Question

When is the Bayes' rule (not the Bayes' theorem) optimal? Explain the meaning of that by using a
2x2 confusion matrix

Answer 1

Bayes Rule is as follows:-

through marginilization we have :-

Where = posterior probability, = likelihood,

= prior probability and = evidence

The optimal decision rule to choose between the two hypothesis is as follows:-

If the prior probabilities are fixed, then decide H_i if > for all ij

we know that optimal decision rule gives the minimum error rate possible.

The bayes rule with more information is as follows

Decide H_i if

Example of Confusion matrix

	=1	=0
l=1	ppv	fdr
l=0	for	npv

represents true disease status (1=disease present, 0=disease absent)

l represents test results(1=test positive, 0= test negative)

ppv = positive predictive value =

npv =negative predictive value =

false discovery rate : fdr = = 1-ppv

false omission rate : for = = 1-npv

ppv, npv, fdr, for are posterior probabilities

sensitivity =

specificity =

sensitivity, specificity, and are the likelihoods

prevalence = =

and are the priors

plr= positive labeling rate=  

nlr= negative labeling rate =

The plr and nlr are the evidence.