A vector diffeomixture (VDM) is a distribution parameterized by a convex
combination of K component loc vectors, loc[k], k = 0,...,K-1, and K
scale matrices scale[k], k = 0,..., K-1. It approximates the following
compound distribution
p(x) = int p(x | z) p(z) dz, where z is in the K-simplex, 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"
)a distribution instance.
float-like Tensor with shape [b1, ..., bB, K-1].
In terms of samples, larger mix_loc[..., k] ==>
Z is more likely to put more weight on its kth component.
float-like Tensor. Broadcastable with mix_loc.
In terms of samples, smaller temperature means one component is more
likely to dominate. I.e., smaller temperature makes the VDM look more
like a standard mixture of K components.
tfp$distributions$Distribution-like instance. Distribution
from which d iid samples are used as input to the selected affine
transformation. Must be a scalar-batch, scalar-event distribution.
Typically distribution$reparameterization_type = FULLY_REPARAMETERIZED
or it is a function of non-trainable parameters. WARNING: If you
backprop through a VectorDiffeomixture sample and the distribution
is not FULLY_REPARAMETERIZED yet is a function of trainable variables,
then the gradient will be incorrect!
Length-K list of float-type Tensors. The k-th element
represents the shift used for the k-th affine transformation. If
the k-th item is NULL, loc is implicitly 0. When specified,
must have shape [B1, ..., Bb, d] where b >= 0 and d is the event
size.
Length-K list of LinearOperators. Each should be
positive-definite and operate on a d-dimensional vector space. The
k-th element represents the scale used for the k-th affine
transformation. LinearOperators must have shape [B1, ..., Bb, d, d],
b >= 0, i.e., characterizes b-batches of d x d matrices
integer scalar representing number of
quadrature points. Larger quadrature_size means q_N(x) better
approximates p(x).
Function taking normal_loc, normal_scale,
quadrature_size, validate_args and returning tuple(grid, probs)
representing the SoftmaxNormal grid and corresponding normalized weight.
normalized) weight.
Default value: quadrature_scheme_softmaxnormal_quantiles.
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.
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 prefixed to Ops created by this class.
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 non-negative 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
K-1 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 non-standard distribution that has properties
particularly useful in variational Bayesian methods.
Conditioned on a draw from the SoftmaxNormal, X|z is a vector whose
components are linear combinations of affine transformations, thus is itself
an affine transformation.
Note: The marginals X_1|v, ..., X_d|v 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 Diffeomixtures 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 gradient-descent (via backprop) to
optimize Monte-Carlo objectives. Such objectives are a finite-sample
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()