Transcript prml-slides-1 - Simon Fraser University

Machine Learning
CMPT 726
Simon Fraser University
CHAPTER 1: INTRODUCTION
Outline
• Comments on general approach.
• Probability Theory.
• Joint, conditional and marginal probabilities.
• Random Variables.
• Functions of R.V.s
• Bernoulli Distribution (Coin Tosses).
• Maximum Likelihood Estimation.
• Bayesian Learning With Conjugate Prior.
• The Gaussian Distribution.
• Maximum Likelihood Estimation.
• Bayesian Learning With Conjugate Prior.
• More Probability Theory.
• Entropy.
• KL Divergence.
Our Approach
• The course generally follows statistics, very
interdisciplinary.
• Emphasis on predictive models: guess the value(s) of
target variable(s). “Pattern Recognition”
• Generally a Bayesian approach as in the text.
• Compared to standard Bayesian statistics:
• more complex models (neural nets, Bayes nets)
• more discrete variables
• more emphasis on algorithms and efficiency
Things Not Covered
• Within statistics:
• Hypothesis testing
• Frequentist theory, learning theory.
• Other types of data (not random samples)
• Relational data
• Scientific data (automated scientific discovery)
• Action + learning = reinforcement learning.
Could be optional – what do you think?
Probability Theory
Apples and Oranges
Probability Theory
Marginal Probability
Joint Probability
Conditional Probability
Probability Theory
Sum Rule
Product Rule
The Rules of Probability
Sum Rule
Product Rule
Bayes’ Theorem
posterior  likelihood × prior
Bayes’ Theorem: Model Version
• Let M be model, E be evidence.
•P(M|E) proportional to P(M) x P(E|M)
Intuition
• prior = how plausible is the event (model, theory) a
priori before seeing any evidence.
• likelihood = how well does the model explain the data?
Probability Densities
Transformed Densities
Expectations
Conditional Expectation
(discrete)
Approximate Expectation
(discrete and continuous)
Variances and Covariances
The Gaussian Distribution
Gaussian Mean and Variance
The Multivariate Gaussian
Reading exponential prob formulas
• In infinite space, cannot just form sum
Σx p(x)  grows to infinity.
• Instead, use exponential, e.g.
p(n) = (1/2)n
• Suppose there is a relevant feature f(x) and I
want to express that “the greater f(x) is, the
less probable x is”.
• Use p(x) = exp(-f(x)).
Example: exponential form sample size
• Fair Coin: The longer the sample size, the
less likely it is.
• p(n) = 2-n.
ln[p(n)]
Sample size n
Exponential Form: Gaussian mean
• The further x is from the mean, the less
likely it is.
ln[p(x)]
2(x-μ)
Smaller variance decreases probability
• The smaller the variance σ2, the less likely x
is (away from the mean).
ln[p(x)]
-σ2
Minimal energy = max probability
• The greater the energy (of the joint state),
the less probable the state is.
ln[p(x)]
E(x)
Gaussian Parameter Estimation
Likelihood function
Maximum (Log) Likelihood
Properties of
and
Curve Fitting Re-visited
Maximum Likelihood
Determine
by minimizing sum-of-squares error,
.
Predictive Distribution
Frequentism vs. Bayesianism
• Frequentists: probabilities are measured as the frequencies
of repeatable events.
• E.g., coin flips, snow falls in January.
• Bayesian: in addition, allow probabilities to be attached to
parameter values (e.g., P(μ=0).
• Frequentist model selection: give performance guarantees
(e.g., 95% of the time the method is right).
• Bayesian model selection: choose prior distribution over
parameters, maximize resulting cost function (posterior).
MAP: A Step towards Bayes
Determine
by minimizing regularized sum-of-squares error,
.
Bayesian Curve Fitting
Bayesian Predictive Distribution
Model Selection
Cross-Validation
Curse of Dimensionality
Rule of Thumb: 10 datapoints per parameter.
Curse of Dimensionality
Polynomial curve fitting, M = 3
Gaussian Densities in
higher dimensions
Decision Theory
Inference step
Determine either
or
.
Decision step
For given x, determine optimal t.
Minimum Misclassification Rate
Minimum Expected Loss
Example: classify medical images as ‘cancer’ or ‘normal’
Truth
Decision
Minimum Expected Loss
Regions
are chosen to minimize
Why Separate Inference and Decision?
• Minimizing risk (loss matrix may change over time)
• Unbalanced class priors
• Combining models
Decision Theory for Regression
Inference step
Determine
.
Decision step
For given x, make optimal
prediction, y(x), for t.
Loss function:
The Squared Loss Function
Generative vs Discriminative
Generative approach:
Model
Use Bayes’ theorem
Discriminative approach:
Model
directly
Entropy
Important quantity in
• coding theory
• statistical physics
• machine learning
Entropy
Entropy
Coding theory: x discrete with 8 possible states; how many
bits to transmit the state of x?
All states equally likely
Entropy
The Maximum Entropy Principle
• Commonly used principle for model selection: maximize
entropy.
• Example: In how many ways can N identical objects be
allocated M bins?
Entropy maximized when
Differential Entropy and the Gaussian
Put bins of width ¢ along the real line
Differential entropy maximized (for fixed
in which case
) when
Conditional Entropy
The Kullback-Leibler Divergence
Mutual Information