The InverseGamma distribution is defined over positive real numbers using parameters concentration (aka "alpha") and scale (aka "beta").

  validate_args = FALSE,
  allow_nan_stats = TRUE,
  name = "InverseGamma"



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


Floating point tensor, the scale params of the distribution(s). Must contain only positive values. This parameter was called rate before release 0.8.


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.


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 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(-beta / x) / Z
Z = Gamma(alpha) beta**-alpha


  • concentration = alpha,

  • scale = beta,

  • 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 [upper incomplete Gamma function](
The parameters can be intuited via their relationship to mean and variance
when these moments exist,

mean = beta / (alpha - 1) when alpha > 1 variance = beta**2 / (alpha - 1)**2 / (alpha - 2) when alpha > 2

i.e., under the same conditions:

alpha = mean2 / variance + 2 beta = mean * (mean2 / variance + 1)

Distribution parameters are automatically broadcast in all functions; see
examples for details.
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](

[gamma function]: R:gamma%20function
[upper incomplete Gamma function]: R:upper%20incomplete%20Gamma%20function
[Michael Figurnov, Shakir Mohamed, Andriy Mnih. Implicit Reparameterization Gradients, 2018]: R:Michael%20Figurnov,%20Shakir%20Mohamed,%20Andriy%20Mnih.%20Implicit%20Reparameterization%20Gradients,%202018

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_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_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()