You might have heard of Markov Decision Processes (MDPs), which are like board games where the computer can see everything and knows exactly what to do at every step. But in real life, things aren’t always that simple. Sometimes you don’t know what state you’re in or what actions will lead to the best outcome. That’s where Partially Observable Markov Decision Processes (POMDPs) come in!

A POMDP is like an MDP with a blindfold on. You can still take actions, but you don’t always know exactly how those actions will affect your state or what the current state actually is. Instead of having perfect information, you have to make decisions based on probabilities and uncertainty.

Let me give you an example: imagine you’re a robot trying to navigate through a maze filled with obstacles. You can see some parts of the maze, but not all of it. Your goal is to reach a specific location in the shortest amount of time possible while avoiding collisions with walls and other objects.

To model this scenario as a POMDP, we’ll need to define our state space (the set of possible states), action space (the set of available actions), observation space (the set of possible observations), and transition function (which describes how the environment changes based on your actions).

In this case, our state space might include things like “in front of a wall,” “at an intersection,” or “near the goal.” Our action space could be things like “move forward,” “turn left,” or “turn right.” And our observation space would depend on what we can see from our current position.

The transition function is where things get interesting. Instead of knowing exactly how each action will affect your state, you have to make probabilistic predictions based on the current state and the chosen action. For example, if you’re in front of a wall and choose to turn left, there’s a certain probability that you’ll end up at an intersection or still be facing the wall (depending on how wide the maze is).

To solve this POMDP, we need to find the optimal policy which actions should we take in each state to reach our goal with the highest possible reward. This can be done using a technique called dynamic programming, specifically value iteration or policy iteration. These algorithms involve iteratively updating values for each state and action based on expected rewards and probabilities of transitioning between states.

But wait there’s more! POMDPs are notoriously difficult to solve because the number of possible states can be enormous (especially in complex environments like mazes or robotic systems). To make things even worse, we have to consider all possible actions and observations at each step, which leads to a huge state-action space.

To deal with this complexity, researchers have developed various techniques for approximating the optimal policy using heuristics or simplified models of the environment. One popular approach is called Monte Carlo Tree Search (MCTS), which involves searching through a tree of possible actions and observations to find the best one based on expected rewards.

They’re like MDPs, but with less information and more uncertainty. But don’t worry, we can still solve them using dynamic programming or approximation techniques. And who knows? Maybe someday robots will be able to navigate through mazes without getting lost!