feat: Implement beta likelihood

[thomaspinder]

Feature Request

Implement the Beta likelihood function. This can be easily achieved as the link function and integration strategy will be identical to the Bernoulli likelihood.

Related Code

https://github.com/JaxGaussianProcesses/GPJax/blob/main/gpjax/likelihoods.py#L170

[brandall25]

@thomaspinder can I have a go at this?

[thomaspinder]

Of course! Do let me know if you need a hand.

[brandall25]

Hi @thomaspinder , this is what I have so far:

@dataclass
class Beta(AbstractLikelihood):
    def link_function(self, alpha: float, beta: float) -> tfd.Distribution:
        r"""The probit link function of the Beta likelihood.

        Args:
            alpha : alpha parameter, indicating mean number of successes
            beta  : beta parameter, indicating mean number of failures

        Returns
        -------
            tfd.Distribution: The likelihood function.
        """
        return tfd.Beta(tf.constant(alpha), tf.constant(beta))

    def predict(self, dist: tfd.Distribution, alpha: float, beta: float) -> tfd.Distribution:
        r"""Evaluate the pointwise predictive distribution.

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

        Args:
            dist (tfd.Distribution): The Gaussian process posterior, evaluated
                at a finite set of test points.
            alpha (float) : alpha parameter, indicating mean number of successes
            beta  (float) : beta parameter, indicating mean number of failures

        Returns
        -------
            tfd.Distribution: The pointwise predictive distribution.
        """
        return self.link_function(alpha, beta)

Couple questions:

1. Should I keep alpha and beta as inputs to the predict function? Or, should I try to infer them from the base distribution class?
2. Do I need to specify any of the other arguments for the call to tfd.Beta()?

Thanks!

[thomaspinder]

Hi @brandall25. This is starting to look good! Are you able to open this as a PR to main? It'll be easier for me to make comments and give feedback there.