tfd_multivariate_student_t_linear_operator: Multivariate Student's t-distribution on 'R^k'

In tfprobability: Interface to 'TensorFlow Probability'

Multivariate Student's t-distribution on R^k

Description

Mathematical Details

Usage

tfd_multivariate_student_t_linear_operator(
  df,
  loc,
  scale,
  validate_args = FALSE,
  allow_nan_stats = TRUE,
  name = "MultivariateStudentTLinearOperator"
)

Arguments

df	A positive floating-point Tensor. Has shape [B1, ..., Bb] where b >= 0.
loc	Floating-point Tensor. Has shape [B1, ..., Bb, k] where k is the event size.
scale	Instance of LinearOperator with a floating dtype and shape [B1, ..., Bb, k, k].
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

The probability density function (pdf) is,

pdf(x; df, loc, Sigma) = (1 + ||y||**2 / df)**(-0.5 (df + k)) / Z
where,
y = inv(Sigma) (x - loc)
Z = abs(det(Sigma)) sqrt(df pi)**k Gamma(0.5 df) / Gamma(0.5 (df + k))

where:

• df is a positive scalar.

• loc is a vector in R^k,

• Sigma is a positive definite shape matrix in R^{k x k}, parameterized as scale @ scale.T in this class,

• Z denotes the normalization constant, and,

• ||y||**2 denotes the squared Euclidean norm of y.

The Multivariate Student's t-distribution distribution is a member of the location-scale family, i.e., it can be constructed as,

X ~ MultivariateT(loc=0, scale=1)   # Identity scale, zero shift.
Y = scale @ X + loc

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_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()

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