Report, 2013.

• Michael Figurnov, Shakir Mohamed, Andriy Mnih. Implicit Reparameterization Gradients. arXiv preprint arXiv:1805.08498, 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_dirichlet(), tfd_empirical(), tfd_exp_gamma(), tfd_exp_inverse_gamma(), tfd_exponential(), 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()