## Bayes’ theorem and charity

Posted by Dheeraj Kattula on April 27, 2009

Lionel Robbins defined economics as “the science which studies human behaviour as a relationship between ends and scarce means which have alternative uses.” When working is resource poor a setting economizing becomes all the more important. A hospital, which wants to serve the poor keeps its general prices low lest the poor never come. Despite this patients would require a dicount in form of ‘charity’. How is it that a hospital can try its best to see that they give charity to the poor and only the poor?

What are the combinations possible? In the descending order of preference for probability of such a combination happening is:-

1. patient poor ( Po )-recieves charity( C )

2. patient not poor( N Po )- recieves no charity (N C )

3. patient not poor- recieves charity

4. patient poor-recieves no charity

Practically it is very difficult to know for sure economic status of a patient. Poor tribals come to hospital in their best dress even when very sick. I have seen the rich go to labour rooms in stinking clothes as they know they would discard that apparel after the delivery!

We know if charity is given or not. Is it possible to use Bayes theorem by looking at conditional probability of the given combinations and apply it to formulate direction of the services?

Bayes’ theorem can be expressed as :

Which can be expanded as :

Each term in Bayes’ theorem has a conventional name:

- P(
*A*) is the prior probability or marginal probability of*A*. It is “prior” in the sense that it does not take into account any information about*B*. - P(
*A*|*B*) is the conditional probability of*A*, given*B*. It is also called the posterior probability because it is derived from or depends upon the specified value of*B*. - P(
*B*|*A*) is the conditional probability of*B*given*A*. - P(
*B*) is the prior or marginal probability of*B*, and acts as a normalizing constant. - The denominator is Probability of
*B*occuring without considering occurance of A. There for it takes conditional probability of B when A occurs and also when non-A occurs.

Intuitively, Bayes’ theorem in this form describes the way in which one’s beliefs about observing ‘A’ are updated by having observed ‘B’.

Let us consider person has received charity. What is the probability that he was poor?

probability of person being poor given that he has recieved charity P ( Po/C )

According to Bayes theorem:-

P (Po/ C) = P ( C / Po ) . P ( Po )

———————————————————-

P ( C / Po ) . P ( Po ) + P ( C / N Po ). P ( N Po )

We know that P (C/Po) cannot be known. If it was then we would not look at this equation at all as P (C/Po) is our main interest. Then P ( Po/C ) would be high when

1. P ( Po ) is high

2. P ( N Po) is small.

That is if one wants to give charity to poor only, he should start a independent entity where only poor come. Then he can give charity with greater confidence! He can have another set up for the non poor where there would not be any charity. Horizontally well integrated organizations can effectively run such systems if they have such a vision.

Ops! In a single set up, there needs to be optimization of some sort. My plan is to think on this and post it in future.

**References**

1.http://en.wikipedia.org/wiki/Economics

2.http://en.wikipedia.org/wiki/Bayes_theorem

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