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# 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 ©.

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# 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)   -logeP(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
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# 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))
 ^CSE454^ << [04] >> 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
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## Theorem H1

if {pi}i=1..N and {qi}i=1..N are probability distributions then

```     N
SUM { - pi log2(qi) }
i=1
```
is minimised when qi=pi.

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# Kullback Leibler distance

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

```     N         pi
SUM pi log2( -- )  >= 0
i=1         qi
```
NB. >=0 by H1. Not necessarily symmetric. (Use an integral for continuous prob' dist'ns.)
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# 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*
```

Average code-word length, i.e. message length, is SUMv in S{P(v) |code(v)|}.
Cannot be less than entropy H. This lower bound is achieved if |code(v)| = -log2P(v).

 ^CSE454^ << [08] >> 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"
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# 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.

 ^CSE454^ << [11] >> . . . MML Sender-receiver paradigm keeps us "honest" ``` Sender -----------> Receiver ``` Sender and receiver get together and agree on priors, code-book, algorithms etc. Can discuss expected data, but have no "real" data. Sender and receiver are separated. Sender gets real data. Sender uses code-book, algorithms etc. to encode data. Receiver gets encoded data. Receiver uses code-book, algorithms etc. to decode data. No hidden information passed "under the table". Safe: Cannot make a hypothesis look better than it really is. Data-compression techniques can be used to design a "good" code if an optimal code is not known.
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# 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
```
e.g
• maximum likelihood estimator
• minimum message length (MML) estimator
• minimum expected Kullback-Leibler distance (min EKL) estimator
• maximum posterior estimator
• etc. ...

different estimators may give different results in different situations.

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# 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.

© 2005 L. Allison, School of Computer Science and Software Engineering, Monash University, Australia 3168.
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