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Transcript of Lecture 2: Statistical learning primer for biologists Alan Qi Purdue Statistics and CS Jan. 15,...
![Page 1: Lecture 2: Statistical learning primer for biologists Alan Qi Purdue Statistics and CS Jan. 15, 2009.](https://reader035.fdocument.pub/reader035/viewer/2022062806/5697bfd11a28abf838cab3b6/html5/thumbnails/1.jpg)
Lecture 2: Statistical learning primer for biologists
Alan QiPurdue Statistics and CS
Jan. 15, 2009
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Outline
• Basics for probability• Regression• Graphical models: Bayesian networks and
Markov random fields• Unsupervised learning: K-means and
Expectation maximization
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Probability Theory
•Sum Rule
Product Rule
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The Rules of Probability
• Sum Rule
• Product Rule
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Bayes’ Theorem
posterior likelihood × prior
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Probability Density & Cumulative Distribution Functions
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Expectations
Conditional Expectation(discrete)
Approximate Expectation(discrete and continuous)
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Variances and Covariances
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The Gaussian Distribution
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Gaussian Mean and Variance
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The Multivariate Gaussian
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Gaussian Parameter Estimation
Likelihood function
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Maximum (Log) Likelihood
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Properties of and
Unbiased
Biased
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Curve Fitting Re-visited
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Maximum Likelihood
Determine by minimizing sum-of-squares error, .
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Predictive Distribution
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MAP: A Step towards Bayes
Determine by minimizing regularized sum-of-squares error, .
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Bayesian Curve Fitting
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Bayesian Networks
• Directed Acyclic Graph (DAG)
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Bayesian Networks
General Factorization
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Generative Models
• Causal process for generating images
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Discrete Variables (1)
• General joint distribution: K 2 -1 parameters
• Independent joint distribution: 2(K-1) parameters
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Discrete Variables (2)
General joint distribution over M variables: KM -1 parameters
M node Markov chain: K-1+(M-1)K(K-1) parameters
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Discrete Variables: Bayesian Parameters (1)
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Discrete Variables: Bayesian Parameters (2)
Shared prior
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Parameterized Conditional Distributions
If are discrete, K-state variables, in general has O(K M) parameters.
The parameterized form
requires only M + 1 parameters
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Conditional Independence
• a is independent of b given c
• Equivalently
• Notation
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Conditional Independence: Example 1
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Conditional Independence: Example 1
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Conditional Independence: Example 2
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Conditional Independence: Example 2
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Conditional Independence: Example 3
• Note: this is the opposite of Example 1, with c unobserved.
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Conditional Independence: Example 3
Note: this is the opposite of Example 1, with c observed.
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“Am I out of fuel?”
B = Battery (0=flat, 1=fully charged)F = Fuel Tank (0=empty, 1=full)G = Fuel Gauge Reading
(0=empty, 1=full)
And hence
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“Am I out of fuel?”
Probability of an empty tank increased by observing G = 0.
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“Am I out of fuel?”
Probability of an empty tank reduced by observing B = 0. This referred to as “explaining away”.
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The Markov Blanket
Factors independent of xi cancel between numerator and denominator.
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Cliques and Maximal Cliques
Clique
Maximal Clique
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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
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Illustration: Image De-Noising (1)
Original Image Noisy Image
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Illustration: Image De-Noising (2)
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Illustration: Image De-Noising (3)
Noisy Image Restored Image (ICM)
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Converting Directed to Undirected Graphs (1)
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Converting Directed to Undirected Graphs (2)
• Additional links: “marrying parents”, i.e., moralization
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Directed vs. Undirected Graphs (2)
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Inference on a Chain
Computational time increases exponentially with N.
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Inference on a Chain
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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.
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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
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K-means Clustering: Goal
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Cost Function
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Two Stage Updates
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Optimizing Cluster Assignment
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Optimizing Cluster Centers
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Convergence of Iterative Updates
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Example of K-Means Clustering
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Mixture of Gaussians• Mixture of Gaussians:
• Introduce latent variables:
• Marginal distribution:
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Conditional Probability
• Responsibility that component k takes for explaining the observation.
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Maximum Likelihood
• Maximize the log likelihood function
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Maximum Likelihood Conditions (1)
• Setting the derivatives of to zero:
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Maximum Likelihood Conditions (2)
• Setting the derivative of to zero:
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Maximum Likelihood Conditions (3)
• Lagrange function:
• Setting its derivative to zero and use the normalization constraint, we obtain:
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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.
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Example
• EM on the Old Faithful data set.
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General EM Algorithm
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EM as Lower Bounding Methods
• Goal: maximize
• Define:
• We have
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Lower Bound
• is a functional of the distribution .
• Since and ,• is a lower bound of the log likelihood
function .
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Illustration of Lower Bound
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Lower Bound Perspective of EM
• Expectation Step:• Maximizing the functional lower bound
over the distribution .
• Maximization Step:• Maximizing the lower bound over the
parameters .
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Illustration of EM Updates