In recent days this picture constantly popped up in my LinkedIn feed. It shows two predictions for the outcome of the 2022 Australian Open Final between Medvedev and Nadal, made in two different points in time: one before the match and the second when the score was 2-0 Medvedev (remember: the first who gets to 3 wins).
In the end Nadal won 2-3 against all odds, but it's not what I'm interested in.
I am rather interested in understanding if it makes sense to apply some Bayesian notions to this scenario. Specifically, what I asked myself is: can I imagine that the model which output the second prediction was a Bayesian model and treat these two sets of predictions as the prior and the posterior? If I can, can I compute the likelihood that made the model output that posterior? And anyway, does it make any sense? I mean, is it something interesting/relevant?
With this in mind, I defined:
- $P(H)$ = prior hypothesis: Medvedev's initial probability of winning = $64/100$
- $P(\neg H)$ = Medvedev's prior propability of losing = $36/100$
- $P(E|H)$ = (not sure how to phrase it, any help is appreciated...) = $\lambda$
- $P(E|\neg H)$ = (not sure how to phrase it, any help is appreciated...) = $\mu$
- $P(H|E)$ = Medvedev's updated probability of winning = $96/100$
Then I applied the formula as follows: $$ P(H|E) = \frac {P(H) P(E|H)}{P(E)} = \frac {P(H) P(E|H)} {P(H) P(E|H) + P(\neg H) P(E|\neg H)} $$ $$ \frac{96}{100} = \frac {\frac{64}{100} \lambda} {\frac{64}{100} \lambda+ \frac{36}{100} \mu} $$
Following some advice, I solved for $\lambda / \mu$: $$ \frac{\lambda}{\mu} = \frac{27}{2} = 13.5 $$
Also, for simplicity, I found two values $\lambda_0$ and $\mu_0$ such that $\lambda_0 + \mu_0 = 1$ and I got: $$ \lambda_0 = 27/29 \approx 93\% \text{ ; } \mu_0 = 2/29 \approx 7\% $$
Given this I'd conclude that a Bayesian model that started with a 64-36% prior would have needed to compute a 93% likelihood in Medvedev's win to update it to 96%. I mean, it had to believe that Medvedev would win 93% of those 64 matches. Is my claim valid? Is this formulation correct? Is this a correct way to apply the Bayesian Theorem in a practical scenario?
I would like to hear and learn from the community! :)
Note You might have encountered this question, slightly differently on CrossValidated but I preferred to move it here since I discovered it is a bigger community and it is possibly even more relevant here.