# 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 ($$\mathbf{f} = \boldsymbol\Phi \mathbf{w}$$) Ey – expected value of y, $$\mathbb{E}[\mathbf{y}|\mathbf{f}]$$. 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 ($$\mathbf{f} = \boldsymbol\Phi \mathbf{w}$$) cdf – Cumulative density function evaluated at y. 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 ($$\mathbf{f} = \boldsymbol\Phi \mathbf{w}$$) df – the derivative $$\partial \log p(y|f) / \partial f$$ 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 ($$\mathbf{f} = \boldsymbol\Phi \mathbf{w}$$) 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. 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 ($$\mathbf{f} = \boldsymbol\Phi \mathbf{w}$$) logp – the log likelihood of each y given each f under this likelihood. 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 Ey – expected value of y, $$\mathbb{E}[\mathbf{y}|\mathbf{f}]$$. 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 cdf – Cumulative density function evaluated at y. 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 df – the derivative $$\partial \log p(y|f) / \partial f$$ 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 logp – the log likelihood of each y given each f under this likelihood. 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. Ey – expected value of y, $$\mathbb{E}[\mathbf{y}|\mathbf{f}]$$. 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. cdf – Cumulative density function evaluated at y. 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. df – the derivative $$\partial \log p(y|f) / \partial f$$ 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. dp – the derivative $$\partial \log p(y|f, \sigma^2)/ \partial \sigma^2$$ where $$sigma^2$$ is the variance. 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. logp – the log likelihood of each y given each f under this likelihood. 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 ($$\mathbf{f} = \boldsymbol\Phi \mathbf{w}$$) Ey – expected value of y, $$\mathbb{E}[\mathbf{y}|\mathbf{f}]$$. 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 ($$\mathbf{f} = \boldsymbol\Phi \mathbf{w}$$) cdf – Cumulative density function evaluated at y. 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 ($$\mathbf{f} = \boldsymbol\Phi \mathbf{w}$$) df – the derivative $$\partial \log p(y|f) / \partial f$$ ndarray
loglike(y, f)

Poisson log likelihood.

Parameters: y (ndarray) – array of integer targets f (ndarray) – latent function from the GLM prior ($$\mathbf{f} = \boldsymbol\Phi \mathbf{w}$$) logp – the log likelihood of each y given each f under this likelihood. ndarray