*“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

Since \[ℙ(A, B) = ℙ(A)⋅ ℙ(B|A)\] and, by symmetry \[ℙ(A, B) = ℙ(B)⋅ℙ(A|B)\] then we can reorganize everything as \[ℙ(B|A) = \frac{ℙ(B)⋅ℙ(A|B)}{ℙ(A)}\]

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

This is the view we will use now

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 can write \(ℙ(A)\) as \[ℙ(A) = ℙ(A,B) + ℙ(A,\textrm{not }B)\] which can be rewritten as \[ℙ(A) = ℙ(B)⋅ℙ(A|B) + ℙ(\textrm{not }B)⋅ℙ(A|\textrm{not }B)\] therefore \[ℙ(B|A) = \frac{ℙ(B)⋅ℙ(A|B)}{ℙ(B)⋅ℙ(A|B) + ℙ(\textrm{not }B)⋅ℙ(A|\textrm{not }B)}\]

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* \(ℙ(\text{COVID}_+)=0.001\) and the *precision* \[
\begin{aligned}
ℙ(\text{test}_+ | \text{COVID}_+)=0.99\\
ℙ(\text{test}_- | \text{COVID}_-)=0.99
\end{aligned}
\]

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

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

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

\[\begin{aligned} ℙ(\text{test}_+, \text{COVID}_+)& =ℙ(\text{test}_+ \vert \text{COVID}_+)\cdotℙ(\text{COVID}_+)\\ & =0.99⋅ 0.001 \end{aligned} \]

and

\[ \begin{aligned} ℙ(\text{test}_+, \text{COVID}_-)& =ℙ(\text{test}_+ \vert \text{COVID}_-)\cdotℙ(\text{COVID}_-) \\ & =(1-0.99)⋅ (1-0.001) \\ & =0.01⋅ 0.999 \end{aligned} \]

\[ \begin{aligned} ℙ(\text{test}_+ \vert \text{COVID}_-) & =1-ℙ(\text{test}_- \vert \text{COVID}_-)\\ & =(1-0.99) =0.01 \end{aligned} \] and \[ \begin{aligned} ℙ(\text{COVID}_-) & =1-ℙ(\text{COVID}_+)\\ & =(1-0.99)⋅ (1-0.001) \\ & =0.01⋅ 0.999 \end{aligned}\]

Therefore \(ℙ(\text{test}_+)= 9.9\times 10^{-4} + 0.00999=0.01098\)

\[ \begin{aligned} ℙ(\text{COVID}_+ \vert \text{test}_+) & = \frac{ℙ(\text{test}_+, \text{COVID}_+)}{ℙ(\text{test}_+)}\\ & = \frac{0.99\cdot 0.001}{0.01098} \\ & = 9.02\% \end{aligned} \]

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.

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