GS 540week 6

HMM basics

• Given a sequence, and state parameters:– Each possible path through the states has a certain

probability of emitting the sequence– P(O|M)

A C G T A G C T T T.04

.10

.02

.06

Probability of taking this state path

given t-probssequence(emissions)

state paths

.01

.04

.03

.08

.0004

.0040

.0006

.0048

Probability of emitting this

sequence from this state path given e-probs

Joint Probability

Viterbi Algorithm

A C G T A G C T T Tsequence

states

Highest weight path

.0004

.0040

.0006

.0048

Joint Probability

…

Viterbi Algorithm

A C G T A G C T T Tsequence

states …

Store at each node:• Likelihood of highest-likelihood path ending here• Traceback to previous node in path

Viterbi Algorithm

A C G T A G C T T Tsequence

states …

Pseudocode for Viterbi Algorithm

• Iterate over positions in sequence• At each position, fill in likelihood and

traceback• Compute emission and transition counts:– number of times state k emits letter s– number of times state k transitions to state k'

• Compute new parameter values

Likelihood function for Viterbi training

• Likelihood function for Viterbi:

• Different than likelihood function for EM:

Machine learning debugging tips• Debugging ML algorithms is much harder than

other code.• Problem: It is hard to verify the output after

each step.

Machine learning debugging tips• Strategies:– Work out a simple example by hand.– Implement a simpler, less efficient algorithm and

compare.– Compute your objective function (i.e. likelihood)

at each iteration.– Stare hard at your code.– Don't use your real data to test.

HW5 Tips

• Template is not a real solution!• Calculate L(M|O) by hand for the first few

observations, compare to your results– For each site, what’s the likelihood of each state?

states

A C G T

HW5 Tips

• Not necessary to explicitly create graph structure [although you may find it helpful]

states

HW5 Tips

• Template is not a real solution!• Calculate L(M|O) by hand for the first few

observations, compare to your results– For each site, what’s the likelihood of each state?

• Make a toy case:– AAAACCCCCCCCCCCCCCC– Easy to calculate L(M|O) by hand

• Use log space computations.

HW6: Baum-Welch

• Goal: learn HMM parameters taking into account all paths:

• Expectation maximization– Forward backward algorithm.– Re-estimate parameter values based on expected

counts.

Bayesian networks, dynamic Bayesian networks, hidden

Markov models

Hidden Markov model

...

Dynamic Bayesian network:More than one (hidden) random variable

per position

...

Bayesian network:Arbitrary structure of random variables

Probabilistic model inference algorithms

Problem: Given a model, what is the probability that a variable X has value x?• Belief propagation• HMMs: Forwards-backwards algorithmsProblem: Given a model, what is the most likely assignment of variables?• Maximum a posteriori (MAP) inference• Viterbi algorithm

Probabilistic model learning algorithms

Problem: Learn model parameters.• EM:

– E step: Use inference to get estimate of hidden variable values.– M step: Re-estimate parameter values.

• HMMs: Baum-Welch algorithm• Use belief propagation for inference: "soft EM"• Use Viterbi inference: "hard EM"

What type of inference should you use to get an estimate of latent variable values?

1. Belief propagation then choose most likely value for each variable.– Might get impossible set of values

2. Viterbi inference:– Might get unrealistic values.