The goal is for you to come away from this video understanding one of the most important
formulas in all of probability, Bayes’ theorem.
This formula is central to scientific discovery, it’s a core tool in machine learning and
AI, and it’s even been used for treasure hunting, when in the 80’s a small team led
by Tommy Thompson used Bayesian search tactics to help uncover a ship that had sunk a century
and half earlier carrying what, in today’s terms, amounts to $700,000,000 worth of gold.
So it's a formula worth understanding.
But of course there were multiple different levels of possible understanding.
At the simplest there’s just knowing what each part means, so you can plug in numbers.
Then there’s understanding why it’s true; and later I’m gonna show you a certain diagram that’s helpful
for rediscovering the formula on the fly as needed.
Then there’s being able to recognize when you need to use it.
With the goal of gaining a deeper understanding, you and I will tackle these in reverse order.
So before dissecting the formula, or explaining the visual that makes it obvious, I’d like
to tell you about a man named Steve. Listen carefully.
Steve is very shy and withdrawn, invariably helpful but with very little interest in people
or in the world of reality. A meek and tidy soul, he has a need for order and structure,
and a passion for detail.
Which of the following do you find more likely: “Steve is a librarian”, or “Steve is
a farmer”?
Some of you may recognize this as an example from a study conducted by the psychologists
Daniel Kahneman and Amos Tversky, whose Nobel-prize-winning work was popularized in books like “Thinking
Fast and Slow”, “The Undoing Project”, . They researched human
judgments, with a frequent focus on when these judgments irrationally contradict what the
laws of probability suggest they should be.
The example with Steve, the maybe-librarian-maybe-farmer, illustrates one specific type of irrationality.
Or maybe I should say “alleged” irrationality; some people debate the conclusion, but more
on all that in a moment.
According to Kahneman and Tversky, after people are given this description of Steve as “meek
and tidy soul”, most say he is more likely to be a librarian than a farmer. After all,
these traits line up better with the stereotypical view of a librarian than that of a farmer.
And according to Kahneman and Tversky, this is irrational.
The point is not whether people hold correct or biased views about the personalities of
librarians or farmers, it’s that almost no one thinks to incorporate information about
ratio of farmers to librarians into their judgments. In their paper, Kahneman and Tversky
said that in the US that ratio is about 20 to 1. The numbers I can find for today put
it much higher than that, but let’s just run with the 20 to 1 ratio since it’s a
bit easier to illustrate, and proves the point just as well.
To be clear, anyone who is asked this question is not expected to have perfect information on the
actual statistics of farmers, librarians, and their personality traits. But the question
is whether people even think to consider this ratio, enough to make a rough estimate. Rationality
is not about knowing facts, it’s about recognizing which facts are relevant.
If you do think to make this estimate, there’s a pretty simple way to reason about the question
– which, spoiler alert, involves all the essential reasoning behind Bayes’ theorem.
You might start by picturing a representative sample of farmers and librarians, say, 200
farmers and 10 librarians. Then when you hear the meek and tidy soul description, let’s
say your gut instinct is that 40% of librarians would fit that description and that 10% of
farmers would. That would mean that from your sample, you’d expect that about 4 librarians
fit it, and that 20 farmers do. The probability that a random person who fits this description
is a librarian is 4/24, or 16.7%.
So even if you think a librarian is 4 times as likely as a farmer to fit this description,
that’s not enough to overcome the fact that there are way more farmers. The upshot, and
this is the key mantra underlying Bayes’ theorem, is that new evidence should not completely
determine your beliefs in a vacuum; it should update prior beliefs.
If this line of reasoning makes sense to you, the way seeing evidence restricts the space
of possibilities, and the ratio you need to consider after that, then congratulations! You understand the heart of Bayes’ theorem.
Maybe the numbers you’d estimate would be a little bit different, but what matters is how you fit
the numbers together to update a belief based on evidence. Here, see if you can take a minute
to generalize what we just did and write it down as a formula.
The general situation where Bayes’ theorem is relevant is when you have some hypothesis,
say that Steve is a librarian, and you see some evidence, say this verbal description
of Steve as a “meek and tidy soul”, and you want to know the probability that the
hypothesis holds given that the evidence is true. In the standard notation, this vertical
bar means “given that”. As in, we’re restricting our view only to the possibilities
where the evidence holds.
The first relevant number is the probability that the hypothesis holds before considering
the new evidence. In our example, that was the 1/21, which came from considering the
ratio of farmers to librarians in the general population. This is known as the prior.
After that, we needed to consider the proportion of librarians that fit this description; the
probability we would see the evidence given that the hypothesis is true. Again, when you
see this vertical bar, it means we’re talking about a proportion of a limited part of the
total space of possibilities, in this cass, limited to the left slide where the hypothesis
holds. In the context of Bayes’ theorem, this value also has a special name, it’s
the “likelihood”.
Similarly, we need to know how much of the other side of our space includes the evidence;
the probability of seeing the evidence given that our hypothesis isn’t true. This little
elbow symbol is commonly used to mean “not” in probability.
Now remember what our final answer was. The probability that our librarian hypothesis
is true given the evidence is the total number of librarians fitting the evidence, 4, divided
by the total number of people fitting the evidence, 24.
Where does that 4 come from? Well it’s the total number of people, times the prior probability
of being a librarian, giving us the 10 total librarians, times the probability that one
of those fits the evidence. That same number shows up again in the denominator, but we
need to add in the total number of people times the proportion who are not librarians,
times the proportion of those who fit the evidence, which in our example gave 20.
The total number of people in our example, 210, gets canceled out – which of course
it should, that was just an arbitrary choice we made for illustration – leaving us finally
with the more abstract representation purely in terms of probabilities. This, my friends,
is Bayes’ theorem.
You often see this big denominator written more simply as P(E), the total probability
of seeing the evidence. In practice, to calculate it, you almost always have to break it down
into the case where the hypothesis is true, and the one where it isn’t.
Piling on one final bit of jargon, this final answer is called the “posterior”; it’s
your belief about the hypothesis after seeing the evidence.
Writing it all out abstractly might seem more complicated than just thinking through the
example directly with a representative sample; and yeah, it is! Keep in mind, though, the
value of a formula like this is that it lets you quantify and systematize the idea of changing
beliefs. Scientists use this formula when analyzing the extent to which new data validates
or invalidates their models; programmers use it in building artificial intelligence, where
you sometimes want to explicitly and numerically model a machine’s belief. And honestly just
for how you view yourself, your own opinions and what it takes for your mind to change,
Bayes’ theorem can reframe how you think about thought itself. Putting a formula to
it is also all the more important as the examples get more intricate.
However you end up writing it, I’d actually encourage you not to memorize the formula,
but to draw out this diagram as needed.
This is sort of the distilled version of thinking with a representative sample where we think
with areas instead of counts, which is more flexible and easier to sketch on the fly.
Rather than bringing to mind some specific number of examples, think of the space of
all possibilities as a 1x1 square. Any event occupies some subset of this space, and the
probability of that event can be thought about as the area of that subset. For example, I
like to think of the hypothesis as filling the left part of this square, with a width
of P(H).
I recognize I’m being a bit repetitive, but when you see evidence, the space of possibilities
gets restricted. Crucially, that restriction may not happen evenly between the left and
the right. So the new probability for the hypothesis is the proportion it occupies in
this restricted subspace.
If you happen to think a farmer is just as likely to fit the evidence as a librarian,
then the proportion doesn’t change, which should make sense. Irrelevant evidence doesn’t
change your belief. But when these likelihoods are very different, that's when your belief changes a lot.
This is actually a good time to step back and consider a few broader takeaways about
how to make probability more intuitive, beyond Bayes’ theorem. First off, there’s the
trick of thinking about a representative sample with a specific number of examples, like our
210 librarians and farmers. There’s actually another Kahneman and Tversky result to this
effect, which is interesting enough to interject here.
They did an experiment similar to the one with Steve, but where people were given the
following description of a fictitious woman named Linda:
Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy.
As a student, she was deeply concerned with issues of discrimination and social justice,
and also participated in anti-nuclear demonstrations.
They were then asked what is more likely: That Linda is a bank teller, or that Linda
is a bank teller and is active in the feminist movement. 85% of participants said the latter
is more likely, even though the set of bank tellers active in the femist movement is a
subset of the set of bank tellers!
But, what’s fascinating is that there’s a simple way to rephrase the question that
dropped this error from 85% to 0. Instead, if participants are told there are 100 people
who fit this description, and asked people to estimate how many of those 100 are bank
tellers, and how many are bank tellers who are active in the feminist movement, no one
makes the error. Everyone correctly assigns a higher number to the first option than to
the second.
Somehow a phrase like “40 out of 100” kicks our intuition into gear more effectively
than “40%”, much less “0.4”, or abstractly referencing the idea of something being more
or less likely.
That said, representative samples don’t easily capture the continuous nature of probability,
so turning to area is a nice alternative, not just because of the continuity, but also
because it’s way easier to sketch out while you’re puzzling over some problem.
You see, people often think of probability as being the study of uncertainty. While that
is, of course, how it’s applied in science, the actual math of probability is really just
the math of proportions, where turning to geometry is exceedingly helpful.
I mean, if you look at Bayes’ theorem as a statement about proportions – proportions
of people, of areas, whatever – once you digest what it’s saying, it’s actually
kind of obvious. Both sides tell you to look at all the cases where the evidence is true,
and consider the proportion where the hypothesis is also true. That’s it. That’s all it’s
saying.
What’s noteworthy is that such a straightforward fact about proportions can become hugely significant
for science, AI, and any situation where you want to quantify belief. You’ll get a better
glimpse of this as we get into more examples.
But before any more examples, we have some unfinished business with Steve. Some psychologists
debate Kahneman and Tversky’s conclusion, that the rational thing to do is to bring
to mind the ratio of farmers to librarians. They complain that the context is ambiguous.
Who is Steve, exactly? Should you expect he’s a randomly sampled American? Or would you
be better to assume he’s a friend of these two psychologists interrogating you?
Or perhaps someone you’re personally likely to know? This assumption determines the prior.
I, for one, run into many more librarians in a given month than farmers. And needless
to say, the probability of a librarian or a farmer fitting this description is highly
open to interpretation.
But for our purposes, understanding the math, notice how any questions worth debating can
be pictured in the context of the diagram. Questions of context shift around the prior,
and questions of personalities and stereotypes shift the relevant likelihoods.
All that said, whether or not you buy this particular experiment the ultimate point that
evidence should not determine beliefs, but update them, is worth tattooing in your mind.
I’m in no position to say whether this does or doesn’t run against natural human intuition,
we’ll leave that to the psychologists. What’s more interesting to me is how we can reprogram
our intuitions to authentically reflect the implications of math, and bringing to mind
the right image can often do just that.