Conjugate Bayes plays a key role in machine learning, ensuring computationally simple posterior updates. We generalize the variational principle behind this computation by using the Fenchel-Young (FY) inequality, which lets us define a generalized evidence lower bound beyond the KL geometry, and at equality, characterizes new generalized posteriors. A generalized notion of conjugacy follows, defined as closure of prior-potential families that are affine under translation by the loss. We show that such families not only preserve additive canonical updating, but also posterior inference with parameter-dependent supports and tuneably heavy tails. This combination is not feasible with standard conjugate Bayes. We derive a new FY generalization of expectation maximization (EM), which recovers classical and Hard EM, and amortized versions. We show examples, including generalized Bayesian linear regression with closed-form bounded support and heavy-tailed posteriors, and a Gaussian mixture with sparse cluster assignments.

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