The process of inverting Markov kernels relates to the important subject of Bayesian modelling and learning. In fact, Bayesian update is exactly kernel inversion. ​We investigate how and when Markov kernels (aka stochastic relations, or probabilistic mappings) can be inverted. We address the question both directly on the category of measurable spaces, and indirectly by interpreting kernels as Markov operators: For the direct option, we introduce a typed version of the category of Markov kernels and use the so-called ‘disintegration of measures’. Here, one has to specialise to measurable spaces borne from a simple class of topological spaces -e.g. Polish spaces (other choices are possible). Our method and result greatly simplify a recent development ​by Culbertson and Sturz. For the operator option, we use a cone version of the category of Markov operators (​aka ​kernels seen as predicate transformers). That is to say, our linear operators are not just continuous, but are required to satisfy the stronger condition of being ω-chain-continuous.​ ​Prior work shows that one obtains an adjunction in the form of a pair of contravariant and inverse functors between the categories of L1- and L∞-cones. Inversion, seen through the operator prism, is just adjunction.​ ​No topological assumption is needed. We​ ​show​ ​that​ ​both​ ​categories​ ​(Markov​ ​kernels​ ​and​ ​ω-chain-continuous Markov operators) are related by a family of contravariant functors Tp for 1 ≤ p ≤ ∞. The Tp’s are Kleisli extensions of (duals of) conditional expectation functors introduced ​before. We​ prove​ ​that​ ​both​ ​notions​ ​of​ ​inversion agree when both defined: if f is a kernel, and f† its direct inverse, then T∞(f)† = T1(f†). This is a joint work with Fredrik Dahlqvist UCL, Ohad Kammar Oxford, Ilias Garnier ENS