# gpytorch.likelihoods¶

## Likelihood¶

class gpytorch.likelihoods.Likelihood(max_plate_nesting=1)[source]

A Likelihood in GPyTorch specifies the mapping from latent function values $$f(\mathbf X)$$ to observed labels $$y$$.

For example, in the case of regression this might be a Gaussian distribution, as $$y(\mathbf x)$$ is equal to $$f(\mathbf x)$$ plus Gaussian noise:

$y(\mathbf x) = f(\mathbf x) + \epsilon, \:\:\:\: \epsilon \sim N(0,\sigma^{2}_{n} \mathbf I)$

In the case of classification, this might be a Bernoulli distribution, where the probability that $$y=1$$ is given by the latent function passed through some sigmoid or probit function:

$\begin{split}y(\mathbf x) = \begin{cases} 1 & \text{w/ probability} \:\: \sigma(f(\mathbf x)) \\ 0 & \text{w/ probability} \:\: 1-\sigma(f(\mathbf x)) \end{cases}\end{split}$

In either case, to implement a likelihood function, GPyTorch only requires a forward method that computes the conditional distribution $$p(y \mid f(\mathbf x))$$.

Calling this object does one of two things:

• If likelihood is called with a torch.Tensor object, then it is assumed that the input is samples from $$f(\mathbf x)$$. This returns the conditional distribution $$p(y|f(\mathbf x))$$.

• If likelihood is called with a MultivariateNormal object, then it is assumed that the input is the distribution $$f(\mathbf x)$$. This returns the marginal distribution $$p(y|\mathbf x)$$.

Parameters

max_plate_nesting (int, default=1) – (For Pyro integration only). How many batch dimensions are in the function. This should be modified if the likelihood uses plated random variables.

expected_log_prob(observations, function_dist, *args, **kwargs)[source]

(Used by VariationalELBO for variational inference.)

Computes the expected log likelihood, where the expectation is over the GP variational distribution.

$\sum_{\mathbf x, y} \mathbb{E}_{q\left( f(\mathbf x) \right)} \left[ \log p \left( y \mid f(\mathbf x) \right) \right]$
Parameters
• observations (torch.Tensor) – Values of $$y$$.

• function_dist (MultivariateNormal) – Distribution for $$f(x)$$.

• args – Additional args (passed to the foward function).

• kwargs – Additional kwargs (passed to the foward function).

Return type

torch.Tensor

abstract forward(function_samples, *args, data={}, **kwargs)[source]

Computes the conditional distribution $$p(\mathbf y \mid \mathbf f, \ldots)$$ that defines the likelihood.

Parameters
• function_samples (torch.Tensor) – Samples from the function ($$\mathbf f$$)

• data (dict {str: torch.Tensor}, optional - Pyro integration only) – Additional variables that the likelihood needs to condition on. The keys of the dictionary will correspond to Pyro sample sites in the likelihood’s model/guide.

Return type

Distribution (with same shape as function_samples )

log_marginal(observations, function_dist, *args, **kwargs)[source]

(Used by PredictiveLogLikelihood for approximate inference.)

Computes the log marginal likelihood of the approximate predictive distribution

$\sum_{\mathbf x, y} \log \mathbb{E}_{q\left( f(\mathbf x) \right)} \left[ p \left( y \mid f(\mathbf x) \right) \right]$

Note that this differs from expected_log_prob() because the $$log$$ is on the outside of the expectation.

Parameters
• observations (torch.Tensor) – Values of $$y$$.

• function_dist (MultivariateNormal) – Distribution for $$f(x)$$.

• args – Additional args (passed to the foward function).

• kwargs – Additional kwargs (passed to the foward function).

Return type

torch.Tensor

marginal(function_dist, *args, **kwargs)[source]

Computes a predictive distribution $$p(y^* | \mathbf x^*)$$ given either a posterior distribution $$p(\mathbf f | \mathcal D, \mathbf x)$$ or a prior distribution $$p(\mathbf f|\mathbf x)$$ as input.

With both exact inference and variational inference, the form of $$p(\mathbf f|\mathcal D, \mathbf x)$$ or $$p(\mathbf f| \mathbf x)$$ should usually be Gaussian. As a result, function_dist should usually be a MultivariateNormal specified by the mean and (co)variance of $$p(\mathbf f|...)$$.

Parameters
• function_dist (MultivariateNormal) – Distribution for $$f(x)$$.

• args – Additional args (passed to the foward function).

• kwargs – Additional kwargs (passed to the foward function).

Returns

The marginal distribution, or samples from it.

Return type

Distribution

pyro_guide(function_dist, target, *args, **kwargs)[source]

(For Pyro integration only).

Part of the guide function for the likelihood. This should be re-defined if the likelihood contains any latent variables that need to be infered.

Parameters
pyro_model(function_dist, target, *args, **kwargs)[source]

(For Pyro integration only).

Part of the model function for the likelihood. It should return the This should be re-defined if the likelihood contains any latent variables that need to be infered.

Parameters

## One-Dimensional Likelihoods¶

Likelihoods for GPs that are distributions of scalar functions. (I.e. for a specific $$\mathbf x$$ we expect that $$f(\mathbf x) \in \mathbb{R}$$.)

One-dimensional likelihoods should extend gpytoch.likelihoods._OneDimensionalLikelihood to reduce the variance when computing approximate GP objective functions. (Variance reduction is accomplished by using 1D Gauss-Hermite quadrature rather than MC-integration).

### GaussianLikelihood¶

class gpytorch.likelihoods.GaussianLikelihood(noise_prior=None, noise_constraint=None, batch_shape=torch.Size([]), **kwargs)[source]

The standard likelihood for regression. Assumes a standard homoskedastic noise model:

$p(y \mid f) = f + \epsilon, \quad \epsilon \sim \mathcal N (0, \sigma^2)$

where $$\sigma^2$$ is a noise parameter.

Note

This likelihood can be used for exact or approximate inference.

Parameters
• noise_prior (Prior, optional) – Prior for noise parameter $$\sigma^2$$.

• noise_constraint (Interval, optional) – Constraint for noise parameter $$\sigma^2$$.

• batch_shape (torch.Size, optional) – The batch shape of the learned noise parameter (default: []).

Variables

noise (torch.Tensor) – $$\sigma^2$$ parameter (noise)

### GaussianLikelihoodWithMissingObs¶

class gpytorch.likelihoods.GaussianLikelihoodWithMissingObs(**kwargs)[source]

The standard likelihood for regression with support for missing values. Assumes a standard homoskedastic noise model:

$p(y \mid f) = f + \epsilon, \quad \epsilon \sim \mathcal N (0, \sigma^2)$

where $$\sigma^2$$ is a noise parameter. Values of y that are nan do not impact the likelihood calculation.

Note

This likelihood can be used for exact or approximate inference.

Parameters
• noise_prior (Prior, optional) – Prior for noise parameter $$\sigma^2$$.

• noise_constraint (Interval, optional) – Constraint for noise parameter $$\sigma^2$$.

• batch_shape (torch.Size, optional) – The batch shape of the learned noise parameter (default: []).

Variables

noise (torch.Tensor) – $$\sigma^2$$ parameter (noise)

### FixedNoiseGaussianLikelihood¶

A Likelihood that assumes fixed heteroscedastic noise. This is useful when you have fixed, known observation noise for each training example.

Note that this likelihood takes an additional argument when you call it, noise, that adds a specified amount of noise to the passed MultivariateNormal. This allows for adding known observational noise to test data.

Note

This likelihood can be used for exact or approximate inference.

Parameters
• noise (torch.Tensor (... x N)) – Known observation noise (variance) for each training example.

• learn_additional_noise (bool, optional) – Set to true if you additionally want to learn added diagonal noise, similar to GaussianLikelihood.

• batch_shape (torch.Size, optional) – The batch shape of the learned noise parameter (default []) if learn_additional_noise=True.

Variables

noise (torch.Tensor) – $$\sigma^2$$ parameter (noise)

Example

>>> train_x = torch.randn(55, 2)
>>> noises = torch.ones(55) * 0.01
>>> pred_y = likelihood(gp_model(train_x))
>>>
>>> test_x = torch.randn(21, 2)
>>> test_noises = torch.ones(21) * 0.02
>>> pred_y = likelihood(gp_model(test_x), noise=test_noises)

Parameters

kwargs

### DirichletClassificationLikelihood¶

class gpytorch.likelihoods.DirichletClassificationLikelihood(targets, alpha_epsilon=0.01, learn_additional_noise=False, batch_shape=torch.Size([]), dtype=torch.float32, **kwargs)[source]

A classification likelihood that treats the labels as regression targets with fixed heteroscedastic noise. From Milios et al, NeurIPS, 2018 [https://arxiv.org/abs/1805.10915].

Note

This likelihood can be used for exact or approximate inference.

Parameters
• targets (torch.Tensor (N).) – classification labels.

• alpha_epsilon (int.) – tuning parameter for the scaling of the likeihood targets. We’d suggest 0.01 or setting via cross-validation.

• learn_additional_noise (bool, optional) – Set to true if you additionally want to learn added diagonal noise, similar to GaussianLikelihood.

• batch_shape (torch.Size, optional) – The batch shape of the learned noise parameter (default []) if learn_additional_noise=True.

Example

>>> train_x = torch.randn(55, 1)
>>> labels = torch.round(train_x).long()
>>> pred_y = likelihood(gp_model(train_x))
>>>
>>> test_x = torch.randn(21, 1)
>>> test_labels = torch.round(test_x).long()
>>> pred_y = likelihood(gp_model(test_x), targets=labels)

Parameters

dtype (torch.dtype, optional) –

### BernoulliLikelihood¶

class gpytorch.likelihoods.BernoulliLikelihood(*args, **kwargs)[source]

Implements the Bernoulli likelihood used for GP classification, using Probit regression (i.e., the latent function is warped to be in [0,1] using the standard Normal CDF $$\Phi(x)$$). Given the identity $$\Phi(-x) = 1-\Phi(x)$$, we can write the likelihood compactly as:

$\begin{equation*} p(Y=y|f)=\Phi(yf) \end{equation*}$

### BetaLikelihood¶

class gpytorch.likelihoods.BetaLikelihood(batch_shape=torch.Size([]), scale_prior=None, scale_constraint=None)[source]

A Beta likelihood for regressing over percentages.

The Beta distribution is parameterized by $$\alpha > 0$$ and $$\beta > 0$$ parameters which roughly correspond to the number of prior positive and negative observations. We instead parameterize it through a mixture $$m \in [0, 1]$$ and scale $$s > 0$$ parameter.

$\begin{equation*} \alpha = ms, \quad \beta = (1-m)s \end{equation*}$

The mixture parameter is the output of the GP passed through a logit function $$\sigma(\cdot)$$. The scale parameter is learned.

$p(y \mid f) = \text{Beta} \left( \sigma(f) s , (1 - \sigma(f)) s\right)$
Parameters
• batch_shape (torch.Size, optional) – The batch shape of the learned noise parameter (default: []).

• scale_prior (Prior, optional) – Prior for scale parameter $$s$$.

• scale_constraint (Interval, optional) – Constraint for scale parameter $$s$$.

Variables

scale (torch.Tensor) – $$s$$ parameter (scale)

### LaplaceLikelihood¶

class gpytorch.likelihoods.LaplaceLikelihood(batch_shape=torch.Size([]), noise_prior=None, noise_constraint=None)[source]

A Laplace likelihood/noise model for GP regression. It has one learnable parameter: $$\sigma$$ - the noise

Parameters
• batch_shape (torch.Size, optional) – The batch shape of the learned noise parameter (default: []).

• noise_prior (Prior, optional) – Prior for noise parameter $$\sigma$$.

• noise_constraint (Interval, optional) – Constraint for noise parameter $$\sigma$$.

Variables

noise (torch.Tensor) – $$\sigma$$ parameter (noise)

### StudentTLikelihood¶

class gpytorch.likelihoods.StudentTLikelihood(batch_shape=torch.Size([]), deg_free_prior=None, deg_free_constraint=None, noise_prior=None, noise_constraint=None)[source]

A Student T likelihood/noise model for GP regression. It has two learnable parameters: $$\nu$$ - the degrees of freedom, and $$\sigma^2$$ - the noise

Parameters
• batch_shape (torch.Size, optional) – The batch shape of the learned noise parameter (default: []).

• noise_prior (Prior, optional) – Prior for noise parameter $$\sigma^2$$.

• noise_constraint (Interval, optional) – Constraint for noise parameter $$\sigma^2$$.

• deg_free_prior (Prior, optional) – Prior for deg_free parameter $$\nu$$.

• deg_free_constraint (Interval, optional) – Constraint for deg_free parameter $$\nu$$.

Variables
• deg_free (torch.Tensor) – $$\nu$$ parameter (degrees of freedom)

• noise (torch.Tensor) – $$\sigma^2$$ parameter (noise)

## Multi-Dimensional Likelihoods¶

Likelihoods for GPs that are distributions of vector-valued functions. (I.e. for a specific $$\mathbf x$$ we expect that $$f(\mathbf x) \in \mathbb{R}^t$$, where $$t$$ is the number of output dimensions.)

A convenient extension of the GaussianLikelihood to the multitask setting that allows for a full cross-task covariance structure for the noise. The fitted covariance matrix has rank rank. If a strictly diagonal task noise covariance matrix is desired, then rank=0 should be set. (This option still allows for a different noise parameter for each task.)

Like the Gaussian likelihood, this object can be used with exact inference.

Note

At least one of has_global_noise or has_task_noise should be specified.

Parameters

• noise_covar – A model for the noise covariance. This can be a simple homoskedastic noise model, or a GP that is to be fitted on the observed measurement errors.

• rank – The rank of the task noise covariance matrix to fit. If rank is set to 0, then a diagonal covariance matrix is fit.

• task_prior – Prior to use over the task noise correlation matrix. Only used when $$\text{rank} > 0$$.

• batch_shape – Number of batches.

• has_global_noise – Whether to include a $$\sigma^2 \mathbf I_{nt}$$ term in the noise model.

• has_task_noise – Whether to include task-specific noise terms, which add $$\mathbf I_n \otimes \mathbf D_T$$ into the noise model.

### SoftmaxLikelihood¶

class gpytorch.likelihoods.SoftmaxLikelihood(num_features=None, num_classes=None, mixing_weights=True, mixing_weights_prior=None)[source]

Implements the Softmax (multiclass) likelihood used for GP classification.

$p(\mathbf y \mid \mathbf f) = \text{Softmax} \left( \mathbf W \mathbf f \right)$

$$\mathbf W$$ is a set of linear mixing weights applied to the latent functions $$\mathbf f$$.

Parameters
• num_features (int) – Dimensionality of latent function $$\mathbf f$$.

• num_classes (int) – Number of classes.

• mixing_weights (bool) – (Default: True) Whether to learn a linear mixing weight $$\mathbf W$$ applied to the latent function $$\mathbf f$$. If False, then $$\mathbf W = \mathbf I$$.

• mixing_weights_prior (Prior, optional) – Prior to use over the mixing weights $$\mathbf W$$.