View source: R/distributions.R
tfd_multivariate_normal_full_covariance | R Documentation |
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_full_covariance( loc = NULL, covariance_matrix = NULL, validate_args = FALSE, allow_nan_stats = TRUE, name = "MultivariateNormalFullCovariance" )
loc |
Floating-point |
covariance_matrix |
Floating-point, symmetric positive definite |
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. |
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
.
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
The batch_shape
is the broadcast shape between loc
and
covariance_matrix
arguments.
The event_shape
is given by last dimension of the matrix implied by
covariance_matrix
. The last dimension of loc
(if provided) must
broadcast with this.
A non-batch covariance_matrix
matrix is a k x k
symmetric positive
definite matrix. In other words it is (real) symmetric with all eigenvalues
strictly positive.
Additional leading dimensions (if any) will index batches.
a distribution instance.
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_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()
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