The Gamma distribution is defined over positive real numbers using
parameters concentration (aka "alpha") and rate (aka "beta").
tfd_gamma( concentration, rate, validate_args = FALSE, allow_nan_stats = TRUE, name = "Gamma" )
| 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. |
| 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. |
a distribution instance.
Mathematical Details
The probability density function (pdf) is,
pdf(x; alpha, beta, x > 0) = x**(alpha - 1) exp(-x beta) / Z Z = Gamma(alpha) beta**(-alpha)
where
concentration = alpha, alpha > 0,
rate = beta, beta > 0,
Z is the normalizing constant, and,
Gamma is the gamma function.
The cumulative density function (cdf) is,
cdf(x; alpha, beta, x > 0) = GammaInc(alpha, beta x) / Gamma(alpha)
where GammaInc is the lower incomplete Gamma function.
The parameters can be intuited via their relationship to mean and stddev,
concentration = alpha = (mean / stddev)**2 rate = beta = mean / stddev**2 = concentration / mean
Distribution parameters are automatically broadcast in all functions; see examples for details.
Warning: The samples of this distribution are always non-negative. However,
the samples that are smaller than np$finfo(dtype)$tiny are rounded
to this value, so it appears more often than it should.
This should only be noticeable when the concentration is very small, or the
rate is very large. See note in tf$random_gamma docstring.
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
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_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()