Lecture 2: Statistical learning primer for biologists Alan Qi Purdue Statistics and CS Jan. 15, 2009
Outline Basics for probability Regression Graphical models: Bayesian networks and Markov random fields Unsupervised learning: K-means and Expectation maximization
Probability Theory Sum Rule Product Rule
The Rules of Probability Sum Rule Product Rule
Bayes’ Theorem posterior likelihood × prior
Probability Density & Cumulative Distribution Functions
Expectations Conditional Expectation (discrete) Approximate Expectation (discrete and continuous)
Variances and Covariances
The Gaussian Distribution
Gaussian Mean and Variance
The Multivariate Gaussian
Gaussian Parameter Estimation Likelihood function
Maximum (Log) Likelihood
Properties of and Unbiased Biased
Curve Fitting Re-visited
Maximum Likelihood Determine by minimizing sum-of-squares error, .
Predictive Distribution
MAP: A Step towards Bayes Determine by minimizing regularized sum-of-squares error, .
Bayesian Curve Fitting
Bayesian Networks Directed Acyclic Graph (DAG)
Bayesian Networks General Factorization
Generative Models Causal process for generating images
Discrete Variables (1) General joint distribution: K 2 -1 parameters Independent joint distribution: 2(K-1) parameters
Discrete Variables (2) General joint distribution over M variables: KM -1 parameters M node Markov chain: K-1+(M-1)K(K-1) parameters
Discrete Variables: Bayesian Parameters (1)
Discrete Variables: Bayesian Parameters (2) Shared prior
Parameterized Conditional Distributions If are discrete, K-state variables, in general has O(K M) parameters. The parameterized form requires only M + 1 parameters
Conditional Independence a is independent of b given c Equivalently Notation
Conditional Independence: Example 1
Conditional Independence: Example 1
Conditional Independence: Example 2
Conditional Independence: Example 2
Conditional Independence: Example 3 Note: this is the opposite of Example 1, with c unobserved.
Conditional Independence: Example 3 Note: this is the opposite of Example 1, with c observed.
“Am I out of fuel?” B = Battery (0=flat, 1=fully charged) And hence B = Battery (0=flat, 1=fully charged) F = Fuel Tank (0=empty, 1=full) G = Fuel Gauge Reading (0=empty, 1=full)
“Am I out of fuel?” Probability of an empty tank increased by observing G = 0.
“Am I out of fuel?” Probability of an empty tank reduced by observing B = 0. This referred to as “explaining away”.
The Markov Blanket Factors independent of xi cancel between numerator and denominator.
Markov Random Fields Markov Blanket
Cliques and Maximal Cliques
Joint Distribution where is the potential over clique C and is the normalization coefficient; note: M K-state variables KM terms in Z. Energies and the Boltzmann distribution
Illustration: Image De-Noising (1) Original Image Noisy Image
Illustration: Image De-Noising (2)
Illustration: Image De-Noising (3) Noisy Image Restored Image (ICM)
Converting Directed to Undirected Graphs (1)
Converting Directed to Undirected Graphs (2) Additional links: “marrying parents”, i.e., moralization
Directed vs. Undirected Graphs (2)
Inference on a Chain Computational time increases exponentially with N.
Inference on a Chain
Supervised Learning Supervised learning: learning with examples or labels, e.g., classification and regression Linear regression (the example we just given), Generalized linear models (e.g, probit classification), Support vector machines, Gaussian processes classifications, etc. Take CS590M-Machine Learning in 2009 fall.
Unsupervised Learning Supervised learning: learning with examples or labels, e.g., classification and regression Unsupervised learning: learning without examples or labels, e.g., clustering, mixture models, PCA, non-negative matrix factorization
K-means Clustering: Goal
Cost Function
Two Stage Updates
Optimizing Cluster Assignment
Optimizing Cluster Centers
Convergence of Iterative Updates
Example of K-Means Clustering
Mixture of Gaussians Mixture of Gaussians: Introduce latent variables: Marginal distribution:
Conditional Probability Responsibility that component k takes for explaining the observation.
Maximum Likelihood Maximize the log likelihood function
Maximum Likelihood Conditions (1) Setting the derivatives of to zero:
Maximum Likelihood Conditions (2) Setting the derivative of to zero:
Maximum Likelihood Conditions (3) Lagrange function: Setting its derivative to zero and use the normalization constraint, we obtain:
Expectation Maximization for Mixture Gaussians Although the previous conditions do not provide closed-form conditions, we can use them to construct iterative updates: E step: Compute responsibilities . M step: Compute new mean , variance , and mixing coefficients . Loop over E and M steps until the log likelihood stops to increase.
Example EM on the Old Faithful data set.
General EM Algorithm
EM as Lower Bounding Methods Goal: maximize Define: We have
Lower Bound is a functional of the distribution . Since and , is a lower bound of the log likelihood function .
Illustration of Lower Bound
Lower Bound Perspective of EM Expectation Step: Maximizing the functional lower bound over the distribution . Maximization Step: Maximizing the lower bound over the parameters .
Illustration of EM Updates