## Introduction

We’ve covered __maximum likelihood estimation____ __in our previous blog. You may follow the link in case you missed it or need a refresher on it. Simply put, it’s a parameter optimisation technique that makes use of probability density functions instead of numerical values. We highly recommend having prior knowledge of Maximum Likelihood Estimation since that would make this a little more intuitive.

## Bayesian Decision Theory

Bayesian Decision theory is a statistical approach to pattern classification. It’s a probabilistic approach to make classifications and measures risks associated with assigning an input to a given class.

It highlights how prior probability by itself isn’t the most efficient way to make predictions. Bayesian decision theory takes into consideration prior probability, likelihood probability, and evidence to compute the posterior probability.

### Prior probability

It is the initial probability of an event before we take into account any new piece of information. For example, we are asked which of the 2 teams, A and B, will win the next match. In the last 5 appearances between the 2, A has won 2 times and B has won 3 times.

So the prior probability of A winning the next match is ⅖. But this may not hold true as there could be various other factors like injured players in team A. So predicting the winner solely on the basis of prior probability may not be the most efficient way to do so.

### Likelihood Probability

Likelihood probability is computed for an event, given some conditions. It is denoted by-

P(B|A_{k}).

Here B is the condition while A_{k} is the outcome. There may be multiple outcomes.

Now, suppose team B has many injured players while team A has their entire squad. This heavily puts odds in favour of A for the next match. While the prior probability said team B was more likely to win the match.

Bayesian Decision theory takes into consideration past results as well as the current situation to make the predictions. It is given by

Source - __link__

P(B) is number of times condition X has occurred. It is referred to as Evidence.

P(A_{k}|B) is the posterior probability. It is the probability of outcome A_{k }^{ }given some condition B.