Likelihood Classes

Bernoulli	Bernoulli likelihood class for (binary) classification tasks.
Binomial	Binomial likelihood class.
Gaussian([var, bounds, shape])	A univariate Gaussian likelihood for general regression tasks.
Poisson([tranfcn])	A Poisson likelihood, useful for various Poisson process tasks.

Likelihood objects for inference within the GLM framework.

class revrand.likelihoods.Bernoulli

Bernoulli likelihood class for (binary) classification tasks.

A logistic transformation function is used to map the latent function from the GLM prior into a probability.

$$p(y_i | f_i) = \sigma(f_i) ^ {y_i} (1 - \sigma(f_i))^{1 - y_i}$$

where $y_i$ is a target, $f_i$ the value of the latent function corresponding to the target, and $\sigma(\cdot)$ is the logistic sigmoid.

Ey(f)

Expected value of the Bernoulli likelihood.

Parameters:f (ndarray) – latent function from the GLM prior ($\mathbf{f} = \boldsymbol\Phi \mathbf{w}$) Returns:Ey – expected value of y, $\mathbb{E}[\mathbf{y}|\mathbf{f}]$. Return type:ndarray

cdf(y, f)

Cumulative density function of the likelihood.

Parameters:

• y (ndarray) – query quantiles, i.e. $P(Y \leq y)$.
• f (ndarray) – latent function from the GLM prior ($\mathbf{f} = \boldsymbol\Phi \mathbf{w}$)

Returns:

cdf – Cumulative density function evaluated at y.

Return type:

ndarray

df(y, f)

Derivative of Bernoulli log likelihood w.r.t. f.

Parameters:

• y (ndarray) – array of 0, 1 valued integers of targets
• f (ndarray) – latent function from the GLM prior ($\mathbf{f} = \boldsymbol\Phi \mathbf{w}$)

Returns:

df – the derivative $\partial \log p(y|f) / \partial f$

Return type:

ndarray

dp(y, f, *args)

Derivative of Bernoulli log likelihood w.r.t.the parameters, $\theta$.

Parameters:

• y (ndarray) – array of 0, 1 valued integers of targets
• f (ndarray) – latent function from the GLM prior ($\mathbf{f} = \boldsymbol\Phi \mathbf{w}$)

Returns:

dp – the derivative $\partial \log p(y|f, \theta)/ \partial \theta$ for each parameter. If there is only one parameter, this is not a list.

Return type:

list, float or ndarray

loglike(y, f)

Bernoulli log likelihood.

Parameters:

• y (ndarray) – array of 0, 1 valued integers of targets
• f (ndarray) – latent function from the GLM prior ($\mathbf{f} = \boldsymbol\Phi \mathbf{w}$)

Returns:

logp – the log likelihood of each y given each f under this likelihood.

Return type:

ndarray

params

Get this object’s Parameter types.

class revrand.likelihoods.Binomial

Binomial likelihood class.

A logistic transformation function is used to map the latent function from the GLM prior into a probability.

$$p(y_i | f_i) = \genfrac(){0pt}{}{n}{y_i} \sigma(f_i) ^ {y_i} (1 - \sigma(f_i))^{n - y_i}$$

where $y_i$ is a target, $f_i$ the value of the latent function corresponding to the target, $n$ is the total possible count, and $\sigma(\cdot)$ is the logistic sigmoid. $n$ can also be applied per observation.

Ey(f, n)

Expected value of the Binomial likelihood.

Parameters:

• f (ndarray) – latent function from the GLM prior ($\mathbf{f} = \boldsymbol\Phi \mathbf{w}$)
• n (ndarray) – the total number of observations

Returns:

Ey – expected value of y, $\mathbb{E}[\mathbf{y}|\mathbf{f}]$.

Return type:

ndarray

cdf(y, f, n)

Cumulative density function of the likelihood.

Parameters:

• y (ndarray) – query quantiles, i.e. $P(Y \leq y)$.
• f (ndarray) – latent function from the GLM prior ($\mathbf{f} = \boldsymbol\Phi \mathbf{w}$)
• n (ndarray) – the total number of observations

Returns:

cdf – Cumulative density function evaluated at y.

Return type:

ndarray

df(y, f, n)

Derivative of Binomial log likelihood w.r.t. f.

Parameters:

• y (ndarray) – array of 0, 1 valued integers of targets
• f (ndarray) – latent function from the GLM prior ($\mathbf{f} = \boldsymbol\Phi \mathbf{w}$)
• n (ndarray) – the total number of observations

Returns:

df – the derivative $\partial \log p(y|f) / \partial f$

Return type:

ndarray

loglike(y, f, n)

Binomial log likelihood.

Parameters:

• y (ndarray) – array of 0, 1 valued integers of targets
• f (ndarray) – latent function from the GLM prior ($\mathbf{f} = \boldsymbol\Phi \mathbf{w}$)
• n (ndarray) – the total number of observations

Returns:

logp – the log likelihood of each y given each f under this likelihood.

Return type:

ndarray

class revrand.likelihoods.Gaussian(var=Parameter(value=1.0, bounds=Positive(upper=None), shape=()))

A univariate Gaussian likelihood for general regression tasks.

No transformation function is needed since this is (conditionally) conjugate to the GLM prior.

$$p(y_i | f_i) = \frac{1}{\sqrt{2 \pi \sigma^2}} \exp\left(- \frac{(y_i - f_i)^2}{2 \sigma^2} \right)$$

where $y_i$ is a target, $f_i$ the value of the latent function corresponding to the target and $\sigma$ is the observation noise (standard deviation).

Parameters:var (Parameter, optional) – A scalar Parameter describing the initial point and bounds for an optimiser to learn the variance parameter of this object.

Ey(f, var)

Expected value of the Gaussian likelihood.

Parameters:

• f (ndarray) – latent function from the GLM prior ($\mathbf{f} = \boldsymbol\Phi \mathbf{w}$)
• var (float, ndarray, optional) – The variance of the distribution, if not input, the initial value of variance is used.

Returns:

Ey – expected value of y, $\mathbb{E}[\mathbf{y}|\mathbf{f}]$.

Return type:

ndarray

cdf(y, f, var)

Cumulative density function of the likelihood.

Parameters:

• y (ndarray) – query quantiles, i.e. $P(Y \leq y)$.
• f (ndarray) – latent function from the GLM prior ($\mathbf{f} = \boldsymbol\Phi \mathbf{w}$)
• var (float, ndarray, optional) – The variance of the distribution, if not input, the initial value of variance is used.

Returns:

cdf – Cumulative density function evaluated at y.

Return type:

ndarray

df(y, f, var)

Derivative of Gaussian log likelihood w.r.t. f.

Parameters:

• y (ndarray) – array of 0, 1 valued integers of targets
• f (ndarray) – latent function from the GLM prior ($\mathbf{f} = \boldsymbol\Phi \mathbf{w}$)
• var (float, ndarray, optional) – The variance of the distribution, if not input, the initial value of variance is used.

Returns:

df – the derivative $\partial \log p(y|f) / \partial f$

Return type:

ndarray

dp(y, f, var)

Derivative of Gaussian log likelihood w.r.t.the variance $\sigma^2$.

Parameters:

• y (ndarray) – array of 0, 1 valued integers of targets
• f (ndarray) – latent function from the GLM prior ($\mathbf{f} = \boldsymbol\Phi \mathbf{w}$)
• var (float, ndarray, optional) – The variance of the distribution, if not input, the initial value of variance is used.

Returns:

dp – the derivative $\partial \log p(y|f, \sigma^2)/ \partial \sigma^2$ where $sigma^2$ is the variance.

Return type:

float

loglike(y, f, var=None)

Gaussian log likelihood.

Parameters:

• y (ndarray) – array of 0, 1 valued integers of targets
• f (ndarray) – latent function from the GLM prior ($\mathbf{f} = \boldsymbol\Phi \mathbf{w}$)
• var (float, ndarray, optional) – The variance of the distribution, if not input, the initial value of variance is used.

Returns:

logp – the log likelihood of each y given each f under this likelihood.

Return type:

ndarray

class revrand.likelihoods.Poisson(tranfcn='exp')

A Poisson likelihood, useful for various Poisson process tasks.

An exponential transformation function and a softplus transformation function have been implemented.

$$p(y_i | f_i) = \frac{g(f_i)^{y_i} e^{-g(f_i)}}{y_i!}$$

where $y_i$ is a target, $f_i$ the value of the latent function corresponding to the target, and $g(\cdot)$ is the tranformation function, which can be either an exponential function, or a softplus function ($\log(1 + \exp(f_i)$).

Parameters:tranfcn (string, optional) – this may be ‘exp’ for an exponential transformation function, or ‘softplus’ for a softplut transformation function.

Ey(f)

Expected value of the Poisson likelihood.

Parameters:f (ndarray) – latent function from the GLM prior ($\mathbf{f} = \boldsymbol\Phi \mathbf{w}$) Returns:Ey – expected value of y, $\mathbb{E}[\mathbf{y}|\mathbf{f}]$. Return type:ndarray

cdf(y, f)

Cumulative density function of the likelihood.

Parameters:

• y (ndarray) – query quantiles, i.e. $P(Y \leq y)$.
• f (ndarray) – latent function from the GLM prior ($\mathbf{f} = \boldsymbol\Phi \mathbf{w}$)

Returns:

cdf – Cumulative density function evaluated at y.

Return type:

ndarray

df(y, f)

Derivative of Poisson log likelihood w.r.t. f.

Parameters:

• y (ndarray) – array of 0, 1 valued integers of targets
• f (ndarray) – latent function from the GLM prior ($\mathbf{f} = \boldsymbol\Phi \mathbf{w}$)

Returns:

df – the derivative $\partial \log p(y|f) / \partial f$

Return type:

ndarray

loglike(y, f)

Poisson log likelihood.

Parameters:

• y (ndarray) – array of integer targets
• f (ndarray) – latent function from the GLM prior ($\mathbf{f} = \boldsymbol\Phi \mathbf{w}$)

Returns:

logp – the log likelihood of each y given each f under this likelihood.

Return type:

ndarray