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_diag_plus_low_rank(
  loc = NULL,
  scale_diag = NULL,
  scale_identity_multiplier = NULL,
  scale_perturb_factor = NULL,
  scale_perturb_diag = NULL,
  validate_args = FALSE,
  allow_nan_stats = TRUE,
  name = "MultivariateNormalDiagPlusLowRank"
)

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_diag	Non-zero, floating-point Tensor representing a diagonal matrix added to scale. May have shape [B1, ..., Bb, k], b >= 0, and characterizes b-batches of k x k diagonal matrices added to scale. When both scale_identity_multiplier and scale_diag are NULL then scale is the Identity.
scale_identity_multiplier	Non-zero, floating-point Tensor representing a scaled-identity-matrix added to scale. May have shape [B1, ..., Bb], b >= 0, and characterizes b-batches of scaled k x k identity matrices added to scale. When both scale_identity_multiplier and scale_diag are NULL then scale is the Identity.
scale_perturb_factor	Floating-point Tensor representing a rank-r perturbation added to scale. May have shape [B1, ..., Bb, k, r], b >= 0, and characterizes b-batches of rank-r updates to scale. When NULL, no rank-r update is added to scale.#'
scale_perturb_diag	Floating-point Tensor representing a diagonal matrix inside the rank-r perturbation added to scale. May have shape [B1, ..., Bb, r], b >= 0, and characterizes b-batches of r x r diagonal matrices inside the perturbation added to scale. When NULL, an identity matrix is used inside the perturbation. Can only be specified if scale_perturb_factor is also specified.
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 = diag(scale_diag + scale_identity_multiplier ones(k)) +
scale_perturb_factor @ diag(scale_perturb_diag) @ scale_perturb_factor.T

where:

• scale_diag.shape = [k],

• scale_identity_multiplier.shape = [],

• scale_perturb_factor.shape = [k, r], typically k >> r, and,

• scale_perturb_diag.shape = [r].

Additional leading dimensions (if any) will index batches. If both scale_diag and scale_identity_multiplier are NULL, then scale is the Identity matrix. 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(), 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()