  Much progress has been made in semi-supervised learning (SSL) by combining
methods that exploit different aspects of the data distribution, e.g.
consistency regularisation relies on properties of $p(x)$, whereas entropy
minimisation pertains to the label distribution $p(y|x)$. Focusing on the
latter, we present a probabilistic model for discriminative SSL, that mirrors
its classical generative counterpart. Under the assumption $y|x$ is
deterministic, the prior over latent variables becomes discrete. We show that
several well-known SSL methods can be interpreted as approximating this prior,
and can be improved upon. We extend the discriminative model to neuro-symbolic
SSL, where label features satisfy logical rules, by showing such rules relate
directly to the above prior, thus justifying a family of methods that link
statistical learning and logical reasoning, and unifying them with regular SSL.
