Exact Formula for Test Diagnosis?

[kondrats]

Hi,
I have been struggling to understand what formula EXACTLY is used for the entropy-based measure in the test diagnostics.
Specifically, can you, please answer the questions (or confirm my guesses) below:

1) Assuming a zero cost, is the entropy based measure whose values are shown in the top-right part of the test diagnosis panel
given by what is known as "mutual information"

MI(X,Y) = Sum_x,y( P(X,Y) * Log_2[ P(X,Y)/P(X)/P(Y) ] ),

where X is the "fault" variable (about which we want to gain more certainty) and Y is the "symptom" variable (whose observation would provide additional certainty about X, P(X,Y) is the JPD of the two nodes, and the sums are over the states {xi} and {yj} of variables X and Y ?

2) If the above formula is correct, then the bars and values in the top right panel pertain to different "symptom" nodes Y. But the top left panel shows different states of each X (i.e. x1, x2, ..., xn) as "faults pursued". So, for a given state xi of variable X, how are the numbers in the top right panel related to the I(X,Y)? Is it a part of one of the sum? As a guess: are those numbers equal to

Sum_y( P(Y|X=xi) * Log_2[ P(X=xi,Y)/P(X=xi)/P(Y) ] ) ?

Thank you in advance for you answers.

Sergiy.

[shooltz[BayesFusion]]

1) SMILE performs diagnosis using 'faults' defined as node/outcome pair. This allows for modelling multiple faults as outcomes of single variable.

2) Assuming single pursued fault and no test cost, the algorithm for calculating test strength looks like this in pseudocode:

Code: Select all

// (F,X) pair is the node/outcome pursued
priorE = Entropy(P(F=X))
for each unobserved test T
	E = 0;
	for each outcome Y of T
		E += Entropy(P(F=X|T=Y)) * P(T=Y)
	end for
	testStrength(T) = abs(priorE - E) / priorE
end for

Entropy(p) = -p * Log2(p) - (1-p) * Log2(1-p)

The pseudocode above assumes that probabilities are calculated with previously observed tests included.

[kondrats]

Thank you very much for the informative answer.
It helps a lot.
Regards.
Sergiy.