PSet 19: Hidden Markov Models

(videos: Hidden Markov Models and Forward Algorithm)

Recall the equation for computing HMM probabilities at any given time step $t$ to be:

$P(X_t | e_{1:t}) = \alpha P(e_t | X_t) \sum_{X_{t-1}} P(X_t | X_{t-1}) P(X_{t-1} | e_{1:t-1})$

Imagine an HMM that models the weather, as we’ve seen before as a Markov chain, the day is either rainy or cloudy or sunny, and the weather one day is a function of the previous. Imagine that you’re in a windowless room, but you have a barometer that is sensing outside.

We know the following:

$P(X_t|X_{t−1})$ for the transition probabilities is...

Using this table, $P(X_t=s | X_{t−1}=r) = 0.1$, not 0.2.

$P(E_t|X_t)$ for the emission probabilities is...

The initial distribution is $P(X1=r)=0.1$, $P(X1=c)=0.2$, $P(X1←s)=0.7$. Over the course of three days, we see three outputs from our barometer: E1=low,E2=med,E3=low.

Compute $P(X3|e_{1:3})$

$P(X3=r | e1,e2,e3) =$ ________
$P(X3=c | e1,e2,e3) =$ ________
$P(X3=s | e1,e2,e3) =$ ________