Markov Decisions Process

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A Markov decision process (MDP), is a discrete-time Stochastic Processes, and are an extension of Markov chains for decision making theory.

An MDP is made up of

• A set of states \(S\)
• A set of actions \(A\)
• A transition kernel \(P_a(s, s\prime) = Pr[s_{t+1} = s\prime | s_t = s, a_t = a]\)
• And the immediate reward function \(R_a(s, s\prime)\)

It is used in Reinforcement Learning as the core data generation process.

Markov Chain

See Probability course for more info.

Properties

Aperiodic

The period of a state is the largest integer \(d\) such that \(P(s_i, s_i)^n = 0\) for every n not divisible by \(d\). The period of i is shown as \(d(s_i)\). If \(P(s_i, s_i)^n = 0\) for all \(d\) then \(d(s_i) = \infty\).

for a state \(s_i\)

• Periodic: if \(d(s_i) > 1\)
• Aperiodic: if \(d(s_i) = 1\).

A Markov Chain is said to be aperiodic if all its states are aperiodic.

Irreducible

An irreducible Markov Chain or Markov Decisions Process is one in which the

Positive Recurrent

Ergodic

An MDP is said to be ergodic if all its states are ergodic. A state \(s_i\) is said to be ergodic if it is aperiodic and positive recurrent. In other words