# MDP Definition

## Terminology and Notation

Let's begin with Markov Decision Process (MDP).

![MDP Notation](/files/-LiqrbUC3RrlCQokBHQf)

We choose action $$a\_t$$ at time $$t$$ when we saw observation $$o\_t$$, and the latent state $$s\_t$$ transfered to $$s\_{t+1}$$, and we got reward $$r(s,t)$$ from the environment.

## Definitions

### Fully Observed

Markov decision process $$\mathcal{M}={\mathcal{S,A,T,r}}$$

* $$\mathcal{S}$$ : state space; states $$s\in\mathcal{S}$$ (discrete or continuous)
* $$\mathcal{A}$$ : action space; actions $$a\in\mathcal{A}$$ (discrete or continuous)
* $$\mathcal{T}$$ : transition operator, a tensor
* $$r$$ : reward function; $$r(s\_t,a\_t):\mathcal{S}\times\mathcal{A}\to \mathbb{R}$$

### Partially Observed

partially observed Markov decision process $$\mathcal{M}={\mathcal{S,A,O,T,E, r}}$$

* $$\mathcal{S}$$ : state space; states $$s\in\mathcal{S}$$ (discrete or continuous)
* $$\mathcal{A}$$ : action space; actions $$a\in\mathcal{A}$$ (discrete or continuous)
* $$\mathcal{O}$$ : observation space; observations $$o\in\mathcal{O}$$ (discrete or continuous)
* $$\mathcal{T}$$ : transition operator, a tensor
* $$\mathcal{E}$$ : emission probability $$p(o\_t|s\_t)$$
* $$r$$ : reward function; $$r(s\_t,a\_t):\mathcal{S}\times\mathcal{A}\to \mathbb{R}$$


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