MarkovKernel
plain-language theorem explainer
A Markov kernel on a finite set consists of nonnegative transition probabilities that sum to one along each row. Analysts of Markov chain mixing via overlap measures cite this interface when establishing contraction bounds. The declaration is a pure structure definition encoding these two properties with no further proof obligations.
Claim. A Markov kernel on a finite set $ι$ is a function $P : ι → ι → ℝ$ such that $P(i,j) ≥ 0$ for all $i,j$ and $∑_j P(i,j) = 1$ for every $i$.
background
The structure supplies the minimal data for row-stochastic matrices on finite state spaces, enabling direct computation of pairwise row overlaps as the sum of entrywise minima. Overlap is shown to be nonnegative and at most one using the nonnegativity and normalization fields. The module context is the derivation of total-variation contraction from lower bounds on these overlaps.
proof idea
The declaration is a structure definition. It introduces the transition map P together with the two axioms nonnegativity and row summation to one. No lemmas are applied and no tactics are used.
why it matters
This interface underpins the overlap computation and the total-variation contraction theorem in the same module. It is instantiated by markovOfMatrix from any row-stochastic nonnegative matrix. Within Recognition Science it supplies the discrete dynamical system needed for contraction arguments, though it remains independent of the phi-ladder and forcing chain.
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