Overview and Probability Theory.

Machine Learning
CMPT 726
Simon Fraser University
• 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
• 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)
• 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
Conditional Expectation
Approximate Expectation
(discrete and continuous)
Expectations are Linear
• Let aX + bY + c be a linear combination of
two random variables (itself a random
• Then E[aX + bY + c] = aE[X] + bE[Y] + c.
• This holds whether or not X and Y are
• Good exercise to prove it.
Variances and Covariances
Think about this difference:
1.Everybody gets a B.
2.10 students get a C, 10 get an A.
The average is the same – how to quantify the difference?
Prove this. Hint: use the linearity of expectation.