What can we say about the Miami heat, one game into the season? Let’s do some Bayesian mathematical analysis…
Priors (as of tipoff last night) [see UPDATES below for alternative priors and good arguments against these]
Probability (Miami is a Juggernaut this year) = 90% = .9
Probability (Miami is a disappointment) = 10% = .1
Probability (Juggernaut Miami beats the Celtics on the road) = 75% = .75
Probability (Disappointment Miami beats the Celtics on the road ) = 30% = .3
Miami did not beat the Celtics on the road last night.
What is the updated probability Miami is a juggernaut?
Use Bayesian inference.
Probability Miami is a juggernaut given that they lost last night = ((Probability they lose if they are a juggernaut)(Probability they are a juggernaut)) / (((Probability they lose last night given they are a juggernaut) (Probability they are a juggernaut)) + ((Probability they lose last night if they are a disappointment) (Probability they are a disappointment)))
Written in simple notation:
P(J | E1) = ( P(E 1| J) P(J) ) / (( P(E1 | J) P(J) ) + (P(E1 | D) P(D)))
P(J|E1) = (.25)(.9 ) / ((.25) (.9) + (.7) (.10))
Updated probability Miami is a juggernaut given last night’s outcome = 76%
Still very high.
Obviously, you can quibble with the estimated priors. I have not fiddled with them to check how sensitive the result is.
You can argue that “disappointment” and “juggernaut” are not specified very well — outside the context of how often a juggernaut or a disappointment beats the celtics, the equation is silent as to the definition of the two terms.
You can also say that this model does not account for teams improving over the course of the year relative to other teams ( as Miami, with virtually a whole new team, will undoubtedly do).
But if you accept the priors as reasonable — that a juggernaut would beat the Celts around 3/4 of the time last night, but a disappointment would beat the Celts only about 1/3 of the time — then the conclusion is that Miami is still likely to be a juggernaut, or at least that it’s way too early to conclude that Miami is going to be a disappointment this year.
Which I bet will stand in contrast to what surely every sportstalk radio station from Pensacola to Havana is saying this morning.
Someone (Tom? Doherty?) check my math. Everyone assess my priors.
UPDATE: There seems to be some confusion over the definition of “juggernaut.” As structured in the model, juggernaut only is definable in reference to the ability to beat the Celtics on the road. Implicitly, the conclusion is that a juggernaut is a team that can beat the Celtics on the road 75% of the time. That seems reasonable to me.
UPDATE #2: My brother-in-law makes a convincing case that some of the priors are off. He notes that the ’96 Bulls were only .800 on the road, and they were the best road team ever, so it’s unlikely that Juggernaut Miami will win 75% of the time against a Top-5 NBA team. So let’s soften that to 55% and rerun the numbers:
P(J|E1) = (.45)(.9 ) / ((.45) (.9) + (.7) (.10))
and this yields an updated probability of Juggernaut at 85%.
Same conclusion: not much can be learned from last night.