gpflow.expectations

Modules

• gpflow.expectations.quadratures

Functions

gpflow.expectations.expectation

gpflow.expectations.expectation(p, obj1, obj2=None, nghp=None)

Compute the expectation <obj1(x) obj2(x)>_p(x) Uses multiple-dispatch to select an analytical implementation, if one is available. If not, it falls back to quadrature.

Parameters:

nghp (int) – passed to _quadrature_expectation to set the number of Gauss-Hermite points used: num_gauss_hermite_points

Return type:

Tensor

Returns:

a 1-D, 2-D, or 3-D tensor containing the expectation

Allowed combinations

• Psi statistics:

  >>> eKdiag = expectation(p, kernel)  (N)  # Psi0
  >>> eKxz = expectation(p, (kernel, inducing_variable))  (NxM)  # Psi1
  >>> exKxz = expectation(p, identity_mean, (kernel, inducing_variable))  (NxDxM)
  >>> eKzxKxz = expectation(p, (kernel, inducing_variable), (kernel, inducing_variable))  (NxMxM)  # Psi2
  

• kernels and mean functions:

  >>> eKzxMx = expectation(p, (kernel, inducing_variable), mean)  (NxMxQ)
  >>> eMxKxz = expectation(p, mean, (kernel, inducing_variable))  (NxQxM)
  

• only mean functions:

  >>> eMx = expectation(p, mean)  (NxQ)
  >>> eM1x_M2x = expectation(p, mean1, mean2)  (NxQ1xQ2)
  .. note:: mean(x) is 1xQ (row vector)
  

• different kernels. This occurs, for instance, when we are calculating Psi2 for Sum kernels:

  >>> eK1zxK2xz = expectation(p, (kern1, inducing_variable), (kern2, inducing_variable))  (NxMxM)
  

Parameters:

• p (Union[ProbabilityDistribution, Tuple[Union[ndarray[Any, Any], Tensor, Variable, Parameter], Union[ndarray[Any, Any], Tensor, Variable, Parameter]]]) –

• obj1 (Union[Kernel, MeanFunction, None, Tuple[Kernel, InducingVariables]]) –

• obj2 (Union[Kernel, MeanFunction, None, Tuple[Kernel, InducingVariables]]) –

• nghp (Optional[int]) –

gpflow.expectations.quadrature_expectation

gpflow.expectations.quadrature_expectation(p, obj1, obj2=None, nghp=None)

Compute the expectation <obj1(x) obj2(x)>_p(x) Uses Gauss-Hermite quadrature for approximate integration.

Parameters:

• num_gauss_hermite_points (int) – passed to _quadrature_expectation to set the number of Gauss-Hermite points used

• p (Union[ProbabilityDistribution, Tuple[Union[ndarray[Any, Any], Tensor, Variable, Parameter], Union[ndarray[Any, Any], Tensor, Variable, Parameter]]]) –

• obj1 (Union[Kernel, MeanFunction, None, Tuple[Kernel, InducingVariables]]) –

• obj2 (Union[Kernel, MeanFunction, None, Tuple[Kernel, InducingVariables]]) –

• nghp (Optional[int]) –

Return type:

Tensor

Returns:

a 1-D, 2-D, or 3-D tensor containing the expectation