Likelihoods

AbstractLikelihood

AbstractLikelihood(
    num_datapoints, integrator=GHQuadratureIntegrator()
)

Bases: Module

Abstract base class for likelihoods.

All likelihoods must inherit from this class and implement the predict and link_function methods.

Parameters:

• num_datapoints (int) –

  the number of data points.

• integrator (AbstractIntegrator, default: GHQuadratureIntegrator() ) –

  The integrator to be used for computing expected log likelihoods. Must be an instance of AbstractIntegrator.

__call__

__call__(dist)

Evaluate the likelihood function at a given predictive distribution.

Parameters:

• dist (Union[MultivariateNormal, GaussianDistribution]) –

  The predictive distribution to evaluate the likelihood at.

Returns:

• Distribution –

  The predictive distribution.

predict abstractmethod

predict(dist)

Evaluate the likelihood function at a given predictive distribution.

Parameters:

• dist (Union[MultivariateNormal, GaussianDistribution]) –

  The predictive distribution to evaluate the likelihood at.

Returns:

• Distribution –

  npd.Distribution: The predictive distribution.

link_function abstractmethod

link_function(f)

Return the link function of the likelihood function.

Parameters:

• f (Float[Array, '...']) –

  the latent Gaussian process values.

Returns:

• Distribution –

  npd.Distribution: The distribution of observations, y, given values of the Gaussian process, f.

expected_log_likelihood

expected_log_likelihood(
    y, mean, variance, mean_g=None, variance_g=None, **_
)

Compute the expected log likelihood.

For a variational distribution \(q(f)\sim\mathcal{N}(m, s)\) and a likelihood \(p(y|f)\), compute the expected log likelihood:

$$\mathbb{E}_{q(f)}\left[\log p(y|f)\right]$$

Parameters:

• y (Float[Array, 'N D']) –

  The observed response variable.

• mean (Float[Array, 'N D']) –

  The variational mean.

• variance (Float[Array, 'N D']) –

  The variational variance.

• mean_g (Float[Array, 'N D'], default: None ) –

  Optional moments of the latent noise process for heteroscedastic likelihoods.

• variance_g (Float[Array, 'N D'], default: None ) –

  Optional moments of the latent noise process for heteroscedastic likelihoods.

• **_ (Any, default: {} ) –

  Unused extra arguments for compatibility with specialised likelihoods.

Returns:

• ScalarFloat ( Float[Array, ' N'] ) –

  The expected log likelihood.

AbstractNoiseTransform

Bases: Module

Abstract base class for noise transformations.

__call__ abstractmethod

__call__(x)

Transform the input noise signal.

moments abstractmethod

moments(mean, variance)

Compute the moments of the transformed noise signal.

LogNormalTransform

Bases: AbstractNoiseTransform

Log-normal noise transformation.

SoftplusTransform

SoftplusTransform(num_points=20)

Bases: AbstractNoiseTransform

Softplus noise transformation.

AbstractHeteroscedasticLikelihood

AbstractHeteroscedasticLikelihood(
    num_datapoints,
    noise_prior,
    noise_transform=SoftplusTransform(),
    integrator=GHQuadratureIntegrator(),
)

Bases: AbstractLikelihood

Base class for heteroscedastic likelihoods with latent noise processes.

predict abstractmethod

predict(dist)

Evaluate the likelihood function at a given predictive distribution.

Parameters:

• dist (Union[MultivariateNormal, GaussianDistribution]) –

  The predictive distribution to evaluate the likelihood at.

Returns:

• Distribution –

  npd.Distribution: The predictive distribution.

link_function abstractmethod

link_function(f)

Return the link function of the likelihood function.

Parameters:

• f (Float[Array, '...']) –

  the latent Gaussian process values.

Returns:

• Distribution –

  npd.Distribution: The distribution of observations, y, given values of the Gaussian process, f.

supports_tight_bound

supports_tight_bound()

Return whether the tighter bound from Lázaro-Gredilla & Titsias (2011) is applicable.

noise_statistics

noise_statistics(mean, variance)

Moment matching of the transformed noise process.

Parameters:

• mean (Float[Array, 'N D']) –

  Mean of the latent noise GP.

• variance (Float[Array, 'N D']) –

  Variance of the latent noise GP.

Returns:

• NoiseMoments ( NoiseMoments ) –

  Expected log variance, inverse variance, and variance.

Gaussian

Gaussian(
    num_datapoints,
    obs_stddev=1.0,
    integrator=AnalyticalGaussianIntegrator(),
)

Bases: AbstractLikelihood

Gaussian likelihood object.

Parameters:

• num_datapoints (int) –

  the number of data points.

• obs_stddev (Union[ScalarFloat, Float[Array, '#N']], default: 1.0 ) –

  the standard deviation of the Gaussian observation noise.

• integrator (AbstractIntegrator, default: AnalyticalGaussianIntegrator() ) –

  The integrator to be used for computing expected log likelihoods. Must be an instance of AbstractIntegrator. For the Gaussian likelihood, this defaults to the AnalyticalGaussianIntegrator, as the expected log likelihood can be computed analytically.

__call__

__call__(dist)

Evaluate the likelihood function at a given predictive distribution.

Parameters:

• dist (Union[MultivariateNormal, GaussianDistribution]) –

  The predictive distribution to evaluate the likelihood at.

Returns:

• Distribution –

  The predictive distribution.

expected_log_likelihood

expected_log_likelihood(
    y, mean, variance, mean_g=None, variance_g=None, **_
)

Compute the expected log likelihood.

For a variational distribution \(q(f)\sim\mathcal{N}(m, s)\) and a likelihood \(p(y|f)\), compute the expected log likelihood:

$$\mathbb{E}_{q(f)}\left[\log p(y|f)\right]$$

Parameters:

• y (Float[Array, 'N D']) –

  The observed response variable.

• mean (Float[Array, 'N D']) –

  The variational mean.

• variance (Float[Array, 'N D']) –

  The variational variance.

• mean_g (Float[Array, 'N D'], default: None ) –

  Optional moments of the latent noise process for heteroscedastic likelihoods.

• variance_g (Float[Array, 'N D'], default: None ) –

  Optional moments of the latent noise process for heteroscedastic likelihoods.

• **_ (Any, default: {} ) –

  Unused extra arguments for compatibility with specialised likelihoods.

Returns:

• ScalarFloat ( Float[Array, ' N'] ) –

  The expected log likelihood.

link_function

link_function(f)

The link function of the Gaussian likelihood.

Parameters:

• f (Float[Array, '...']) –

  Function values.

Returns:

• Normal –

  npd.Normal: The likelihood function.

predict

predict(dist)

Evaluate the Gaussian likelihood.

Evaluate the Gaussian likelihood function at a given predictive distribution. Computationally, this is equivalent to summing the observation noise term to the diagonal elements of the predictive distribution's covariance matrix.

Parameters:

• dist (Distribution) –

  The Gaussian process posterior, evaluated at a finite set of test points.

Returns:

• MultivariateNormal –

  npd.Distribution: The predictive distribution.

Bernoulli

Bernoulli(
    num_datapoints, integrator=GHQuadratureIntegrator()
)

Bases: AbstractLikelihood

Parameters:

• num_datapoints (int) –

  the number of data points.

• integrator (AbstractIntegrator, default: GHQuadratureIntegrator() ) –

  The integrator to be used for computing expected log likelihoods. Must be an instance of AbstractIntegrator.

__call__

__call__(dist)

Evaluate the likelihood function at a given predictive distribution.

Parameters:

• dist (Union[MultivariateNormal, GaussianDistribution]) –

  The predictive distribution to evaluate the likelihood at.

Returns:

• Distribution –

  The predictive distribution.

expected_log_likelihood

expected_log_likelihood(
    y, mean, variance, mean_g=None, variance_g=None, **_
)

Compute the expected log likelihood.

For a variational distribution \(q(f)\sim\mathcal{N}(m, s)\) and a likelihood \(p(y|f)\), compute the expected log likelihood:

$$\mathbb{E}_{q(f)}\left[\log p(y|f)\right]$$

Parameters:

• y (Float[Array, 'N D']) –

  The observed response variable.

• mean (Float[Array, 'N D']) –

  The variational mean.

• variance (Float[Array, 'N D']) –

  The variational variance.

• mean_g (Float[Array, 'N D'], default: None ) –

  Optional moments of the latent noise process for heteroscedastic likelihoods.

• variance_g (Float[Array, 'N D'], default: None ) –

  Optional moments of the latent noise process for heteroscedastic likelihoods.

• **_ (Any, default: {} ) –

  Unused extra arguments for compatibility with specialised likelihoods.

Returns:

• ScalarFloat ( Float[Array, ' N'] ) –

  The expected log likelihood.

link_function

link_function(f)

The probit link function of the Bernoulli likelihood.

Parameters:

• f (Float[Array, '...']) –

  Function values.

Returns:

• BernoulliProbs –

  npd.Bernoulli: The likelihood function.

predict

predict(dist)

Evaluate the pointwise predictive distribution.

Evaluate the pointwise predictive distribution, given a Gaussian process posterior and likelihood parameters.

Parameters:

• dist ([npd.MultivariateNormal, GaussianDistribution].) –

  The Gaussian process posterior, evaluated at a finite set of test points.

Returns:

• BernoulliProbs –

  npd.Bernoulli: The pointwise predictive distribution.

Poisson

Poisson(
    num_datapoints, integrator=GHQuadratureIntegrator()
)

Bases: AbstractLikelihood

Parameters:

• num_datapoints (int) –

  the number of data points.

• integrator (AbstractIntegrator, default: GHQuadratureIntegrator() ) –

  The integrator to be used for computing expected log likelihoods. Must be an instance of AbstractIntegrator.

__call__

__call__(dist)

Evaluate the likelihood function at a given predictive distribution.

Parameters:

• dist (Union[MultivariateNormal, GaussianDistribution]) –

  The predictive distribution to evaluate the likelihood at.

Returns:

• Distribution –

  The predictive distribution.

expected_log_likelihood

expected_log_likelihood(
    y, mean, variance, mean_g=None, variance_g=None, **_
)

Compute the expected log likelihood.

For a variational distribution \(q(f)\sim\mathcal{N}(m, s)\) and a likelihood \(p(y|f)\), compute the expected log likelihood:

$$\mathbb{E}_{q(f)}\left[\log p(y|f)\right]$$

Parameters:

• y (Float[Array, 'N D']) –

  The observed response variable.

• mean (Float[Array, 'N D']) –

  The variational mean.

• variance (Float[Array, 'N D']) –

  The variational variance.

• mean_g (Float[Array, 'N D'], default: None ) –

  Optional moments of the latent noise process for heteroscedastic likelihoods.

• variance_g (Float[Array, 'N D'], default: None ) –

  Optional moments of the latent noise process for heteroscedastic likelihoods.

• **_ (Any, default: {} ) –

  Unused extra arguments for compatibility with specialised likelihoods.

Returns:

• ScalarFloat ( Float[Array, ' N'] ) –

  The expected log likelihood.

link_function

link_function(f)

The link function of the Poisson likelihood.

Parameters:

• f (Float[Array, '...']) –

  Function values.

Returns:

• Poisson –

  npd.Poisson: The likelihood function.

predict

predict(dist)

Evaluate the pointwise predictive distribution.

Evaluate the pointwise predictive distribution, given a Gaussian process posterior and likelihood parameters.

Parameters:

• dist (Union[MultivariateNormal, GaussianDistribution]) –

  The Gaussian process posterior, evaluated at a finite set of test points.

Returns:

• Poisson –

  npd.Poisson: The pointwise predictive distribution.

inv_probit

inv_probit(x)

Compute the inverse probit function.

Parameters:

• x (Float[Array, '*N']) –

  A vector of values.

Returns

Float[Array, "*N"]: The inverse probit of the input vector.