7. Joint closure of prediction and Bayesian inference
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The finite categorical case admits an equally explicit construction that joins dynamics with Paper VI's inference refinement. This is useful because a state can be closed for prediction but too coarse for a new observation, or sufficient for an update but not for the next transition.
Let \(\Omega\) be a finite state set. Let \(K_u(i,j)\) be known row-stochastic transition matrices. Let \(\ell_y(i)>0\) be a finite family of observation likelihoods. A partition \(\pi\) has aggregation matrix
Distributions are column vectors. Prediction is \(p\mapsto K_u^\top p\), and a Bayesian update is
The coarse state must commute with both operations:
Fix a mandatory initial partition \(\pi_0\) with at least two blocks. This preserves distinctions that the intended application requires and excludes the trivial one-state quotient.
Proposition 9: Coarsest joint finite closure
First split \(\pi_0\) by the likelihood vectors \((\ell_y(i))_y\). Then repeatedly split each current block by the signatures
where \(B\) ranges over the current partition. The process terminates after at most \(|\Omega|-|\pi_{\mathrm{init}}|\) strict refinement rounds. The final partition is the unique coarsest refinement of \(\pi_0\) on which all declared predictions and Bayesian updates close for every prior.
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Proof
At a stable partition, states in a common block have the same transition probability into every coarse block. Those probabilities define \(\bar K_u\) and give the first commutation identity, by checking it on point-mass priors. Likelihood constancy on each block defines \(\bar\ell_y\) and gives the second identity by direct summation.
For minimality, every admissible partition refining \(\pi_0\) has at least two blocks. By Proposition 12 below, exact updating for all priors requires each fine likelihood to be constant within each block. Thus it refines the first likelihood split. Suppose it refines a current partition. Each current block is a union of its blocks. Exact prediction makes the probability of reaching each such union constant within any admissible block, so it also refines the next split. Induction proves it refines the final partition. Each strict refinement increases the number of blocks, which proves termination. \(\square\)
The transition condition is the finite lumpability criterion [R4], expressed directly as a commutation identity. The joint construction adds the explicitly declared likelihood distinctions. Its output is a quotient of a known finite model, not an automatically learned model of an unknown world.
The accompanying rational example starts with six states. Likelihoods create three blocks, transition distinctions force two further rounds of splitting, and the final model has five states. Two original states remain genuinely equivalent for every action and observation in the declared family. The example verifies both commuting identities exactly.