But hey, maybe you’re just looking for some light reading material while you wait in line at the grocery store.

So let’s say you have a robot that needs to navigate through an environment and pick up objects along the way. Sounds simple enough, right? Well, it would be if there weren’t any obstacles or other agents in the environment making things more complicated. And what about the fact that your robot can only perceive its surroundings partially at best? This is where POMDPs come in they allow us to model and solve problems like this by taking into account all of these factors.

But how do we actually go about solving a POMDP? Well, one popular approach involves using policy iteration to find an optimal solution. And that’s where policy graphs come in handy! A policy graph is essentially just a fancy way of visualizing the different actions your robot can take and the resulting states it might end up in.

Here’s how you can create a policy graph for our little navigation problem:
1. Start by defining your POMDP model, which includes things like the state space, observation space, action space, transition probabilities, and reward function.
2. Use this model to generate a set of possible policies these are essentially just sequences of actions that your robot can take in response to different observations.
3. Evaluate each policy using some sort of performance metric (like expected cumulative reward) and choose the best one(s).
4. Iterate over steps 2-3 until you’ve converged on an optimal solution.
5. Use a policy graph to visualize your final solution, which should look something like this:

# This script represents a policy graph for a robot navigating through obstacles to pick up an object and reach a goal.

# The script starts with the [Start] node, which is the starting point for the robot.
[Start]

# The robot has two options at this point, either go to Obstacle1 or Obstacle2.
# The forward slash represents a decision point.
/ \

# Obstacle1 is represented by a node and the robot must navigate around it to reach the Pickup Object.
Obstacle1 --- Pickup Object

# Once the robot has picked up the object, it can move towards the Goal.
# The arrow represents the direction of movement.
--- Goal

# At this point, the robot has reached the Goal and the policy is complete.
# However, the script does not include a way for the robot to return to the starting point.

--- Goal
|       ^
v        |
[Obstacle2]     [Pickup Object]
|         |
v         v
[Goal]          [Start]

In this example, the policy graph shows that there are two possible paths for our robot to take one involves going around Obstacle1 and picking up an object before heading towards the goal, while the other involves avoiding both obstacles but missing out on the opportunity to pick up the object. The optimal solution (which is highlighted in green) takes into account all of these factors and finds a way to maximize expected cumulative reward over time.

For one thing, they can be pretty complex (especially if your problem involves lots of states or actions), which means that they might not always be the best way to represent your solution. And since policy iteration is an iterative process, it can take a long time to converge on an optimal solution especially for large POMDPs!

But hey, at least you’re learning something new today, right? Maybe next week we’ll talk about how to optimize our policies using reinforcement learning… or maybe not. Who knows what kind of crazy ideas I might come up with in the meantime?!