Bayesian Decision Rules

Given the task of deciding which of two possible phenomena may have caused a particular observation, we can minimize our probability of error by computing which is most likely to have given rise to the observation.

If we are choosing between Spanish and English as possible ``diseases'' which may have caused the following ``symptom''

immunotechno

we must compare

$$P({\tt immunotechno},{\tt Spanish})$$

with

$$P({\tt immunotechno},{\tt English})$$

. Bayes' theorem tells us that we could instead compare

$$P({\tt immunotechno}\vert{\tt Spanish})p({\tt Spanish})$$

with

$$P({\tt immunotechno}\vert{\tt English})p({\tt English})$$

.