In our system the evidence used during the reasoning is given by an annotation file. The annotation file describes the state of each manifestation in a given retinal image. The reasoning process also uses the opthalmologist's knowledge about the problem domain. This knowledge is summarized in the belief table as probability values of the form P(M=S/D). We will look at three formualtions that best suit our problem domain.

Bayes' Rule is given by P(A/B) = P(B/A)*P(A)/P(B). Substituting A by D to represent a diagnosis and substituting B by M to represent a manifestation, we apply the Bayes' Rule to our system as follows:

This formulation essentially means that the probability of a diagnosis being present given a manifestation in a particular state is equal to the probability of the manifestation in that particular state multiplied by the apriori probability of the diagnosis and scaled by the overall probability of the manifestation being in that particular state given all of the diagnoses. This formualtion can also be extended to include multiple manifestations. The formulation is then given by

In the formulation j=1..J refers to each of the total J manifestations and for each manifestation j, k represents the state of the manifestation. I refers to the total I diagnoses being considered.

Strict application of Bayes' Rule is problematic. The problems are as follows: The belief table contains many zeros. Since the Bayes' Rule is a multiplicative combination of all the evidence. Due to this fact even if one of the manifestations presented as evidence has zero probability, it invalidates the evidence provided by the other manifestations. Bayes' Rule also does not account for the influence of multiple diagnoses on the presence of a single manifestation. Bayes' Rule assumes that the values in the belief table are globally uniform, but this might not be the case in reality.

2.Noisy-Max-In any system like ours we observe that a single diagnosis need not be the only cause for the presence of the various manifestations.To include the possibility of multiple diagnoses we used the noisy MAX formulation. To diagnose the system using the noisy-max method we need the probability of a manifestation in a particular state under the influence of these multiple diagnoses.These combined probabilities have been denoted as the noisy MAX probabilities. In order to calculate the noisy MAX probabilities we consider all the possible combinations of diseases. We exclude mutually exclusive diagnoses from any diagnosis combination. We end up with 867 total diagnosis combinations. The formula to find these combined probabilities is as follows:-

In the formualtion C denotes the maximum number of diagnosis combinations. The noisy MAX probabilities so found are then used to diagnose the retinal image. This formualtion is computationally expensive. The computational complexity grows exponentially as the number of diagnoses is increased.

3.Normalized sums- The values in the belief table are assumed to be globally relative. Given the number of values in the belief table we question this assumption. The normalized sums method disregards the assumption of global relativity in the belief table. There are two steps involved in calculating the normalized sums. The first step is to normalize the probability values in the belief table. This normalization is done independantly for each state of each manifestation across the set of diagnoses. The belief values are normalized on a scale from zero to one.

The second step is to use the normalized values as indicators or contributors to the presence of a diagnosis. This concept is formalized as follows:

The term P(D_i/M_j=S) is no longer a probability. But the notation is used to indicate the similarity of the result to the belief in a diagnosis given a set of evidence. This method is summarized in the example below.

Let us assume that the original belief table for a manifestation in a particular state is as follows:

Diag1	Diag2	Diag3	Diag4	Diag5	Diag6	Diag7	Diag8	Diag9	Diag10	Diag11	Diag12	Diag13
0.2	0.1	0	0	0	0	0	0	0.5	0.3	0	0.1	0.1

Normalizing these values we get

Diag1	Diag2	Diag3	Diag4	Diag5	Diag6	Diag7	Diag8	Diag9	Diag10	Diag11	Diag12	Diag13
0.4	0.2	0	0	0	0	0	0	1	0.6	0	0.2	0.2

These values in the above table are the normalized values of P(M=S/D). The normalized sums for each of the 13 diagnoses is found by adding the corresponding normalized probabilities for all the 39 manifestations.

The normalized sums formulation is a novel reasoning approach. We believe that this is the first time the formulation has been used.

Fisher's Linear Discriminant Test:- The Fishers test is as follows: The 13 belief values are sorted in a descending order. These sorted values are partitioned into 2 sets A and B with the first 13 elements into the first set and the last element into the second set. Then the mean and the standard deviation for both the sets are calculated and the Fishers criterion is found as

In succeeding iterations the size of set A is decreased and that of set B is increased. The largest Fishers criterion so found is considered as the breakpoint and all the diagnoses in set B are considered as the system diagnosis. Sometimes more than one diagnosis can emerge as the cause for the presence of the manifestations. There is a maximum limit of 3 diagnoses that can be so indicated. This is to ensure that the system makes a definitive diagnosis rather than one which indicates all the diagnoses in the database as the cause. The results of the Fishers test are the resultant system diagnoses.