Information etc.

Introduction

• Information
• Entropy
• Kullback Leibler distance
• Coding
• Minimum Message Length (MML) inference
• Estimators
• Inference v. prediction

CSE454 2005 : This document is online at   http://www.csse.monash.edu.au/~lloyd/tilde/CSC4/CSE454/   and contains hyper-links to other resources - Lloyd Allison ©.

Information

Defn: The information in learning of an event `A' of probability P(A) is I(A) = -log₂P(A) bits.

• P(A) = 1 => I(A) = 0
• P(B) = 0 => I(B) = ∞ bits(!)
• P(C) = 1/2 => I(C) = 1 bit
• also measured in nits (aka nats)   -log_eP(A)     NB. Maths easier
  (Alan Turing used base₁₀, the `ban' 3+ bits, and the deciban -- Hodges, Alan T' the Enigma, p197.)
• Learning: Zero bits of information are learned if event is known already

Entropy

Entropy is the average information in a probability distribution over a sample (data) space S.
Defn:

   H = - SUMv in S {P(v) log2 P(v)}

• -log₂ P(v)} is the information in v
• H is an average (weighted by P(v))

e.g. Fair coin: P(head) = P(tail) = 1/2;
H = -1/2 log₂(1/2) - 1/2 log₂(1/2)
= 1/2 + 1/2 = 1 bit

e.g. Biased coin: P(head) = 3/4, P(tail) = 1/4;
H = -3/4 log₂(3/4) - 1/4 log₂(1/4)
= 3/4{2 - log₂3} + 2/4
= 2 - 3/4 log₂3 < 1 bit

Theorem H1

if {p_i}_i=1..N and {q_i}_i=1..N are probability distributions then

     N
  SUM { - pi log2(qi) }
   i=1

Kullback Leibler distance

Defn: The K-L distance (also relative entropy) of prob' dist'n {q_i} from {p_i} is

     N         pi
  SUM pi log2( -- )  >= 0
   i=1         qi

Coding

We work with prefix codes: No code word is a prefix of another code word.

      Sender     ----------->   Receiver

                         +
 source     ----->  (0|1)   ----->   source
 symbols*   encode          decode   symbols*

Shannon (1948) Mathematical Theory of Communication

1. Any uniquely decodable code for S has avg length >= H

2. There exists a prefix code for S with avg length <=H+1.

Huffman (1952)

algorithm to construct code

Arithmetic coding

gets arbitrarily close to coding lower bound

compression = modelling + coding,   and coding is "solved"

Minimum Message Length (MML)

If we want to learn (infer) a hypothesis (theory | model) or parameter estimate(s), H, from data D

devise code for shortest expected two-part message

(i) H and then (ii) D|H

Note trade-off: complexity of H   v. complexity of D|H, both measured in bits, prevents over-fitting;

this issue includes precision used to state continuous-valued parameters.

Sender-receiver paradigm keeps us "honest"

      Sender     ----------->   Receiver

1. Sender and receiver get together and agree on priors, code-book, algorithms etc. Can discuss expected data, but have no "real" data.
   
2. Sender and receiver are separated.
   
3. Sender gets real data.
   
4. Sender uses code-book, algorithms etc. to encode data.
   
5. Receiver gets encoded data.
   
6. Receiver uses code-book, algorithms etc. to decode data.

Estimators

An estimator is a function from (a training set, i.e.) the sample (data) space, S, to the hypothesis (model, parameter) space H.

  e: S ---> H

• maximum likelihood estimator
• minimum message length (MML) estimator
• minimum expected Kullback-Leibler distance (min EKL) estimator
• maximum posterior estimator
• etc. ...

Prediction

Prediction is different from inference.

Inference (learning) is about finding an explanation (hypothesis, model, parameter est') for the data, prediction is about predicting future data and an explanation might or might not help.

The best thing that can be done in prediction is to use an average over all hypotheses weighted by their posterior probabilities.