ExpGamma distribution.

The ExpGamma distribution is defined over the real line using parameters concentration (aka "alpha") and rate (aka "beta"). This distribution is a transformation of the Gamma distribution such that X ~ ExpGamma(..) => exp(X) ~ Gamma(..).

tfd_exp_gamma(
  concentration,
  rate = NULL,
  log_rate = NULL,
  validate_args = FALSE,
  allow_nan_stats = TRUE,
  name = "ExpGamma"
)

Arguments

concentration	Floating point tensor, the concentration params of the distribution(s). Must contain only positive values.
rate	Floating point tensor, the inverse scale params of the distribution(s). Must contain only positive values. Mutually exclusive with log_rate.
log_rate	Floating point tensor, natural logarithm of the inverse scale params of the distribution(s). Mutually exclusive with rate.
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) can be derived from the change of variables rule (since the distribution is logically equivalent to tfb_log()(tfd_gamma(..))):

pdf(x; alpha, beta > 0) = exp(x)**(alpha - 1) exp(-exp(x) beta) / Z + x
Z = Gamma(alpha) beta**(-alpha)

where:

• concentration = alpha, alpha > 0,

• rate = beta, beta > 0,

• Z is the normalizing constant of the corresponding Gamma distribution, and

• Gamma is the gamma function.

The cumulative density function (cdf) is,

cdf(x; alpha, beta, x) = GammaInc(alpha, beta exp(x)) / Gamma(alpha)

where GammaInc is the lower incomplete Gamma function.

Distribution parameters are automatically broadcast in all functions. Samples of this distribution are reparameterized (pathwise differentiable). The derivatives are computed using the approach described in Figurnov et al., 2018.

References

• Michael Figurnov, Shakir Mohamed, Andriy Mnih. Implicit Reparameterization Gradients. arXiv preprint arXiv:1805.08498, 2018.

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_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()