How would you like to be able to tell the future? Well, you can't
But, billions of dollars and millions of lives depend on
scientists predicting the future outcome of events
every day.
From the results of elections to the paths of hurricanes
to the likelihood of terrorist attacks
We rely on predictions based on statistical analysis to help us assess risks,
invest our money, and make potentially life-altering decisions.
But how do you determine the odds of something that is inherently uncertain?
Here's a thought exercise for you.
Imagine a woman in her 40s who wants to know if she has breast cancer.
If she does, statistics show that there is a 75% chance that a mammogram will detect it
But if she doesn't, then there is a 10% chance that the mammogram will give her a false positive.
That is, it will say she has cancer and she doesn't.
Add this to the fact that we also know that 1.4% of all women in their forties have breast cancer.
Now, imagine our hypothetical woman gets a mammogram and
it comes back positive.
What are the odds that the test is right?
You are probably thinking, about 75%
Right?
Wrong!
The odds that she actually has cancer are only 9.6%!
Now don't feel bad!
Most doctors get this wrong too!
This is a problem that calls for Bayesian reasoning,
which, while really counter-intuitive, is the basis for a lot of modern scientific thinking.
Bayesian reasoning is a method by which you can determine the importance
of a particular piece of information when trying to prove or disprove a hypothesis.
To do that, Bayes's Theorem asks for 3 things.
First, the probability that your hypothesis was true before you conducted the experiment.
In this case, our hypothesis is that the woman has cancer when she gets the mammogram.
Second, you have to weigh the odds of a given outcome if your hypothesis is true
In our example, the odds of getting a positive test result.
And third, the odds of that same outcome if your hypothesis is false.
Meaning, that the woman gets a positive test, but doesn't have cancer.
If you, like most doctors the first time they read this question, thought that a positive test
result meant 75% chance of cancer,
You made the very human mistake of thinking that the odds
of a woman with cancer getting a positive result
were the same as the odds of a woman with a positive result having cancer.
But they are not!
They are in fact, totally, drastically different!
To think in Bayesian terms is to think this way:
Say you got a group of 1000 women in their forties.
14 of them have cancer,
cause we said that 1.4% of women in this age group have cancer,
remember?
that means 986 of them do not have cancer.
Of the 14 who have cancer, as we established,
75% of them will test positive in a mammogram.
75% of 14 is 10.5.
And of the 986 who don't have cancer,
10% of them will receive a positive test result,
and 10% of 986 is 98.6.
So, after administering a thousand tests,
there will be
10.5 true positives
and 98.6 false positives.
The total number of positives
is 109.1,
so 10.6 true positives
out of a 109.1 total positives means
that there is a 9.6% chance
that a woman in her forties who recieves
the positive mammogram result
will actually have cancer.
Thinking like this does not come naturally to most of us
but it's the only way to measure
the real significance of experimental results.
If you're not thinking like a Bayesian
you can wildly overestimate the relevance of your findings.
That is a big problem in science.
Consider for example that 80% of experimental results
published in academic journals in the United States
turn out to be impossible to replicate.
Bayesian reasoning makes us think rationally,
which humans are not always great at,
and better reasoning means better science
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