# Posterior probabilities: A Bayesian update of the Miami Heat

October 27, 2010

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

Event

Miami did not beat the Celtics on the road last night.

Question

What is the updated probability Miami is a juggernaut?

Solution

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.

### 7 Responses to Posterior probabilities: A Bayesian update of the Miami Heat

1. Chris on October 27, 2010 at 9:58 am

Prior:
Probability Miami sports radio will scream the sky is falling every time the ‘invincible’ Miami Heat loses? 110%

Event: Miami loses to the Celtics

Solution: Yep, complete meltdown this morning. In case you were wondering, the world is ending, or if not this is certainly worse than the foreclosure crisis, 11% unemployment, and Deep Horizon – combined!

2. LD on October 27, 2010 at 10:01 am

What are you basing your priors on?

• admin on October 27, 2010 at 11:17 am

As I say in the update, Juggernaut is implicitly defined as a team that can beat the Celts 75% of the time on the road — I haven’t done the math, but a vegas line of 65-68 wins on the season is probability associated with that kind of chance against the C’s in Boston. The 90% Juggernaut is my estimation of popular opinion about the Heat, reflecting the common view that they will be an all-time great time this year, and the small minority who think that’s crazy talk.

3. John on October 27, 2010 at 10:28 am

Your math is right (And yes, I am qualified to check it! Genetic analysis of pedigrees is based on Bayesian inference).

As for the conclusion, I think the Heat are not and will not be a juggernaut, because they don’t have a center, and you need one, especially in the playoffs when the other power teams in your conference are the Celtics (Perk/Shaq/Baby) and Magic (Howard). That’s not to say the Heat can’t beat either of those teams, as they too have flaws (former: age; latter: lack of other talent), but the Heat are not on par with the best edition of Bird’s Celtics or Jordan’s Bulls or Kobe & Shaq’s Lakers (hmm, discuss, should that list include a particular edition of Magic’s Lakers?)

• Dan D on October 27, 2010 at 12:51 pm

As noted, the math is right.

John, how could one not include Magic’s Lakers in that list? I hated them, but they did beat Boston 2 of 3 times in the finals, right?

• John on October 28, 2010 at 2:03 pm

I should have been more clear: I was having a hard time thinking of a particular year for Magic and the Lakers. I mean, the 86 Celtics were clearly Bird’s best team, and the 96 Bulls were probably Jordan’s best team. But which Lakers team was Magic’s best?

4. Matt on October 27, 2010 at 12:20 pm

Your math is right, provided P(D) = P(~J) = 1-P(J), which I believe it does (.9 and .1). I have no comment on the priors, but it would be interesting to see what would happen if you lowered P(J).