Mathematical Framework of Partially Observable Markov Decision Process
The decision process in a POMDP is a cycle of states, actions, and observations. At each time step, the agent:
- Observes a signal that partially reveals the state of the environment.
- Chooses an action based on the accumulated observations.
- Receives a reward dependent on the action and the underlying state.
- Moves to a new state based on the transition model.
The key challenge in a POMDP is that the agent does not know its exact state but has a belief or probability distribution over the possible states. This belief is updated using the Bayes’ rule as new observations are made, forming a belief update rule:
[Tex]Bel(s’) =\frac{ P(o|s’,a) \sum_s P(s’|s,a) Bel(s)}{P(o|a, Bel)}[/Tex]
Where:
- Bel(s) is the prior belief of being in state s.
- Bel(s′) is the updated belief after observing o and taking action a.
Partially Observable Markov Decision Process (POMDP) in AI
Partially Observable Markov Decision Process (POMDP) is a mathematical framework employed for decision-making in situations of uncertainty, where the decision-maker lacks complete information or noisy information regarding the current state of the environment. POMDPs have broad applicability in diverse domains such as robotics, healthcare, finance, and others.
This article provides an in-depth overview of Partially Observable Markov Decision Processes (POMDPs), their components, mathematical framework, solving strategies, and practical application in maze navigation using Python.
Table of Content
- What is Partially Observable Markov Decision Process (POMDP)?
- Mathematical Framework of Partially Observable Markov Decision Process
- Markov Decision Process vs POMDP
- Strategies for Solving Partially Observable Markov Decision Processes
- Exploring Maze Navigation with Partially Observable Markov Decision Processes in Python
- Conclusion
Pre-Requisites
- Probability theory: Probability theory is applied to POMDPs to model the uncertainty surrounding the observations made by the agent and the changes in state within the environment.
- Markov processes: A Markov process, sometimes referred to as a Markov chain, is a stochastic model that depicts how a system changes over time. It assumes that the system’s future state is solely dependent on its current state and not on the preceding set of events.
- Decision theory: Taking into account the trade-offs between various actions and their possible outcomes, decision theory offers a framework for making decisions under uncertainty.
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