So, we're told that 25.9% of this country's population is infected with HIV. So the probability of having HIV is 0.259. We also know something about the accuracy of the test, which seems to vary depending on whether the person has HIV or not. This is very common for medical tests. They tend to have different accuracy rates the, different accuracy rates for whether the patient has the disease or does not have the disease.
This statement, for those who carry HIV, the ELISA test is 99.7% accurate, basically means that probability of testing positive, because that's what an accurate result would be if a person has HIV, so probability of positive given HIV is 0.997.
This statement, for those who do not carry HIV, the test is 92.6% accurate, means probability of testing negative, because that's what accurate would mean in this case given that the patient does not have HIV, is 0.926.
The question says, if an individual from Swaziland has tested positive, what is the probability that he carries HIV? So what we know is that the person tested positive. We're looking to see what is the probability that they have HIV. What we can see here is that we have a situation where we're asked for a conditional probability, and the condition has been reversed from one of the things that we are given, and we should really think about a tree diagram in this case. Those tend to be the most effective methods for getting to the answer. There are definitely other ways that you can solve this problem, and you can organize the information that's given to you. But a tree diagram tends to be one where you can really efficiently and effectively organize the information that you're given. And you're going to get to the right answer if you do it the right way.

So, the first branch in the tree is always made up of marginal probabilities, since we're dividing up our population without conditioning on any other attributes. Some people in the population have HIV. That's the top branch. And others don't. That's the bottom branch. So probability of having HIV, as we saw, was 0.259 in Swaziland.
And the probability of not having HIV is the complement of that, 1 minus 0.259 is going to give us 0.741. So about 74.1% of the population in Swaziland does not have HIV. Note that probabilities on a set of branches always add up to 1.
Next, we move on to conditional probabilities. Let's start with the part of the population who has HIV, so we're going to be working with the top branch here. When these people take the test, they may get a positive or a negative result, because the test isn't 100% accurate. Therefore, we divide up the HIV population into two, those who test positive, and those who test negative. Based on information on the test that we were provided earlier, we know that the probability of testing positive, if someone has HIV is 0.997.

Then, probability of testing negative if someone has HIV, this would be a false negative, would be the complement of that, 0.003. Similarly, among those who don't have HIV, some still test positive, and some test negative. Probability of accurately testing negative if the patient doesn't have HIV is 0.926.

And the probability of a false positive, that's testing positive even though the patient does not have HIV is the complement of that, 0.074. Remember, our goal is to find the probability of having HIV, given that the patient has tested positive. Which based on Bayes theorem should be probability of HIV and positive divided by probability of testing positive. Remember, the numerator is always the joint probability, and the denominator is the marginal probability of what we're conditioning on. So far, we don't have the building blocks we need to calculate the probability that we're interested in.

To get the join probabilities, like the one in the numerator, using the probability true, all we need to do is multiply across the branches. This is why a probability tree is useful. Because it organizes the information for you in a way where you'd no longer have to think, what should I multiply with what. And you, all you need to do is carry along the branches and pick up the building blocks along the way.

We start with the marginal probability of having HIV and we multiply it by the probability of testing positive, given that the patient has HIV. So I'm following the first, the very top branch here, which is going to yield us the joint probability of having HIV and testing positive. So what we get is 0.259 from the first branch times 0.997 from the second branch, which gives up 0.2582. So, there's a 25.82% chance that a randomly drawn person from the Swaziland population has HIV and tests positive.

Similarly, probability of HIV and negative is going to be the probability of HIV, 0.259, times the probability of negative given HIV, 0.003. That's a really tiny probability, 0.0008. We can keep going and calculate similar probabilities for the lower branch, the no HIV population as well.
Probability of no HIV and positive comes out to be 5.48% and probability of no HIV and negative comes out to be 68.60, 68.62%. We've done a bunch of calculations so far, but let's go back to the task at hand. We're only interested in those who test positive, because that's what our given are. And among these, we're especially interested in those who actually have the HIV.

Screenshot taken from Coursera 10:00

Bayesian Inference

Screenshot taken from Coursera 1:05

before you collect data

Screenshot taken from Coursera 5:25

data collection

Screenshot taken from Coursera 7:23

posterior

Screenshot taken from Coursera 12:13

Screenshot taken from Coursera 13:12

recap

Screenshot taken from Coursera 13:30

Examples of Bayesian Inference

Screenshot taken from Coursera 5:33

Screenshot taken from Coursera 7:22

Table of Contents