Suppose we have a robot that can be in one of two states: state 1 or state 2. The robot moves between states according to the following rules:
If the robot is in state 1, it has a 70% chance of staying in state 1 and a 30% chance of transitioning to state 2.
If the robot is in state 2, it has a 60% chance of staying in state 2 and a 40% chance of transitioning to state 1.
The robot has two sensors: sensor 1 and sensor 2. The sensors are not perfect and sometimes give incorrect readings. The emission probabilities for each sensor are as follows:
If the robot is in state 1, sensor 1 gives the correct reading with a probability of 0.8 and an incorrect reading with a probability of 0.2. If the robot is in state 2, sensor 1 gives the correct reading with a probability of 0.4 and an incorrect reading with a probability of 0.6.
If the robot is in state 1, sensor 2 gives the correct reading with a probability of 0.3 and an incorrect reading with a probability of 0.7. If the robot is in state 2, sensor 2 gives the correct reading with a probability of 0.9 and an incorrect reading with a probability of 0.1.
Assume that the robot starts in state 1 with equal probability and generates the following sensor readings over time:
sensor 1: correct, correct, incorrect
sensor 2: incorrect, correct, correct
What is the most likely sequence of states that the robot went through to generate these observations?