Class 26: Bayes’ Theorem

Methodology of Scientific Research

Andrés Aravena, PhD

May 5, 2023

Why you should learn (some) math

70% of the world population got one or more dosis of COVID-19 vaccine

Today, according to “Our World in Data”

In other words, more than 30% of people has not been vaccinated


There are many reasons

  • Only 29.7% of people in low-income countries have been vaccinated

  • In rich countries we observe vaccine hesitance

Why people do not want to be vaccinated?

Again, there are many reasons

  • Political

  • Belief

  • Distrust of Science

Some people distrust Science

Besides the Anti-vaccine people, we have

  • Climate change denial

  • Flat Earthers

  • and several others

Can Science be trusted?

PLOS Medicine 2005

Most Scientists are honest

In 2009, 2% of scientists admitted to falsifying studies at least once

14% admitted to personally knowing someone who did

The problem is not dishonesty

Most people do not lie

Replicability crisis

A 2016 poll of 1,500 scientists reported that 70% of them had failed to reproduce at least one other scientist’s experiment

50% had failed to reproduce one of their own experiments

There are problems with the experiments and their analysis

Science August 2015•vol 349 issue 62519


Repeated the top 100 studies in psychology

  • Only 36% of the replications gave significant findings
    • compared to 97% of the original studies
  • The effect size in the replications was half of the effect size in the original studies, on average

This is not limited to psychology

Why does this happens?

Journal of the Royal Statistical Society and American Statistical Association

Cargo cults

This has been said before

Richard Feynman

  • Physicist
  • Excellent professor
  • Worked in the Manhattan Project at 25 years old
  • Nobel Prize on Physics in 1965
  • He was talking about USA in the 1970s

Cargo-cult statistics

“the ritualistic miming of statistics rather than conscientious practice”

Cargo-cult statistics and scientific crisis

“practitioners go through the motions of fitting models, computing p-values or confidence intervals, or simulating posterior distributions”

“They invoke statistical terms and procedures as incantations, with scant understanding of the assumptions or relevance of the calculations, or even the meaning of the terminology”

The authors say

“We believe that poor statistical education and practice are symptoms of and contributors to problems in science as a whole”

My opinion

As someone who teaches maths and statistics, I say

We are bad at teaching maths

We need to find better ways to teach math

(in particular to math teachers)

Math is a superpower

Math allow us to travel in time, and see the invisible

That is why people who knows do not want to teach math:

It gives power to the people

“They” do not want you to know math

Quick example

Example: diagnosis

Imagine that you are randomly chosen for a test of COVID-19

The result is “positive”. It says that you have the virus

But this test fails 2% of times, giving a false positive or a false negative

Then the question:

What is the probability that you have COVID-19 given that the test said “positive”?

It is not 98%

The test is correct 98% of times

That is, the probability of a positive test given that you have COVID

But we really want to know the probability of COVID given that the test is positive

They are not the same

We need to know the base rate

To answer this question we must know the prevalence of COVID

That is, what is the proportion of the population with COVID

There were 728,692 active cases in Turkey last year
(only 1,128 were serious)

Population of Turkey is 85,828,516

Dividing both numbers we find that the prevalence is 0.86%

Let’s see why

Let’s assume that:

  • There are one million people
    (this will make it easy to calculate)
  • The test has a precision of 98%
  • The prevalence of COVID in the population is 0.86%
  • The people to test is chosen randomly from the population

Let’s fill this matrix

  Test- Test+ Total
COVID- . . .
COVID+ . . .
Total . . .

COVID reality in the rows and test results in the columns

Let’s assume that there is a million people

  Test- Test+ Total
COVID- . . .
COVID+ . . .
Total . . 1000000

We will fill this matrix in the following slides

Assuming one million people makes the math easier

0.86% of them are COVID positive

  Test- Test+ Total
COVID- . . 991400
COVID+ . . 8600
Total . . 1000000

Prevalence is the percentage of the population that has COVID.
In other words, it is the probability of (COVID+)

98% are correctly diagnosed

  Test- Test+ Total
COVID- 971572 . 991400
COVID+ . 8428 8600
Total . . 1000000

Precision is the probability of a correct diagnostic

(Here we assumed that the error rates are the same for positive and negative. That may not be the case always)

2% are wrongly diagnosed

  Test- Test+ Total
COVID- 971572 19828 991400
COVID+ 172 8428 8600
Total . . 1000000

(this error rate is only an example. Real tests are usually better)

Total people diagnosed

  Test- Test+ Total
COVID- 971572 19828 991400
COVID+ 172 8428 8600
Total 971744 28256 1000000

We sum and fill the empty boxes

28256 people got positive test, but only 8428 of them have COVID

Probability of having COVID if the text is positive: 29.83%


More References

Who do you want to be?


Bayes Theorem

“An Essay towards solving a Problem in the Doctrine of Chances” is a work on the mathematical theory of probability by the Reverend Thomas Bayes, published in 1763, two years after its author’s death

The use of the Bayes theorem has been extended in science and in other fields

From Wikipedia, the free encyclopedia

Bayes rule

Since we have \[ℙ(A, B) = ℙ(A)⋅ ℙ(B|A)\] and, by symmetry, we also have \[ℙ(A, B) = ℙ(B)⋅ℙ(A|B)\] then we conclude that \[ℙ(A)⋅ ℙ(B|A) = ℙ(B)⋅ℙ(A|B)\]

Dividing by \(ℙ(A)\)

We can reorganize everything as \[ℙ(B|A) = \frac{ℙ(B)⋅ℙ(A|B)}{ℙ(A)}\]

This is known as Bayes’ Rule or Bayes’ Theorem

One of the most famous theorems

What does it mean

It can be understood as \[ℙ(B|A) = ℙ(A|B)⋅\frac{ℙ(B)}{ℙ(A)}\] which is a rule to reverse the order of the conditional probability

It shows us how to go from \(ℙ(A|B)\) to \(ℙ(B|A)\)

This is the view we will use now

How do we learn

Another point of view is \[ℙ(B|A) = \frac{ℙ(A|B)}{ℙ(A)}⋅ℙ(B)\] which is a rule to update our opinions

  • \(ℙ(B)\) is the a priori probability
  • \(ℙ(B|A)\) is a posteriori probability

Bayes says how to change \(ℙ(B)\) when we learn \(A\)

“When the facts change, I change my opinion. What do you do, sir?”

John Maynard Keynes (1883 – 1946), English economist, “father” of macroeconomics

We need \(ℙ(A)\)

We can write \(ℙ(A)\) as \(ℙ(A,B) + ℙ(A,\text{not } B)\)
which can be rewritten as \[ℙ(A) = ℙ(B)⋅ℙ(A|B) + ℙ(\text{not } B)⋅ℙ(A|\text{not } B)\] therefore we have a second formula for Bayes’ rule \[ℙ(B|A) = \frac{ℙ(B)⋅ℙ(A|B)}{ℙ(B)⋅ℙ(A|B) + ℙ(\text{not } B)⋅ℙ(A|\text{not } B)}\]

We did it before

When we filled the “diagnostic” matrix, we did this

  not A A Total
not B \(ℙ(\text{not }A,\text{not }B)\) \(ℙ(A,\text{not } B)\) \(ℙ(\text{not } B)\)
B \(ℙ(\text{not } A, B)\) \(ℙ(A|B)\) \(ℙ(B)\)
Total \(ℙ(\text{not } A)\) \(ℙ(A)\) \(1\)

To get \(ℙ(A, B)\) we used \(ℙ(B)⋅ℙ(A|B)\)

Here \(A\) is “test positive” and \(B\) is “having COVID”

Diagnostics Using Bayes Theorem

We want to evaluate the probability of having COVID, given that the test is positive \[ℙ(\text{COVID}_+ | \text{test}_+)\] given that we know the prevalence \(ℙ(\text{COVID}_+)=0.0086\) and the precision \[ \begin{aligned} ℙ(\text{test}_+ | \text{COVID}_+)=0.98\\ ℙ(\text{test}_- | \text{COVID}_-)=0.98 \end{aligned} \]

What is the probability of being sick given that the test is positive?

\[ℙ(\text{COVID}_+ | \text{test}_+)=\frac{ℙ(\text{test}_+ | \text{COVID}_+)⋅ℙ(\text{COVID}_+)}{ℙ(\text{test}_+)}\]

We need to know \(ℙ(\text{test}_+),\) which we can get as

\[ℙ(\text{test}_+)= ℙ(\text{test}_+, \text{COVID}_+)+ ℙ(\text{test}_+, \text{COVID}_-)\]

We can calculate them

\[\begin{aligned} ℙ(\text{test}_+, \text{COVID}_+)& =ℙ(\text{test}_+ \vert \text{COVID}_+)\cdotℙ(\text{COVID}_+)\\ & =0.98⋅ 0.0086 = 0.008428 \end{aligned} \] and \[ \begin{aligned} ℙ(\text{test}_+, \text{COVID}_-)& =ℙ(\text{test}_+ \vert \text{COVID}_-)\cdotℙ(\text{COVID}_-) \\ & =(1-0.98)⋅ (1-0.0086) \\ & =0.02⋅0.9914 = 0.019828 \end{aligned} \]

Finally, we find the value we were looking for

\[ \begin{aligned} ℙ(\text{COVID}_+ | \text{test}_+) & = \frac{ℙ(\text{test}_+ | \text{COVID}_+)⋅ℙ(\text{COVID}_+)}{ℙ(\text{test}_+)}\\ & = \frac{0.98⋅0.0086}{0.98⋅0.0086 + 0.02⋅0.9914} \\ & = \frac{0.008428}{0.008428 + 0.019828} \\ & = 0.2983 \end{aligned} \]

Prob of having COVID given positive test is near 30%
(assuming there are no other symptoms, or other information)


We would like to have no errors, but errors are usually unavoidable. The technology used will limit the precision of diagnostics.

In practice, diagnostic test designers know that telling that you are healthy when you are sick is much worse than telling that you are sick when you are healthy. In other words, the cost of a false positive is usually lower than the cost of a false negative.

Of course the costs depend on the disease and in the technique.

What happens if the false positive rate is different from the false negative rate?

\[\begin{aligned} ℙ(\text{test}_+ \vert \text{COVID}_+)=a\\ ℙ(\text{test}_- \vert \text{COVID}_-)=b \end{aligned} \]