Bayesian Inference is an important econometric tool. Over the next few days, we will review some of the basic Bayesian inference methods.
Economicitis occurs in 300 out of every 100,000 adults. Recently, however, a test has been developed to screen for the disease. Of 1000 individuals with economicitis who were tested, only 40 had an erroneous negative test. Out of 1000 healthy individuals, 20 out of 1000 individuals had an erroneous positive test result.
My friend Ron received the sad news that his test result shows that he has economicitis. Ron wants to know that given the test result is positive, what is the actual chance that he has the disease.
One way to estimate this is using Bayesian inference. According to Bayesian theory:
- Posterior Odds = prior odds x likelihood ratio
- Posterior Odds=p(θ1)/p(θ2) * [p(X1|θ1)/p(X1|θ2)]
The prior odds are having the disease are 300/99,700. This is equivalent to the prior probability Ron has the disease (300/100,000) divided by the prior probability he does not have the disease (99,700/100,000). The likelihood ratio is equal to the probability of having a positive test given the person has the disease (1-40/1000) divided by the probability of having a positive test given that the person is healthy (20/1000). Thus we have:
- Posterior Odds = (300/99,700) x [(960/1000)/(20/1000)] = .144
This means that the chance the individuals who test positive for economicitis actually have the disease is about one in seven. To calculate the posterior probability, simply use the following formula:
- Posterior Probability = (posterior odds)/(1 + posterior odds)=.144/1.144=12.6%
Thus, Ron should not be too worried about having the disease. Using the prior, Ron only had a 0.3% change of having the disease, but even after having tested positive for economicitis, Ron still only has a 12.6% chance of being stricken by this deadly disease.