# MAP and ML in practice on the New Jersey Turnpike

Since I might be teaching detection and estimation next semester, I’ve been thinking a little bit about decision rules during my commute down the New Jersey Turnpike. The following question came to mind:

Suppose you see a car on the Turnpike who is clearly driving dangerously (weaving between cars, going 90+ MPH, tailgating an ambulance, and the like). You have to decide whether the car has New Jersey or New York plates [*]?

This is a hypothesis testing problem. I will assume for simplicity that New York drivers have cars with New York plates and New Jersey drivers have New Jersey plates [**]:
$H_0$: New Jersey driver
$H_1$: New York driver
Let $Y$ be a binary variable indicating whether or not I observe dangerous driving behavior. Based on my entirely subjective experience, I would say the in terms of likelihoods,
$\mathbb{P}(Y = 1 | H_1) > \mathbb{P}(Y = 1 | H_0)$
so the maximum likelihood (ML) rule would suggest that the driver is from New York.

However, if I take into account my (also entirely subjective) priors on the fraction of drivers $P(H_0), P_H(1)$ from New Jersey and New York, respectively, I would have to say
$\mathbb{P}(Y = 1 | H_1) P(H_1) < \mathbb{P}(Y = 1 | H_0) P(H_0)$
so the maximum a-posteriori probability (MAP) rule would suggest that the driver is from New Jersey.

Which is better?

[*] I am assuming North Jersey here, so Pennsylvania plates are negligible.
[**] This may be a questionable modeling assumption given suburban demographics.