Question

Let us use Bayes to run an assessment for two future hypotheses:

H1: we should open the economy asap
H2: we should wait until the signs become clearer that it is safe to open the economy.

As leader of the free world, you are gung-ho on H1, so Pr(H1) = .8; and so Pr(H2) = .2
Let’s suppose medical experts give you the following evidence (E):

If H1, then the chance of avoiding a restart of the pandemic is (E) is 10%.
If H2, then the chance of avoiding a restart of the pandemic is (E) is 70%.

a) First draw a diagram, starting with each of the hypotheses, and then for each one, the
evidence associated with them:

b) Now calculate, reading off the diagram, Pr (H1/E) and Pr(H2/E). Show Bayes, show
the parts of the formula, and then each part corresponding to the parts of the diagram. Lastly
calculate (use a calculator if needed) the probabilities.

c) Some new evidence comes in, F (from Fauci): if we open the economy asap, it will be
nearly impossible to avoid a restart of the pandemic (and thus probably necessitating having to
go through all of this all over again). The likelihoods are: Pr(F/H1&E) = 1%; Pr(F/H2&E) = 90%.
What is Pr(H1/E&F)? Show the new Bayes, and each part corresponding to the parts of the
diagram (first draw the new diagram that adds to the old one).

Answer 1

(a)

The probability diagram is as follows.

(b)

By Bayes' theorem we have,

By Bayes' theorem we have,

(c)

The probability diagram is as follows.

By Bayes' theorem we have,