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How many parameters are needed to describe this distribution?

Consider a distribution consisting of three binary variables which admit the factorisation: \( p(a, b, c) = p(a \mid b)p(b \mid c)p(c) \)

How many parameters are needed to specify distributions of this form?

Note that this is different to the general case where all variables might be dependent: \( p(a, b, c) = p(a \mid b, c)p(b \mid c)p(c) \)

In the case where we have no information about the variable dependency, we need to specify 8-1=7 parameters to define the distribution. There are \( 2^3 = 8 \) possible outcomes, but as the probabilities add to 1, we only need 7 parameters to fully define the distribution.

In the given case, b's value is sufficient to know the distribution of a. So some simplification is possible. A way to think about the situation is to consider 5 coins: \( c, {b_1, b_2}, {a_1, a_2} \). We will flip 3 coins starting with coin \( c \). The outcome of flipping \( c \) determines with coin \( b \) from \( {b_1, b_2} \) will be flipped. The outcome of flipping coin \( b \) determines which coin \( a \) from \( {a_1, a_2} \). As the coins can be represented as bernoilli random variables, they can be described with a single parameter representing the probability of landing heads. As there are 5 coins, we need 5 parameters to fully describe the distribution of the whole system.

Drawing a graph is another way to model this problem.


Bayesian Reasoning and Machine Learning
David Barber
Q 1.6