Now suppose our decision is d2.

Again, we might be right or we might be wrong.

If we decide that the patient has HIV, and indeed they do,

then the decision is right.

And the loss associated with d2 is zero, in other words, no loss.

If we decide the patient has HIV but they actually do not,

then the decision is wrong and the loss associated with d2 is some value w2.

The consequences of making a wrong decision d1 or d2 are different.

If the decision is d1 and it is wrong,

then you decided that the patient does not have HIV when in reality they do.

This is a false negative, potential consequences are no treatment and

premature death.

These are grave consequences.

If the decision is d2 and it is wrong then you decided that the patient

has HIV when in reality they don’t.

This is a false positive.

Potential consequences are distrust and unnecessary further investigation.

These consequences are certainly not ideal but

they are much less grave than the consequences of a false negative decision.

Let's put these definitions in the context of the HIV testing example with ELISA.

The loss function for d1 can take on the values zero or w1 say 1000 and

the loss function for d2 can take on the values zero or ten.

The values of w1 or w2 are arbitrarily chosen.

But the important thing to realize is that w1,

the loss associated with a false negative determination.

Is much higher than w2,

the loss associated with a false positive determination.

Remember from the earlier video that our patient had tested positive on the ELISA.

In that video, we also calculated the posterior probability of the patient not

having HIV given positive ELISA result.

As approximately 0.88 and

the posterior probability of having HIV given the positive ELISA result.

As the complement of this value, approximately .12.

Then the expected loss for d1 can be calculated as the sum of the two

posterior probabilities weighted by their associated losses.

That is, the posterior possibility of the patient not having HIV.

Given positive ELISA times zero, since the loss for

deciding the patient does not have HIV would be the right decision in this case.

Times the posterior probability of having HIV given positive ELISA times 1,000,

the loss associated with a false negative.

We can similarly calculate the expected loss for d2.

Since the expected loss for d2 is lower, we should make that decision.

That is, we should decide that the patient has HIV.