markovflow.models.pep

Module containing a model for CVI

Module Contents

class PowerExpectationPropagation(kernel: markovflow.kernels.SDEKernel, input_data: Tuple[tf.Tensor, tf.Tensor], likelihood: markovflow.likelihoods.PEPScalarLikelihood, mean_function: Optional[markovflow.mean_function.MeanFunction] = None, learning_rate=1.0, alpha=1.0)

Bases: markovflow.models.variational_cvi.GaussianProcessWithSitesBase

This is an approximate inference called Power Expectation Propagation.

Approximates a the posterior of a model with GP prior and a general likelihood using a Gaussian posterior parameterized with Gaussian sites.

The following notation is used:

• x - the time points of the training data.

• y - observations corresponding to time points x.

• s(.) - the latent state of the Markov chain

• f(.) - the noise free predictions of the model

• p(y | f) - the likelihood

• t(f) - a site (indices will refer to the associated data point)

• p(.) the prior distribution

• q(.) the variational distribution

We use the state space formulation of Markovian Gaussian Processes that specifies: the conditional density of neighbouring latent states: p(xₖ₊₁| xₖ) how to read out the latent process from these states: fₖ = H xₖ

The likelihood links data to the latent process and p(yₖ | fₖ). We would like to approximate the posterior over the latent state space model of this model.

We parameterize the joint posterior using sites tₖ(fₖ)

p(x, y) = p(x) ∏ₖ tₖ(fₖ)

where tₖ(fₖ) are univariate Gaussian sites parameterized in the natural form

t(f) = exp(𝞰ᵀφ(f) - A(𝞰)), where 𝞰=[η₁,η₂] and 𝛗(f)=[f,f²]

(note: the subscript k has been omitted for simplicity)

The site update of the sites are given by the classic EP update rules as described in:

@techreport{seeger2005expectation,

title={Expectation propagation for exponential families}, author={Seeger, Matthias}, year={2005}

}

Parameters

• kernel – A kernel that defines a prior over functions.

• input_data – A tuple of (time_points, observations) containing the observed data: time points of observations, with shape batch_shape + [num_data], observations with shape batch_shape + [num_data, observation_dim].

• likelihood – A likelihood. with shape batch_shape + [num_inducing].

• mean_function – The mean function for the GP. Defaults to no mean function.

• learning_rate – the learning rate of the algorithm

• alpha – the power as in Power Expectation propagation

local_objective(Fmu, Fvar, Y)

Local objective of the PEP algorithm : log E_q(f) p(y|f)ᵃ

local_objective_gradients(Fmu, Fvar)

Gradients of the local objective of the PEP algorithm wrt to the predictive mean

mask_indices(exclude_indices)

Binary mask (cast to float), 0 for the excluded indices, 1 for the rest

compute_cavity_from_marginals(marginals)

Compute cavity from marginals :param marginals: list of tensors

compute_cavity()

The cavity distributions for all data points. This corresponds to the marginal distribution qᐠⁿ(fₙ) of qᐠⁿ(f) = q(f)/tₙ(fₙ)ᵃ

compute_log_norm()

Compute log normalizer

update_sites(site_indices=None)

Compute the site updates and perform one update step :param site_indices: list of indices to be updated

elbo() → tf.Tensor

Computes the marginal log marginal likelihood of the approximate joint p(s, y)

energy()

PEP Energy

predict_log_density(input_data: Tuple[tf.Tensor, tf.Tensor], full_output_cov: bool = False) → tf.Tensor

Compute the log density of the data at the new data points.

Parameters

• input_data – A tuple of time points and observations containing the data at which to calculate the loss for training the model: a tensor of inputs with shape batch_shape + [num_data], a tensor of observations with shape batch_shape + [num_data, observation_dim].

• full_output_cov – Either full output covariance (True) or marginal variances (False).

gradient_correction(inputs, grads)

Transforms vectors g=[g1,g2] and i=[i1,i2] into h=[h1, h2] where h2 = 1/2 * 1/(i2 + 1/g2) and h1 = 2 * h2 * (g1/g2 - i1)

Parameters

• inputs – a tensor of inputs with shape batch_shape + [num_data],

• grads – a tensor of gradients with shape batch_shape + [num_data],

Returns

a tensor of modified gradients with shape batch_shape + [num_data],