- The late great astronomer and science popularizer,
Carl Sagan, had a famous saying:
"Extraordinary claims require extraordinary evidence."
In this, he was echoing a argument by David Hume.
Hume said, "Well, what's more likely, that the laws
of the Universe as we've always experienced them are wrong,
or that some guy misremembered something?"
And these are all versions of a kind of reasoning
that is called 'Bayesian,' after the Reverend Thomas Bayes.
It just means after you've seen all of the evidence,
how much should you believe something?
And it assumes that you don't just believe something
or disbelieve it, you assign a degree of belief.
We all want that, we don't wanna be black and white,
dichotomous absolutists.
We wanna calibrate our degree of belief
to the strength of the evidence.
Bayes' theorem is how you ought to do that.
Bayes' theorem, at first glance, looks kinda scary
'cause it's got all of these letters and symbols,
but more important, conceptually, it's simple-
and at some level, I think we all know it.
Posterior probability, that is credence in an idea
after looking at the evidence can be estimated by the prior:
that is, how much credence did the idea have even
before you looked at that evidence?
The prior should be based on everything that we know so far,
on data gathered in the past, our best-established theories,
anything that's relevant
to how much you should believe something
before you look at the new evidence.
Second term is sometimes called the likelihood,
and that refers to if the hypothesis is true,
how likely is it that you will see the evidence
that you are now seeing?
You just divide that product- the prior, the likelihood-
by the commonness of the data, probability of the data,
which is, how often do you expect
to see that evidence across the board,
whether the idea you're testing is true or false?
If something is very common, so for example,
lots of things that give people headaches and back pain,
you don't diagnose some exotic disease
whose symptoms happen to be back pain and headaches
just because so many different things
can give you headaches and back pain.
There's a cliche in medical education:
If you hear hoof beats outside the window,
don't conclude that it's a zebra,
it's much more likely to be a horse.
And that's another way of getting people
to take into account Bayesian priors.
There are many realms in life
in which if all we cared about was
to be optimal statisticians, we should apply Bayes' theorem-
just plug the numbers in.
But there are things in life
other than making the best possible statistical prediction.
And sometimes we legitimately say, "Sorry, you can't look
at the Bayes rate:
rates of criminal violence
or rates of success in school."
It's true you may not have
the same statistical predictive power,
but predictive power isn't the only thing in life.
You may also want fairness.
You may want to not perpetuate a vicious circle
where some kinds of people,
through disadvantage, might succeed less often,
but then if everyone assumes they'll succeed less often,
they'll succeed even less often.
It could also go too far just by saying, "Well, only 10%
of mechanical engineers are women, so there must be a lot
of sexism in mechanical engineering programs
that cause women to fail."
And you might say, "Well, wait a second,
what is the Bayes rate of women
who wanna be mechanical engineers in the first place?"
There, if you're accusing lots of people of sexism
without looking at the Bayes rate,
you might be making a lot of false accusations.
I think we've got to think very carefully about the realms
in which, morally, we want not to be Bayesians
and the realms in which we do wanna be Bayesian,
such as journalism and social science
where we just wanna understand the world.
It's one of the most touchy and difficult
and politically sensitive hot buttons that are out there.
And that's a dilemma that faces us with all taboos,
including forbidden Bayes rates.
Still, we can't evade the responsibility
of deciding when are Bayes rates permissible,
when are they forbidden?
What Bayes' theorem just says is that your degree of belief
in a hypothesis should be determined
by how likely the hypothesis is beforehand,
before you even look at the evidence.
If the hypothesis is true, what are the odds
that you would see the evidence that you are seeing,
scaled by how common is that evidence across the board
whether the hypothesis is true or false?
If you could follow what I just said,
you understand Bayes' theorem.
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