Simplified use of Baye's Theorem in Computer Networks

**Notation:**

1. P(X) means the probability of occurence of event X.

2. P(X|Y) means the probability of occurence of event X provided event Y has already happened

3. An means A with subscript n.

4. 'sum'is the summation function. **∑**

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Introduction to Baye's Theorem

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Suppose for the occurence of an event A, n hypotheses are proposed like A1,A2,......,An. Now for 1≤k≤n,

**P(Ak|A)= [P(Ak)*P(A|Ak)] / [∑ (i=1 to n) {P(Ai)*P(A|Ai)} ]**

The probabilites P(Ai), i=1,2,.....,n are called *a priori probabilities* and are known before beginning the experiment. Note these are the probabilities of occurence of the hypotheses.

The probabilities P(A|Ai) are called *likelihoods* as they show how likely A is to occur given a priori probabilities.

The probabilites P(Ai|A) are called *posteriori probabilities* as they are obtained after the experiment.

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The Mathematical Model

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http://img295.imageshack.us/img295/1549/bayes0pg.png

The objective is to send data from A to B through the available nodes on the network.

Now P(Ci|A) implies probability of data being received correctly by Ci provided A sends it and P(A|Ci) implies probability of A sending the data provided Ci acknowledges the transmission. The acknowledgment will depend upon the state of the machine at node Ci. For e.g. if it is engaged in another transmission or is performing a resource hogging task it may not acknowledge the transmission. So P(A|Ci) is to be found out in real time by the network software.

Now applying the Baye's Theorem to the segment of data flow from A to Ci,

**P(Ci|A)= [P(Ci)*P(A|Ci)] / [∑ (i=1 to 4) {P(Ci)*P(A|Ci)} ]**

Before the beginning of data flow, the prbability of all Ci receiving data is equal i.e. P(C1)=P(C2)=P(C3)=P(C4)=**1/4**

From this the probability of Ci receiving the data correctly will be known and the node with P(Ci|A)=1 will recieve the datagrams. If none of them provides with absolute prbability i.e. 1 then the data will not be sent, A will wait for a few minutes or try other nodes in the network.

In the same way data from Ci will be sent to Di and at last to B.

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