Decorated Linear Relations: Extending Gaussian Probability with Uninformative Priors

We introduce extended Gaussian distributions as a precise and principled way of combining Gaussian probability uninformative priors, which indicate complete absence of information. To give an extended Gaussian distribution on a finite-dimensional vector space $X$ is to give a subspace $D$, along which no information is known, together with a Gaussian distribution on the quotient $X/D$. We show that the class of extended Gaussians remains closed under taking conditional distributions. We then introduce decorated linear maps and relations as a general framework to combine probability with nondeterminism on vector spaces, which includes extended Gaussians as a special case. This enables us to apply methods from categorical logic to probability, and make connections to the semantics of probabilistic programs with exact conditioning.