markovflow.posterior

Module containing posterior processes for GP models.

Module Contents

class PosteriorProcess(name=None)

Bases: tf.Module, abc.ABC

Abstract class for forming a posterior process.

Posteriors that extend this class must implement the sample_state_trajectories(), sample_f(), predict_state() and predict_f() methods.

sample_state(new_time_points: tf.Tensor, sample_shape: markovflow.base.SampleShape, *, input_data: Optional[Tuple[tf.Tensor, tf.Tensor]] = None) → tf.Tensor

Generate joint state samples at new_time_points.

Parameters

• new_time_points – Time points to generate sample trajectories for, with shape batch_shape + [num_time_points].

• sample_shape – A SampleShape that is the shape (or number of) sampled trajectories to draw.

• input_data –

  A tuple of time points and observations containing the data:

  • A tensor of inputs with shape batch_shape + [num_data]

  • A tensor of observations with shape batch_shape + [num_data, observation_dim]

  This is an optional argument only passed in for inference with an importance-weighted posterior.

Returns

A tensor containing:

• Sampled trajectories at new points, with shape sample_shape + batch_shape + [num_time_points, state_dim]

• Sampled trajectories at conditioning points, with shape sample_shape + batch_shape + [num_conditioning_points, state_dim]

abstract sample_state_trajectories(new_time_points: tf.Tensor, sample_shape: markovflow.base.SampleShape, *, input_data: Optional[Tuple[tf.Tensor, tf.Tensor]] = None) → Tuple[tf.Tensor, tf.Tensor]

Generate joint sampled state trajectories evaluated both at new_time_points and at some points that we condition on for obtaining the posterior.

Parameters

• new_time_points – Time points to generate sample trajectories for, with shape batch_shape + [num_time_points].

• sample_shape – A SampleShape that is the shape (or number of) sampled trajectories to draw.

• input_data –

  A tuple of time points and observations containing the data:

  • A tensor of inputs with shape batch_shape + [num_data]

  • A tensor of observations with shape batch_shape + [num_data, observation_dim]

  This is an optional argument only passed in for inference with an importance-weighted posterior.

Returns

A tensor containing:

• Sampled trajectories at new points, with shape sample_shape + batch_shape + [num_time_points, state_dim]

• Sampled trajectories at conditioning points, with shape sample_shape + batch_shape + [num_conditioning_points, state_dim]

Raises

NotImplementedError – Must be implemented in derived classes.

abstract sample_f(new_time_points: tf.Tensor, sample_shape: markovflow.base.SampleShape, *, input_data: Optional[Tuple[tf.Tensor, tf.Tensor]] = None) → tf.Tensor

Generate joint function evaluation samples (projected states) at new_time_points.

Parameters

• new_time_points – Time points to generate sample trajectories for, with shape batch_shape + [num_time_points].

• sample_shape – A SampleShape that is the shape (or number of) sampled trajectories to draw.

• input_data –

  A tuple of time points and observations containing the data:

  • A tensor of inputs with shape batch_shape + [num_data]

  • A tensor of observations with shape batch_shape + [num_data, observation_dim]

  This is an optional argument only passed in for inference with an importance-weighted posterior.

Returns

A tensor containing sampled trajectories, with shape sample_shape + batch_shape + [num_time_points, output_dim].

Raises

NotImplementedError – Must be implemented in derived classes.

abstract predict_state(new_time_points: tf.Tensor) → Tuple[tf.Tensor, tf.Tensor]

Predict state at new_time_points. Note these time points should be sorted.

Parameters

new_time_points – Time points to generate observations for, with shape batch_shape + [num_new_time_points,].

abstract predict_f(new_time_points: tf.Tensor, full_output_cov: bool = False) → Tuple[tf.Tensor, tf.Tensor]

Predict marginal function values at new_time_points. Note these time points should be sorted.

Parameters

• new_time_points – Time points to generate observations for, with shape batch_shape + [num_new_time_points].

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

class ConditionalProcess(posterior_dist: markovflow.gauss_markov.GaussMarkovDistribution, kernel: markovflow.kernels.SDEKernel, conditioning_time_points: tf.Tensor, mean_function: Optional[markovflow.mean_function.MeanFunction] = None)

Bases: PosteriorProcess

Represents a posterior process indexed on the real line.

This means \(q(s(.))\) is built by combining the marginals \(q(s(Z))\) and the conditional process \(p(s(.)|s(Z))\) into:

\[q(s(.)) = ∫p(s(.)|s(Z))q(s(Z)) ds(Z)\]

The marginals at discrete time inputs are available in closed form (see the predict_f() method).

It also includes methods for sampling from the posterior process.

Parameters

• posterior_dist – The posterior represented by a Gauss-Markov distribution used for inference. For variational models this is the model defined by the variational distribution.

• kernel – The kernel of the prior process.

• conditioning_time_points – The time points to condition on for inference, with shape batch_shape + [num_time_points].

• mean_function – The mean function of the process that is added to fs.

predict_state(new_time_points: tf.Tensor) → Tuple[tf.Tensor, tf.Tensor]

Predict state at new_time_points. Note these time points should be sorted.

Parameters

new_time_points – Time points to generate observations for, with shape batch_shape + [num_new_time_points,].

Returns

Predicted mean and covariance for the new time points, with respective shapes batch_shape + [num_new_time_points, state_dim] batch_shape + [num_new_time_points, state_dim, state_dim].

predict_f(new_time_points: tf.Tensor, full_output_cov: bool = False) → Tuple[tf.Tensor, tf.Tensor]

Predict marginal function values at new_time_points. Note these time points should be sorted.

Note

new_time_points that are far outside the self.conditioning_time_points specified when instantiating the class will revert to the prior.

Parameters

• new_time_points – Time points to generate observations for, with shape batch_shape + [num_new_time_points].

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

Returns

Predicted mean and covariance for the new time points, with respective shapes batch_shape + [num_new_time_points, output_dim] and either batch_shape + [num_new_time_points, output_dim, output_dim] or batch_shape + [num_new_time_points, output_dim].

sample_state_trajectories(new_time_points: tf.Tensor, sample_shape: markovflow.base.SampleShape, *, input_data: Optional[Tuple[tf.Tensor, tf.Tensor]] = None) → Tuple[tf.Tensor, tf.Tensor]

Generate joint state samples at new_time_points and the self.conditioning_time_points specified when instantiating the class.

See Appendix 2 of “Doubly Sparse Variational Gaussian Processes” for a derivation.

The following notation is used:

• \(t\) - a vector of new time points

• \(z\) - a vector of the conditioning time points

• \(sₚ/uₚ\) - prior state sample at \(t/z\)

• \(sₒ/Uₒ\) - posterior state sample at \(t/z\)

• \(p(.)\) - the prior

• \(q(.)\) - the posterior

Jointly sample from the prior at new and conditioning points:

\[[sₚ, uₚ] ~ p(s([t, z]))\]

And sample from the posterior at the conditioning points:

\[uₒ ~ q(s(z))\]

A sample from the posterior state is given by:

\[sₒ = sₚ - E[s(t) | s(z) = uₒ - uₚ]\]

Noting \(z₋,z₊\), for each new point \(tₖ\) the points in \(z\) closest to \(tₖ\) and \(vₖ = [s(z₋),s(z₊)]\) are:

\[E[s(tₖ)|s(z)] = E[s(tₖ)|vₖ] = Pₖ vₖ\]

That is, the conditional mean is local; it only depends on the nearing conditioning states.

Parameters

• new_time_points – Time points to generate sample trajectories for, with shape batch_shape + [num_time_points].

• sample_shape – A SampleShape that is the shape of sampled trajectories to draw. This can be either an integer or a tuple/list of integers.

• input_data –

  A tuple of time points and observations containing the data:

  • A tensor of inputs with shape batch_shape + [num_data]

  • A tensor of observations with shape batch_shape + [num_data, observation_dim]

  Note this argument will be ignored if your posterior is an AnalyticPosteriorProcess.

Returns

A tensor containing:

• Sampled trajectories at new points, with shape sample_shape + batch_shape + [num_time_points, state_dim]

• Sampled trajectories at conditioning points, with shape sample_shape + batch_shape + [num_conditioning_points, state_dim]

sample_f(new_time_points: tf.Tensor, sample_shape: markovflow.base.SampleShape, *, input_data: Optional[Tuple[tf.Tensor, tf.Tensor]] = None) → tf.Tensor

Generate joint function evaluation samples (projected states) at new_time_points.

Parameters

• new_time_points – Time points to generate sample trajectories for, with shape batch_shape + [num_time_points].

• sample_shape – A SampleShape that is the shape (or number of) sampled trajectories to draw.

• input_data –

  A tuple of time points and observations containing the data:

  • A tensor of inputs with shape batch_shape + [num_data]

  • A tensor of observations with shape batch_shape + [num_data, observation_dim]

  Note this argument will be ignored if your posterior is an AnalyticPosteriorProcess.

Returns

A tensor containing the sampled trajectories, with shape sample_shape + batch_shape + [num_time_points, output_dim].

class AnalyticPosteriorProcess(posterior_dist: markovflow.gauss_markov.GaussMarkovDistribution, kernel: markovflow.kernels.SDEKernel, conditioning_time_points: tf.Tensor, likelihood: Likelihood, mean_function: Optional[markovflow.mean_function.MeanFunction] = None)

Bases: ConditionalProcess

Represents the (approximate) posterior process of a GP model.

It inherits the marginal prediction and sampling methods from the parent ConditionalProcess class.

It also includes a method to predict the observations (see predict_y()).

Parameters

• posterior_dist – The posterior represented by a Gauss-Markov distribution used for inference. For variational models this is the model defined by the variational distribution.

• kernel – The kernel of the prior process.

• conditioning_time_points – The time points to condition on for inference, with shape batch_shape + [num_time_points].

• likelihood – Likelihood defining how to project from f-space to an observation.

• mean_function – The mean function of the process that is added to fs.

predict_y(new_time_points: tf.Tensor, full_output_cov=False) → Tuple[tf.Tensor, tf.Tensor]

Predict observation marginals at new_time_points. Note these time points should be sorted.

Parameters

• new_time_points – Time points to generate observations for, with shape batch_shape + [num_new_time_points].

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

Returns

Predicted mean and covariance for the new time points, with respective shapes batch_shape + [num_new_time_points, output_dim] and either batch_shape + [num_new_time_points, output_dim, output_dim] or batch_shape + [num_new_time_points, output_dim].

class ImportanceWeightedPosteriorProcess(num_importance_samples: int, proposal_dist: markovflow.gauss_markov.GaussMarkovDistribution, kernel: markovflow.kernels.SDEKernel, conditioning_time_points: tf.Tensor, likelihood: Likelihood, mean_function: Optional[markovflow.mean_function.MeanFunction] = None)

Bases: PosteriorProcess

Represents the approximate posterior process of a GP model.

The approximate posterior process is inferred via importance-weighted variational inference.

Parameters

• num_importance_samples – The number of importance-weighted samples.

• proposal_dist – The proposal represented by a Gauss-Markov distribution, from which we draw samples. This is the model defined by the variational distribution.

• kernel – The kernel of the prior process.

• conditioning_time_points – Time points to condition on for inference, with shape batch_shape + [num_time_points].

• likelihood – Likelihood defining how to project from f-space to an observation.

• mean_function – The mean function of the process that is added to fs.

_log_qu_density(samples_u: tf.Tensor, stop_gradient: bool = False)

Log density of the posterior process evaluated at the conditioning points.

Parameters

• samples_u – State samples at the conditioning time points, with shape sample_shape + [num_conditioning_points, state_dim].

• stop_gradient – Whether to stop the gradient flow through the samples. It is useful to do so when optimising the proposal distribution with control variates for reduced variance.

Returns

log q(u) [num_samples]

log_importance_weights(samples_s: tf.Tensor, samples_u: tf.Tensor, input_data: Tuple[tf.Tensor, tf.Tensor], stop_gradient: bool = False) → tf.Tensor

Compute the log-importance weights for some state samples.

The importance weights are given by:

\[w = p(Y | s) p(s, u) / q(s, u)\]

Because it is assumed that \(q(s | u) = p(s | u)\), the weights reduce to:

\[w = p(Y | s) p(u) / q(u)\]

We evaluate this ratio for some tensors of samples_s and samples_u, which are assumed to have been drawn from \(q(s, u)\). To do this, samples_s are projected to \(f\) before being passed to the likelihood object.

Parameters

• samples_s – A tensor of samples drawn from \(p(s|u)\), with shape sample_shape + batch_shape + [num_data, state_dim].

• samples_u – A tensor of samples drawn from \(q(u)\), with shape sample_shape + batch_shape + [num_inducing, state_dim].

• input_data –

  A tuple of time points and observations containing the data:

  • A tensor of inputs with shape batch_shape + [num_data]

  • A tensor of observations with shape batch_shape + [num_data, observation_dim]

• stop_gradient – Whether to call stop gradient on \(q(u)\). This is useful for control variate schemes.

Return log_weights

A tensor with shape [sample_shape].

_iwvi_samples_and_weights(new_time_points: tf.Tensor, input_data: Tuple[tf.Tensor, tf.Tensor], sample_shape: markovflow.base.SampleShape) → Tuple[tf.Tensor, tf.Tensor, tf.Tensor]

Sample from q(states) indexed by new_time_points and compute the log weights associated.

Parameters

• new_time_points – ordered time input where to sample with shape batch_shape + [num_new_time_points]

• input_data –

  A tuple of time points and observations containing the data:

  • A tensor of inputs with shape batch_shape + [num_data]

  • A tensor of observations with shape batch_shape + [num_data, observation_dim]

• sample_shape – A SampleShape that specifies how many samples to draw, with shape (..., num_importance_samples).

Returns

state samples from the posterior and the log-weights with shapes: sample_shape + batch_shape + [num_new_time_points, state_dim] sample_shape sample_shape + batch_shape + [num_conditioning_points, state_dim]

sample_state_trajectories(new_time_points: tf.Tensor, sample_shape: markovflow.base.SampleShape, *, input_data: Tuple[tf.Tensor, tf.Tensor] = None) → Tuple[tf.Tensor, tf.Tensor]

Sample the importance-weighted posterior over states.

Parameters

• new_time_points – Ordered time input from which to sample, with shape batch_shape + [num_new_time_points].

• sample_shape – A SampleShape that is the shape (or number of) sampled trajectories to draw.

• input_data –

  A tuple of time points and observations containing the data:

  • A tensor of inputs with shape batch_shape + [num_data]

  • A tensor of observations with shape batch_shape + [num_data, observation_dim]

Returns

The ordered samples states, with shape sample_shape + batch_shape + [num_new_time_points, state_dim].

sample_f(new_time_points: tf.Tensor, sample_shape: markovflow.base.SampleShape, *, input_data: Tuple[tf.Tensor, tf.Tensor] = None) → tf.Tensor

Sample the importance-weighted (IWVI) posterior over functions.

Note that to compute the expected value of some function under the iwvi posterior, it is likely to be more efficient to use expected_value().

Parameters

• new_time_points – Ordered time input from which to sample, with shape batch_shape + [num_new_time_points].

• sample_shape – A SampleShape that is the shape (or number of) sampled trajectories to draw.

• input_data –

  A tuple of time points and observations containing the data:

  • A tensor of inputs with shape batch_shape + [num_data]

  • A tensor of observations with shape batch_shape + [num_data, observation_dim]

Returns

The ordered samples on latent functions, with shape [num_samples] + batch_shape + [num_new_time_points, num_outputs].

expected_value(new_time_points: tf.Tensor, input_data: Tuple[tf.Tensor, tf.Tensor], func=tf.identity) → tf.Tensor

Compute the expected value of the function func acting on a random variable \(f\).

\(f\) is represented by a GP in this case, using importance sampling at the times given in new_time_points. That is:

\[∫qₚ(f) func(f) df = Σₖ wₖ func(fₖ)\]

…where:

• \(qₚ\) is the importance-weighted approximate posterior distribution of \(f\)

• \(wₖ\) are the importance weights

For example, to compute the posterior mean we set func = tf.identify.

Parameters

• new_time_points – Ordered time input from which to sample, with shape batch_shape + [num_new_time_points].

• input_data –

  A tuple of time points and observations containing the data:

  • A tensor of inputs with shape batch_shape + [num_data]

  • A tensor of observations with shape batch_shape + [num_data, observation_dim]

• func – The function to compute the expected value of. func should act on the last dimension of a tensor. That last dimension will have length as specified by the output_dim of the underlying emission model. The return shape of func need not be the same, but we expect all other dimensions to broadcast.

Returns

A tensor with shape batch_shape + [num_new_time_points, output_dim].

abstract predict_state(new_time_points: tf.Tensor) → Tuple[tf.Tensor, tf.Tensor]

Not applicable to ImportanceWeightedPosteriorProcess. The marginal state predictions are not available in closed form.

Parameters

new_time_points – Time points to generate observations for, with shape batch_shape + [num_new_time_points,].

abstract predict_f(new_time_points: tf.Tensor, full_output_cov: bool = False) → Tuple[tf.Tensor, tf.Tensor]

Not applicable to ImportanceWeightedPosteriorProcess. The marginal function predictions are not available in closed form.

Parameters

• new_time_points – Time points to generate observations for, with shape batch_shape + [num_new_time_points].

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

_correct_mean_shape(mean: tf.Tensor, kernel: markovflow.kernels.SDEKernel) → tf.Tensor

Helper function that checks if the state space model is defined over a StackKernel so that it can bring the output of the mean function to the right shape. In any other case, the mean is returned unaltered.

Parameters

• mean –

  the mean value that has been computed via a MeanFunction, with shape

  batch_shape + [num_data, output_dim]

  or batch_shape + [num_data, 1] in the case of a StackKernel,

  where batch_shape[-1] = output_dim

• kernel – the corresponding kernel of the GaussMarkovDistribution

Returns

the mean value with the correct shape which is batch_shape + [num_data, output_dim] or batch_shape[:-1] + [num_data, output_dim] in the case of a StackKernel as the last dimension of the batch_shape is the output_dim.