A vector diffeomixture (VDM) is a distribution parameterized by a convex
combination of K
component loc
vectors, loc[k], k = 0,...,K1
, and K
scale
matrices scale[k], k = 0,..., K1
. It approximates the following
compound distribution
p(x) = int p(x  z) p(z) dz
, where z is in the Ksimplex, and
p(x  z) := p(x  loc=sum_k z[k] loc[k], scale=sum_k z[k] scale[k])
tfd_vector_diffeomixture( mix_loc, temperature, distribution, loc = NULL, scale = NULL, quadrature_size = 8, quadrature_fn = tfp$distributions$quadrature_scheme_softmaxnormal_quantiles, validate_args = FALSE, allow_nan_stats = TRUE, name = "VectorDiffeomixture" )
mix_loc 


temperature 

distribution 

loc  Length 
scale  Length 
quadrature_size 

quadrature_fn  Function taking 
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. 
a distribution instance.
The integral int p(x  z) p(z) dz
is approximated with a quadrature scheme
adapted to the mixture density p(z)
. The N
quadrature points z_{N, n}
and weights w_{N, n}
(which are nonnegative and sum to 1) are chosen such that
q_N(x) := sum_{n=1}^N w_{n, N} p(x  z_{N, n}) > p(x)
as N > infinity
.
Since q_N(x)
is in fact a mixture (of N
points), we may sample from
q_N
exactly. It is important to note that the VDM is defined as q_N
above, and not p(x)
. Therefore, sampling and pdf may be implemented as
exact (up to floating point error) methods.
A common choice for the conditional p(x  z)
is a multivariate Normal.
The implemented marginal p(z)
is the SoftmaxNormal
, which is a
K1
dimensional Normal transformed by a SoftmaxCentered
bijector, making
it a density on the K
simplex. That is,
Z = SoftmaxCentered(X)
, X = Normal(mix_loc / temperature, 1 / temperature)
The default quadrature scheme chooses z_{N, n}
as N
midpoints of
the quantiles of p(z)
(generalized quantiles if K > 2
).
See Dillon and Langmore (2018) for more details.
About Vector
distributions in TensorFlow.
The VectorDiffeomixture
is a nonstandard distribution that has properties
particularly useful in variational Bayesian methods.
Conditioned on a draw from the SoftmaxNormal, Xz
is a vector whose
components are linear combinations of affine transformations, thus is itself
an affine transformation.
Note: The marginals X_1v, ..., X_dv
are not generally identical to some
parameterization of distribution
. This is due to the fact that the sum of
draws from distribution
are not generally itself the same distribution
.
About Diffeomixture
s and reparameterization.
The VectorDiffeomixture
is designed to be reparameterized, i.e., its
parameters are only used to transform samples from a distribution which has no
trainable parameters. This property is important because backprop stops at
sources of stochasticity. That is, as long as the parameters are used after
the underlying source of stochasticity, the computed gradient is accurate.
Reparametrization means that we can use gradientdescent (via backprop) to
optimize MonteCarlo objectives. Such objectives are a finitesample
approximation of an expectation and arise throughout scientific computing.
WARNING: If you backprop through a VectorDiffeomixture sample and the "base"
distribution is both: not FULLY_REPARAMETERIZED
and a function of trainable
variables, then the gradient is not guaranteed correct!
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_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_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()