Directed graphs

In,

information channel with input X and output Y
symbol outputs Y may be the same as the input symbols X, i.e., X = Y.

However in most cases and hence in general,

X ≠ Y.
A convenient and useful way to represent the information channel is by directed graphs.

Example use of Directed Graphs:

If,

information channel with input X and output Y
such that, probabilities p(xi) are
p(x0) = 0.4
p(x1) = 0.6
and transitional probabilities p(yj|xi) are
p(y0|x0) = 0.80          p(y0|x1) = 0.00
p(y1|x0) = 0.15          p(y1|x1) = 0.05
p(y2|x0) = 0.05          p(y2|x1) = 0.15
p(y3|x0) = 0.00          p(y3|x1) = 0.80
Then, representing x0, x1, y0, y1, y2 and y3 by ⬤ such that,
raw (without arrows) directed graph

❶

The known transitional probabilities can help us draw directed arrows between ⬤ of X and ⬤ of Y.

Therefore,

directed graph with arrows with conditional probability labels
Also since p(xi) and p(yj|xi) are known, p(yj) is given by,
p(yj) = transitional probability × p(xi)
Hence,
p(y sub 0) = 0.32, p(y sub 1) = 0.09, p(y sub 2) = 0.11 and p(y sub 3) = 0.48
Therefore, the completed directed graph is
completed directed graph

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Next:

Determining information gained or lost (p:3) ➽

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