1. Example and Bayesian Treatment

” John is software engineer with 15+ years of technical experiences in IT. He applied to the CTO position in a startup company. In the final interview, and among a short list of 5 applicants , John was double qualified than the other four candidates. Do you think that John got the job? “

If I am willing to guess, I will assume that John definitely got the job, says with 90% chance. I think also that you agree with me, do you? Based on the available information, John has over-performed all other applicants in the last round of interviews. Accordingly, he is the most likely to get the job. But does Bayes’ Theory shares us the same opinion? Let us see how Bayes’ Theorem guesses the answer!

The first step in Bayes’ Theory is to estimate the probability that John got the job. This assumption is usually called the hypothesis (H). As all applicants in the short list have equal opportunity to get the job, John – as well as the other four – has equal probability of 1/5 (or 20%) to get the job. This estimation is called the prior and can be written as:

\begin{equation}
P(H) = \frac{1}{5} = 20\%
\end{equation}

There is another piece of information; that is, John is double qualified than the other four applicants after the final evaluation. This piece of information is called the evidence (E), and it is the information piece that should update our prior believe (John got the job). The goal now is to calculate the probability that John got the job (i.e. P(H)), given the fact that John was double-qualified for the job than all other applicants (i.e. E). This probability is called the posterior and it is mathematically written as P(H|E).

But how can we update our hypothesis (H) using the evidence (E)? Here comes Bayes’ Theorem into the picture. Bayes’ Theory binds the prior and the posterior by a third component called the likelihood. The likelihood is nothing but the opposite conditional probability to the posterior (i.e. P(E|H)) or the probability that John was double-qualified than all other applicants, given that John got the Job.

In order to calculate the likelihood of our example, we need to quantify our evidence with numbers. Suppose the last interview round includes an evaluation sheet of 10 points. In this evaluation sheet John got 8/10 while each other applicant got only 4/10. We can translate these ratios into probabilities as follows:

\begin{equation*}
P(\textit{John was double-qualified $|$ John got the job}) = P(E|H) = 0.8
\end{equation*}

\begin{equation*}
P(\textit{John was double-qualified $|$ John did not get the job}) = P(E|\neg H) = 0.4
\end{equation*}

Now we can write Bayes’ Theorem as:
\begin{equation*}
Posterior \propto Likelihood \times Prior
\end{equation*}

or:
\begin{equation*}
P(H|E) \propto P(E|H) \times P(H)
\end{equation*}

Note that both sides are proportional to each other! in order to equalize them, we need to normalize the right side by using the marginal probability. Marginal probability is the total probability of seeing the evidence E, whether the hypothesis H holds or not. It is calculated as follows:
\begin{equation*}
\begin{split}
\textit{marginal prob.} & = P(E \cap H) + P(E \cap\neg H)\\
& = P(E|H) \times P(H) + P(E|\neg H) \times P(\neg H)
\end{split}
\end{equation*}

Now we can re-write Bayes’ Theorem as:

\begin{equation*}
P(H|E) =\dfrac{P(E|H) \times P(H)}{P(E|H) \times P(H) + P(E|\neg H) \times P(\neg H)}
\end{equation*}

For simplicity and symmetry, the denominator is usually written as P(E):

\begin{equation*}
{\boxed{P(H|E) =\dfrac{P(E|H) \times P(H)}{P(E)} } }
\end{equation*}

Let’s plug in our numbers and calculate the posterior given the above data:

\begin{equation}
\begin{split}
P(H|E) & =\dfrac{P(E|H) \times P(H)}{P(E|H) \times P(H) + P(E|\neg H) \times P(\neg H)}\\\\
& = \frac{0.8 \times 0.2}{0.8 \times 0.2 + 0.4 \times 0.8} = \frac{0.16}{0.16+0.32} = \frac{0.16}{0.48} \\\\
&= \frac{1}{3}   \approx 33\%
\end{split}
\end{equation}

The above result says that according to Bayes’ Theorem, John has 33% chance to be the CTO, given that he double-qualified all other applicants! Does this result make sense to you?