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Markov Processes & Random Generation

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Abstractions about stochastic transitions between states — Markov kernels, chains, operators, and their categorical formalization as the category of Markov kernels — together with decision-making under partial observability via POMDPs, and the generation of randomness through pseudorandom seeds and physical entropy.

8 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

  • Category of Markov kernels — A category whose objects are measurable spaces and whose morphisms are Markov kernels, composed by integrating one conditional probability kernel through another.
  • Discrete-time Markov chain — A stochastic sequence whose next-state distribution depends on the current state and transition step but not on the earlier path once the present is known.
  • Markov kernel — A measurable assignment sending each source point to a probability measure on a target space, generalizing a stochastic transition matrix to arbitrary measurable spaces.
  • Markov operator — A positive mass-preserving operator that propagates probability densities, measures or observables through a stochastic transition.
  • Partially observable Markov decision process — A sequential decision model with Markovian hidden states, stochastic observations and actions chosen from observation histories or belief-state probability distributions.
  • Pushforward measure — The measure on a target measurable space obtained by assigning each target set the original measure of its preimage under a measurable map.
  • Random number generation — Produce symbols intended to be unpredictable or distributionally random by sampling physical entropy or evolving a deterministic pseudorandom state under an explicit seeding and output convention.
  • Random seed — The initial state value supplied to a pseudorandom generator so that it deterministically produces a reproducible sequence.