Multivariate Student's t-distribution on R^k

Source: R/distributions.R

Mathematical Details

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.

Value

a distribution instance.

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

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()