The Nature of Information II

The Nature of Information II

George Hrabovsky

MAST

Introduction

The previous article was concerned primary with a sequences of events that were mutually exclusive. What happens when more than one such sequence, or scheme, are combined?

Combining Schemes—Mutually Independent Sets

Let’s say that we have two finite schemes,

and

Say that the propbability of the joint occurence of events

 and

 

is

This property is called a mutually independent probability. In fact the set of events

forms another finite scheme. The entropy for such a scheme is,

Combining Schemes—Mutually Dependent Sets

So what if A and B are not mutually independent? Here we say that the probability of event

 
of the scheme B occurs assuming that event

 
occurs from scheme A is

This gives us the scheme

with entropy

It turns out that

 
is a random variable called the expectation of H(B) in the scheme A. It is also written

.

What does It all Mean?

The amount of information delivered by a scheme never decreases unless a another scheme is realized beforehand.