The multivariate normal distribution on R^k

The Multivariate Normal distribution is defined over R^k`` and parameterized by a (batch of) length-k loc vector (aka "mu") and a (batch of) k x kscale matrix;covariance = scale @ scale.Twhere@` denotes matrix-multiplication.

tfd_multivariate_normal_tri_l(
  loc = NULL,
  scale_tril = NULL,
  validate_args = FALSE,
  allow_nan_stats = TRUE,
  name = "MultivariateNormalTriL"
)

Arguments

loc	Floating-point Tensor. If this is set to NULL, loc is implicitly 0. When specified, may have shape [B1, ..., Bb, k] where b >= 0 and k is the event size.
scale_tril	Floating-point, lower-triangular Tensor with non-zero diagonal elements. scale_tril has shape [B1, ..., Bb, k, k] where b >= 0 and k is the event size.
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

Mathematical Details

The probability density function (pdf) is,

pdf(x; loc, scale) = exp(-0.5 ||y||**2) / Z
y = inv(scale) @ (x - loc)
Z = (2 pi)**(0.5 k) |det(scale)|

where:

• loc is a vector in R^k,

• scale is a linear operator in R^{k x k}, cov = scale @ scale.T,

• Z denotes the normalization constant, and,

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

A (non-batch) scale matrix is:

scale = scale_tril

where scale_tril is lower-triangular k x k matrix with non-zero diagonal, i.e., tf$diag_part(scale_tril) != 0. Additional leading dimensions (if any) will index batches.

The MultivariateNormal distribution is a member of the location-scale family, i.e., it can be constructed as,

X ~ MultivariateNormal(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_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()