# What family is the posterior predictive distribution in when the likelihood is a Bernoulli and the prior (and posterior) is Gaussian?

I have a problem where the hidden random variables I want to know about are continuous (i.e. my parameters can take any real number), and so I have modeled them as having a Gaussian p.d.f. (because this is typically done in my context). However, the likelihood is a Bernoulli (or binomial), given that I want to perform binary classification. In other words, you can imagine that there's the data-generating process $$\tilde{x} \sim p(x \mid \theta)$$, where $$x$$ is discrete and $$\theta$$ is continuous, so $$p(\theta)$$ is a Gaussian and the likelihood $$\theta \mapsto p(\tilde{x} \mid \theta)$$ is a Bernoulli.

Now, I know that a Bernoulli likelihood with a beta prior leads to a beta posterior and that a Gaussian prior and a Gaussian likelihood lead to a Gaussian posterior (i.e. these are the conjugate priors).

However, what family is the posterior predictive distribution (p.p.d.) in when the likelihood is a Bernoulli (or binomial) and the prior and sampling distributions (i.e. the distribution I use to compute the p.p.d.) are Gaussians? Why do I want to know this? Because I want to estimate the p.p.d. by sampling and I want to know which moments I can estimate. For example, would it make sense to compute the mean or variance of the samples if the p.p.d. is not a Gaussian? Probably not.

Now, in my case, I am approximating the posterior with a Gaussian, so the posterior I will use the compute the p.p.d. will be a Gaussian, but the likelihood will still be a Bernoulli.