Last Uncertainty Wednesday, I introduced sensitivity and specificity as measures of how good a test is (or using the language of our framework, how strong a signal is). We derived the following formula
P(B | H) = P(B)/P(H) * P(H | B)
which relates P(B | H), i.e. the probability of the state of the world being B conditional on receiving signal H, to P(H | B), i.e. the probability of receiving signal H when the world is B, aka the sensitivity of the test.
Let’s rewrite this slightly to get a better handle on what it means
P(B | H) = [P(H | B) / P(H)] * P(B)
So now we see that the formula let’s us turn an unconditional probability P(B) into a conditional one. Or put differently, it relates the probability of the world being in state B *before* we have observed a signal to the probability *after* we have observed signal H. That’s why P(B) is also sometimes referred to as the “prior” and P(B | H) as the “posterior” probability.
Let’s plug in all the numbers we have from the previous posts based on the PSA Test Example
P(B | H) = [0.51 / 0.091302] * 0.0031 = 5.585858 * 0.0031 = 0.017316
Thankfully this foots with the number for P(B | H) we had found originally and it now shows clearly how this test gives us about a 5.58x lift, meaning the posterior probability is 5.58x as high as the one before we received the signal.
We can also see that this “lift” number is linear in the sensitivity. A 20% better test / stronger signal in the sense of a 20% higher sensitivity, will give us 20% more lift above our prior probability.
But we also clearly see that the absolute effect depends equally on P(H), that is how likely the signal itself is. A less likely signal, meaning lower P(H), gives us much more lift.
So what makes for a really good test? A really good test has high sensitivity (big numerator) on an unlikely signal (small denominator).
