gpflow.conditionals#

Modules#

Functions#

gpflow.conditionals.base_conditional#

gpflow.conditionals.base_conditional(Kmn, Kmm, Knn, f, *, full_cov=False, q_sqrt=None, white=False)[source]#

Given a g1 and g2, and distribution p and q such that:

p(g2) = N(g2; 0, Kmm)

p(g1) = N(g1; 0, Knn)
p(g1 | g2) = N(g1; Knm (Kmm⁻¹) g2, Knn - Knm (Kmm⁻¹) Kmn)

And:

q(g2) = N(g2; f, q_sqrt q_sqrtᵀ)

This method computes the mean and (co)variance of:

q(g1) = ∫ q(g2) p(g1 | g2)
Parameters

  • Kmn (Tensor) – [M, …, N]

  • Kmm (Tensor) – [M, M]

  • Knn (Tensor) – […, N, N] or N

  • f (Tensor) – [M, R]

  • full_cov (bool) – bool

  • q_sqrt (Optional[Tensor]) – If this is a Tensor, it must have shape [R, M, M] (lower triangular) or [M, R] (diagonal)

  • white (bool) – bool

Return type

Tuple[Tensor, Tensor]

Returns

[N, R] or [R, N, N]

gpflow.conditionals.conditional#

This function uses multiple dispatch, which will depend on the type of argument passed in:

gpflow.conditionals.conditional( object, InducingVariables, Kernel, object )
# dispatch to -> gpflow.conditionals.conditionals._sparse_conditional(...)
gpflow.conditionals.conditionals._sparse_conditional(Xnew, inducing_variable, kernel, f, *, full_cov=False, full_output_cov=False, q_sqrt=None, white=False)[source]#

Single-output GP conditional.

The covariance matrices used to calculate the conditional have the following shape:

  • Kuu: [M, M]

  • Kuf: [M, N]

  • Kff: [N, N]

Further reference:

  • See gpflow.conditionals._dense_conditional (below) for a detailed explanation of conditional in the single-output case.

  • See the multiouput notebook for more information about the multiouput framework.

Parameters

  • Xnew (Tensor) – data matrix: [N, D]

  • f (Tensor) – data matrix: [M, R]

  • full_cov (bool) – return the covariance between the datapoints

  • full_output_cov (bool) – return the covariance between the outputs. NOTE: as we are using a single-output kernel with repetitions these covariances will be zero.

  • q_sqrt (Optional[Tensor]) – matrix of standard-deviations or Cholesky matrices, size [M, R] or [R, M, M].

  • white (bool) – boolean of whether to use the whitened representation

  • inducing_variable (InducingVariables) –

  • kernel (Kernel) –

Return type

Tuple[Tensor, Tensor]

Returns

  • mean: [N, R]

  • variance: [N, R], [R, N, N], [N, R, R] or [N, R, N, R]

Please see gpflow.conditional._expand_independent_outputs for more information about the shape of the variance, depending on full_cov and full_output_cov.

gpflow.conditionals.conditional( object, object, Kernel, object )
# dispatch to -> gpflow.conditionals.conditionals._dense_conditional(...)
gpflow.conditionals.conditionals._dense_conditional(Xnew, X, kernel, f, *, full_cov=False, full_output_cov=False, q_sqrt=None, white=False)[source]#

Given f, representing the GP at the points X, produce the mean and (co-)variance of the GP at the points Xnew.

Additionally, there may be Gaussian uncertainty about f as represented by q_sqrt. In this case f represents the mean of the distribution and q_sqrt the square-root of the covariance.

Additionally, the GP may have been centered (whitened) so that:

p(v) = 𝒩(𝟎, 𝐈)
f = 𝐋v

thus:

p(f) = 𝒩(𝟎, 𝐋𝐋ᵀ) = 𝒩(𝟎, 𝐊).

In this case f represents the values taken by v.

The method can either return the diagonals of the covariance matrix for each output (default) or the full covariance matrix (full_cov=True).

We assume R independent GPs, represented by the columns of f (and the first dimension of q_sqrt).

Parameters

  • Xnew (Tensor) – data matrix, size [N, D]. Evaluate the GP at these new points

  • X (Tensor) – data points, size [M, D].

  • kernel (Kernel) – GPflow kernel.

  • f (Tensor) – data matrix, [M, R], representing the function values at X, for R functions.

  • q_sqrt (Optional[Tensor]) – matrix of standard-deviations or Cholesky matrices, size [M, R] or [R, M, M].

  • white (bool) – boolean of whether to use the whitened representation as described above.

  • full_cov (bool) –

  • full_output_cov (bool) –

Return type

Tuple[Tensor, Tensor]

Returns

  • mean: [N, R]

  • variance: [N, R] (full_cov = False), [R, N, N] (full_cov = True)

gpflow.conditionals.conditional( object, SharedIndependentInducingVariables, SharedIndependent, object )
# dispatch to -> gpflow.conditionals.multioutput.conditionals.shared_independent_conditional(...)
gpflow.conditionals.multioutput.conditionals.shared_independent_conditional(Xnew, inducing_variable, kernel, f, *, full_cov=False, full_output_cov=False, q_sqrt=None, white=False)[source]#

Multioutput conditional for an independent kernel and shared inducing inducing. Same behaviour as conditional with non-multioutput kernels. The covariance matrices used to calculate the conditional have the following shape:

  • Kuu: [M, M]

  • Kuf: [M, N]

  • Kff: N or [N, N]

Further reference:

  • See gpflow.conditionals._conditional for a detailed explanation of conditional in the single-output case.

  • See the multioutput notebook for more information about the multioutput framework.

Parameters

  • Xnew (Tensor) – data matrix, size [N, D].

  • f (Tensor) – data matrix, [M, P]

  • full_cov (bool) – return the covariance between the datapoints

  • full_output_cov (bool) – return the covariance between the outputs. Note: as we are using a independent kernel these covariances will be zero.

  • q_sqrt (Optional[Tensor]) – matrix of standard-deviations or Cholesky matrices, size [M, P] or [P, M, M].

  • white (bool) – boolean of whether to use the whitened representation

  • inducing_variable (SharedIndependentInducingVariables) –

  • kernel (SharedIndependent) –

Return type

Tuple[Tensor, Tensor]

Returns

  • mean: [N, P]

  • variance: [N, P], [P, N, N], [N, P, P] or [N, P, N, P]

Please see gpflow.conditional._expand_independent_outputs for more information about the shape of the variance, depending on full_cov and full_output_cov.

gpflow.conditionals.conditional( object, SeparateIndependentInducingVariables, SharedIndependent, object )
gpflow.conditionals.conditional( object, SharedIndependentInducingVariables, SeparateIndependent, object )
gpflow.conditionals.conditional( object, SeparateIndependentInducingVariables, SeparateIndependent, object )
# dispatch to -> gpflow.conditionals.multioutput.conditionals.separate_independent_conditional(...)
gpflow.conditionals.multioutput.conditionals.separate_independent_conditional(Xnew, inducing_variable, kernel, f, *, full_cov=False, full_output_cov=False, q_sqrt=None, white=False)[source]#
Parameters
Return type

Tuple[Tensor, Tensor]

gpflow.conditionals.conditional( object, FallbackSharedIndependentInducingVariables, IndependentLatent, object )
gpflow.conditionals.conditional( object, FallbackSeparateIndependentInducingVariables, IndependentLatent, object )
# dispatch to -> gpflow.conditionals.multioutput.conditionals.fallback_independent_latent_conditional(...)
gpflow.conditionals.multioutput.conditionals.fallback_independent_latent_conditional(Xnew, inducing_variable, kernel, f, *, full_cov=False, full_output_cov=False, q_sqrt=None, white=False)[source]#

Interdomain conditional with independent latents. In this case the number of latent GPs (L) will be different than the number of outputs (P) The covariance matrices used to calculate the conditional have the following shape: - Kuu: [L, M, M] - Kuf: [M, L, N, P] - Kff: [N, P, N, P], [N, P, P], [N, P]

Further reference:

  • See gpflow.conditionals._conditional for a detailed explanation of conditional in the single-output case.

  • See the multioutput notebook for more information about the multioutput framework.

  • See above for the parameters and the return value.

Parameters
Return type

Tuple[Tensor, Tensor]

gpflow.conditionals.conditional( object, InducingPoints, MultioutputKernel, object )
# dispatch to -> gpflow.conditionals.multioutput.conditionals.inducing_point_conditional(...)
gpflow.conditionals.multioutput.conditionals.inducing_point_conditional(Xnew, inducing_variable, kernel, f, *, full_cov=False, full_output_cov=False, q_sqrt=None, white=False)[source]#

Multi-output GP with fully correlated inducing variables. The inducing variables are shaped in the same way as evaluations of K, to allow a default inducing point scheme for multi-output kernels. The covariance matrices used to calculate the conditional have the following shape: - Kuu: [M, L, M, L] - Kuf: [M, L, N, P] - Kff: [N, P, N, P], [N, P, P], [N, P]

Further reference:

  • See gpflow.conditionals._conditional for a detailed explanation of conditional in the single-output case.

  • See the multioutput notebook for more information about the multioutput framework.

Parameters

  • f (Tensor) – variational mean, [L, 1]

  • q_sqrt (Optional[Tensor]) – standard-deviations or cholesky, [L, 1] or [1, L, L]

  • Xnew (Tensor) –

  • inducing_variable (InducingPoints) –

  • kernel (MultioutputKernel) –

  • full_cov (bool) –

  • full_output_cov (bool) –

  • white (bool) –

Return type

Tuple[Tensor, Tensor]

gpflow.conditionals.conditional( object, SharedIndependentInducingVariables, LinearCoregionalization, object )
gpflow.conditionals.conditional( object, SeparateIndependentInducingVariables, LinearCoregionalization, object )
# dispatch to -> gpflow.conditionals.multioutput.conditionals.coregionalization_conditional(...)
gpflow.conditionals.multioutput.conditionals.coregionalization_conditional(Xnew, inducing_variable, kernel, f, *, full_cov=False, full_output_cov=False, q_sqrt=None, white=False)[source]#

Most efficient routine to project L independent latent gps through a mixing matrix W. The mixing matrix is a member of the LinearCoregionalization and has shape [P, L]. The covariance matrices used to calculate the conditional have the following shape: - Kuu: [L, M, M] - Kuf: [L, M, N] - Kff: [L, N] or [L, N, N]

Further reference:

  • See gpflow.conditionals._conditional for a detailed explanation of conditional in the single-output case.

  • See the multioutput notebook for more information about the multioutput framework.

Parameters
Return type

Tuple[Tensor, Tensor]

gpflow.conditionals.sample_conditional#

This function uses multiple dispatch, which will depend on the type of argument passed in:

gpflow.conditionals.sample_conditional( object, SharedIndependentInducingVariables, LinearCoregionalization, object )
# dispatch to -> gpflow.conditionals.multioutput.sample_conditionals._sample_conditional(...)
gpflow.conditionals.multioutput.sample_conditionals._sample_conditional(Xnew, inducing_variable, kernel, f, *, full_cov=False, full_output_cov=False, q_sqrt=None, white=False, num_samples=None)[source]#

sample_conditional will return a sample from the conditinoal distribution. In most cases this means calculating the conditional mean m and variance v and then returning m + sqrt(v) * eps, with eps ~ N(0, 1). However, for some combinations of Mok and Mof more efficient sampling routines exists. The dispatcher will make sure that we use the most efficent one. :rtype: Tuple[Tensor, Tensor, Tensor] :return: [N, P] (full_output_cov = False) or [N, P, P] (full_output_cov = True)

Parameters
gpflow.conditionals.sample_conditional( object, InducingVariables, Kernel, object )
gpflow.conditionals.sample_conditional( object, object, Kernel, object )
# dispatch to -> gpflow.conditionals.sample_conditionals._sample_conditional(...)
gpflow.conditionals.sample_conditionals._sample_conditional(Xnew, inducing_variable, kernel, f, *, full_cov=False, full_output_cov=False, q_sqrt=None, white=False, num_samples=None)[source]#

sample_conditional will return a sample from the conditional distribution. In most cases this means calculating the conditional mean m and variance v and then returning m + sqrt(v) * eps, with eps ~ N(0, 1). However, for some combinations of Mok and Mof more efficient sampling routines exists. The dispatcher will make sure that we use the most efficient one.

Return type

Tuple[Tensor, Tensor, Tensor]

Returns

samples, mean, cov samples has shape [num_samples, N, P] or [N, P] if num_samples is None mean and cov as for conditional()

Parameters

  • Xnew (Tensor) –

  • inducing_variable (InducingVariables) –

  • kernel (Kernel) –

  • f (Tensor) –

  • full_cov (bool) –

  • full_output_cov (bool) –

  • q_sqrt (Optional[Tensor]) –

  • white (bool) –

  • num_samples (Optional[int]) –

gpflow.conditionals.uncertain_conditional#

gpflow.conditionals.uncertain_conditional(Xnew_mu, Xnew_var, inducing_variable, kernel, q_mu, q_sqrt, *, mean_function=None, full_output_cov=False, full_cov=False, white=False)[source]#

Calculates the conditional for uncertain inputs Xnew, p(Xnew) = N(Xnew_mu, Xnew_var). See conditional documentation for further reference.

Parameters

  • Xnew_mu (Tensor) – mean of the inputs, size [N, D]in

  • Xnew_var (Tensor) – covariance matrix of the inputs, size [N, n, n]

  • inducing_variable (InducingVariables) – gpflow.InducingVariable object, only InducingPoints is supported

  • kernel (Kernel) – gpflow kernel object.

  • q_mu (Tensor) – mean inducing points, size [M, Dout]

  • q_sqrt (Tensor) – cholesky of the covariance matrix of the inducing points, size [t, M, M]

  • full_output_cov (bool) – boolean wheter to compute covariance between output dimension. Influences the shape of return value fvar. Default is False

  • white (bool) – boolean whether to use whitened representation. Default is False.

  • mean_function (Optional[MeanFunction]) –

  • full_cov (bool) –

Return fmean, fvar

mean and covariance of the conditional, size fmean is [N, Dout], size fvar depends on full_output_cov: if True f_var is [N, t, t], if False then f_var is [N, Dout]

Return type

Tuple[Tensor, Tensor]