Chance, Spread and What to Expect
What Is Left After the News
Last timeTwo Things at Once
Conditioning restricts the table to the row you now know you are in, and reversing the direction of a conditional requires the base rate, which never quietly cancels.
Something has been learned. A request failed, a test came back positive, a user
clicked. The question is what is now true of everything else. This is the single
most useful operation in the subject, and it is also where trained people make
confident errors, so it is worth doing slowly.
Start from the table of the last lesson. Learning a piece of news means that
some rows or columns are no longer possible. Delete them. What remains no longer
sums to one, because weight has been thrown away, so divide by what survived.
- the weight that is consistent with both, the part of A that survives the news
- the total weight that survived the news, the normalising divisor
- the weight on A within the restricted world, read as the chance of A given B
- the division is the whole trick, and it is why conditioning on something unlikely can change a number enormously
Multiply both sides by the divisor and you have a statement worth naming in its
own right: a joint weight is a conditional times a margin.
- the joint weight, which is what the table actually holds
- how often the second follows when the first has happened, the mechanism
- how often the first thing happens at all, the base rate
Turning a conditional around
Here is the step that matters. The mechanism is almost always known in one
direction and wanted in the other. A test is designed and validated by giving it
to people whose condition is known, so what is measured is the chance of a
positive given illness. What a patient wants is the chance of illness given a
positive. These are different numbers.
Write the joint weight both ways, once with each quantity in the conditioning
position. Both expressions equal the same joint weight, so they equal each other,
and solving for the one that is wanted gives the reversal rule. The base rate
appears in it because it has to: the two directions genuinely differ, and the
base rate is exactly the information that separates them.
It is worth being clear about what that derivation is and is not. It is two
lines of algebra applied to a definition, with no new assumption anywhere, which
is why the result holds universally and why there is nothing to argue with. The
controversy that has attached itself to the name belongs to a different question,
namely where the base rate comes from when nobody has counted it. The rule
itself is bookkeeping, and the three lessons that precede this one already
contain everything it uses.
The lesson stops here
3 more paragraphs to go
You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.
The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.
See the planThe contentsThis is the reading half
Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.
The contents