tfd_poisson_log_normal_quadrature_compound: 'PoissonLogNormalQuadratureCompound' distribution

View source: R/distributions.R

tfd_poisson_log_normal_quadrature_compoundR Documentation

PoissonLogNormalQuadratureCompound distribution

Description

The PoissonLogNormalQuadratureCompound is an approximation to a Poisson-LogNormal compound distribution, i.e.,

p(k|loc, scale) = int_{R_+} dl LogNormal(l | loc, scale) Poisson(k | l)
approx= sum{ prob[d] Poisson(k | lambda(grid[d])) : d=0, ..., deg-1 }

Usage

tfd_poisson_log_normal_quadrature_compound(
  loc,
  scale,
  quadrature_size = 8,
  quadrature_fn = tfp$distributions$quadrature_scheme_lognormal_quantiles,
  validate_args = FALSE,
  allow_nan_stats = TRUE,
  name = "PoissonLogNormalQuadratureCompound"
)

Arguments

loc

float-like (batch of) scalar Tensor; the location parameter of the LogNormal prior.

scale

float-like (batch of) scalar Tensor; the scale parameter of the LogNormal prior.

quadrature_size

integer scalar representing the number of quadrature points.

quadrature_fn

Function taking loc, scale, quadrature_size, validate_args and returning tuple(grid, probs) representing the LogNormal grid and corresponding normalized weight. Default value: quadrature_scheme_lognormal_quantiles.

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

By default, the grid is chosen as quantiles of the LogNormal distribution parameterized by loc, scale and the prob vector is [1. / quadrature_size]*quadrature_size.

In the non-approximation case, a draw from the LogNormal prior represents the Poisson rate parameter. Unfortunately, the non-approximate distribution lacks an analytical probability density function (pdf). Therefore the PoissonLogNormalQuadratureCompound class implements an approximation based on quadrature. Note: although the PoissonLogNormalQuadratureCompound is approximately the Poisson-LogNormal compound distribution, it is itself a valid distribution. Viz., it possesses a sample, log_prob, mean, variance, etc. which are all mutually consistent.

Mathematical Details

The PoissonLogNormalQuadratureCompound approximates a Poisson-LogNormal compound distribution. Using variable-substitution and numerical quadrature (default: based on LogNormal quantiles) we can redefine the distribution to be a parameter-less convex combination of deg different Poisson samples. That is, defined over positive integers, this distribution is parameterized by a (batch of) loc and scale scalars.

The probability density function (pdf) is,

pdf(k | loc, scale, deg) = sum{ prob[d] Poisson(k | lambda=exp(grid[d])) : d=0, ..., deg-1 }

Note: probs returned by (optional) quadrature_fn are presumed to be either a length-quadrature_size vector or a batch of vectors in 1-to-1 correspondence with the returned grid. (I.e., broadcasting is only partially supported.)

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_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(), 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()


tfprobability documentation built on Sept. 1, 2022, 5:07 p.m.