Markov decision process

Key takeaways

Markov decision process belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Markov decision process to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Markov decision process from memory before moving on to harder problems.

Reference excerpt

A Markov decision process (MDP) is a mathematical model for sequential decision making when outcomes are uncertain. It is a type of stochastic decision process, and is often solved using the methods of stochastic dynamic programming. Originating from operations research in the 1950s, MDPs have since gained recognition in a variety of fields, including ecology, economics, healthcare, telecommunications and reinforcement learning. Reinforcement learning utilizes the MDP framework to model the interaction between a learning agent and its environment. In this framework, the interaction is characterized by states, actions, and rewards. The MDP framework is designed to provide a simplified representation of key elements of artificial intelligence challenges. This modeling framework incorporates the understanding of cause and effect, the management of uncertainty and nondeterminism, and the pursuit of explicit goals. The name comes from its connection to Markov chains, a concept developed by the Russian mathematician Andrey Markov. The "Markov" in "Markov decision process" refers to the underlying structure of state transitions that still follow the Markov property. The process is called a "decision process" because it involves making decisions that influence these state transitions, extending the concept of a Markov chain into the realm of decision-making under uncertainty.

Definition

A Markov decision process is a 4-tuple ( S , A , P a , R a ) {\displaystyle (S,A,P_{a},R_{a})} , where:

S {\displaystyle S} is a set of states called the state space. The state space may be discrete or continuous, like the set of real numbers.

A {\displaystyle A} is a set of actions called the action space (alternatively, A s {\displaystyle A_{s}} is the set of actions available from state s {\displaystyle s} ). As for state, this set may be discrete or continuous.

P a ( s , s ′ ) {\displaystyle P_{a}(s,s')} is the probability that action a {\displaystyle a} in state s {\displaystyle s} at time t {\displaystyle t} will lead to state s ′ {\displaystyle s'} at time t + 1 {\displaystyle t+1} . In general, this probability transition is defined to satisfy Pr ( s t + 1 ∈ S ′ ∣ s t = s , a t = a ) = ∫ S ′ P a ( s , s ′ ) d s ′ , {\displaystyle \Pr(s_{t+1}\in S'\mid s_{t}=s,a_{t}=a)=\int _{S'}P_{a}(s,s')ds',} for every S ′ ⊆ S {\displaystyle S'\subseteq S} measurable. In case the state space is discrete, the integral is intended with respect to the counting measure, so that the latter simplifies as P a ( s , s ′ ) = Pr ( s t + 1 = s ′ ∣ s t = s , a t = a ) {\displaystyle P_{a}(s,s')=\Pr(s_{t+1}=s'\mid s_{t}=s,a_{t}=a)} ; in case S ⊆ R d {\displaystyle S\subseteq \mathbb {R} ^{d}} , the integral is usually intended with respect to the Lebesgue measure.

R a ( s , s ′ ) {\displaystyle R_{a}(s,s')} is the immediate reward (or expected immediate reward) received after action a {\displaystyle a} is taken to transition from state s {\displaystyle s} to state s ′ {\displaystyle s'} . The reward is in general a random variable. A policy function π {\displaystyle \pi } is a (potentially probabilistic) mapping from state space ( S {\displaystyle S} ) to action space ( A {\displaystyle A} ).

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Markov decision process

Start with the simplest possible case. Write down what Markov decision process claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Markov decision process before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Markov decision process ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Markov decision process

In research: Markov decision process appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Markov decision process in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Markov decision process is common in secondary-school and first-year university syllabi. It links to neighbouring topics Dynamic programming, Markov processes, Optimal decisions, so understanding it makes those chapters shorter.
In everyday life: Look for Markov decision process outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Markov decision process in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Markov decision process means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Markov decision process out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Markov decision process in simple terms?

A Markov decision process (MDP) is a mathematical model for sequential decision making when outcomes are uncertain. It is a type of stochastic decision process, and is often solved using the methods of stochastic dynamic programming.

Why does Markov decision process matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Markov decision process?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Markov decision process.