1: Introduction

• Learning: Given data, D, learn (or fit, or estimate) a hypothesis, H (or model, or parameter(s)), for D.
  
  
• Prediction: Given input attribute(s) (or variable(s)), possibly trivial, predict output attribute(s).
  
  
• Data: Observations (values) drawn from some (sample|data) space.
  
  
• Typically
  • a statistical model is a formal mathematical model,
  • a machine learning method may be a complex model & use approximations,
  • data mining emphasises (very) large data sets, efficient & maybe ad hoc methods.
  Much overlap, and different terminology!

CSE454 2002 : This document is online at   http://www.csse.monash.edu.au/~lloyd/tilde/CSC4/CSE454/   and contains hyper-links to other resources - Lloyd Allison ©.

• Sample space is set of possible outcomes of some experiment
  
  
• e.g. examine 1st element of a gene;
  sample space = Base = {A, C, G, T}
  
  
• An event is a subset, possibly singular, of the sample space
  
  
• e.g. purine = {A, G}
  
  

NB. The term data space is often used, in machine learning.

• A random variable, X, takes values, with probabilities, from the sample space
  
  
• Write P(X=A) or just P(A) etc.
  
  
• e.g. P(X=A) = 0.4

Inference

People often distinguish between

• selecting a model class,
• selecting a model from a class,
• estimating the parameters of a model.

• model class = polynomials
• model = quadratic
• fully-parametrized model = 3 x² - 4.5 x + 7.2

Bayes

If B₁, B₂, ..., B_k is a partition of a set B (of causes) then

             P(A|Bi) P(Bi)
P(Bi|A) = -------------------     i=1, 2, ..., k
             k
          SUM   P(A|Bj) P(Bj)
           j=1

   k
SUM P(D|Hj) P(Hj) = P(D)
 j=1

P(Hi|D) = P(D|Hi) P(Hi) / P(D)   posterior

P(Hi|D)    P(D|Hi) P(Hi)
------- = --------------        posterior odds-ratio
P(Hj|D)    P(D|Hj) P(Hj)

• P(H_i)     prior probability of H_i
  
  
• P(H_i|D)     posterior probability of H_i
  
  
• P(D|H_i)     likelihood

NB. Can ignore P(H_i) in posterior odds-ratio if, and only if, P(H_i)=P(H_j). Maximum likelihood may can cause problems when we have inequality.

Example

C₁, a fair coin, P(H) = P(T) = 0.5.

C₂, a biased coin, P(H) = 2/3, P(T) = 1/3.

One of the coins is thrown 4 times, giving H, T, T, H.

Which coin was thrown? H₁ : was C₁.   H₂ : was C₂.

Prior, P(C₁) = P(C₂) = 0.5.

Likelihood, P(HTTH | C₁) = 1/16

and   P(HTTH | C₂) = 4/9 . 1/9 = 4/81.

Posterior odds-ratio, P(C₁|HTTH)/P(C₂|HTTH) = (1/16 . 1/2) / (4/81 . 1/2) = 81/64.

Now, P(C₁|HTTH) + P(C₂|HTTH) = 1

and if x/(1-x) = 81/64, then 64.x = 81 - 81.x, x = 81/145

P(C₁|HTTH) = 81/145.

This case is simple because the model space is discrete, in fact finite (2).

e.g. prediction

Know P(C₁) = 81/145,   P(C₂) = 64/145.

The more likely coin is C₁.

If we assumed the coin really was C₁, would predict P(H) = 0.5 in future.

But the coin might be C₂.

Should predict   P(H) = 81/145 . 1/2 + 64/145 . 2/3 = (243 + 256) / (145 . 6) = 499 / 870 = 0.57

i.e. use a weighted average of the hypotheses.

Conclusion

• data
• models, parameters
• priors, likelihood, posterior
• inference
• prediction