Lecture 20: Learning Probabilistic Models

Lecture 20: Learning Probabilistic Models¶

AIMA Chapter 20 — 1 hour¶

Learning Objectives¶

• Learn Bayesian network parameters (ML, Bayesian)

• Apply EM for hidden variables

• Learn HMM parameters

• Learn network structure

Maximum Likelihood¶

• Data: D = {x¹,...,xᵐ}

• Likelihood: L(θ) = P(D|θ)

• ML estimate: θ* = argmax L(θ)

Bayesian Parameter Learning¶

• Prior: P(θ)

• Posterior: P(θ|D) ∝ P(D|θ) P(θ)

• Predict: P(x|D) = ∫ P(x|θ) P(θ|D) dθ

EM Algorithm¶

• Hidden variables: Z unobserved

• E-step: P(Z|X,θ)

• M-step: θ = argmax E[log P(X,Z|θ)]

• Convergence: Local optimum

EM: Mixture of Gaussians¶

• Components: K Gaussians

• Hidden: Which component each point

• E: Soft assignment

• M: Update means, covariances

Learning HMMs¶

• Baum-Welch: EM for HMM

• Parameters: A, B, π

• E: Forward-backward

• M: Update parameters

Summary¶

• ML: Maximize likelihood

• Bayesian: Posterior over parameters

• EM: Hidden variables

• HMM: Baum-Welch

References¶

• AIMA Ch. 20

• Russell & Norvig, AIMA 4e, Ch. 20

• Chapter PDF: chapters/chapter-20.pdf

Questions?¶

Next lecture: Deep Learning (Chapter 21)