### Will the Cubs Be Swept?

Major league baseball analysts were asked the above question after the Mets had won the first two games against the Cubs.

Several of them respond, "No, the Cubs are too good a team to be swept."

I think they were not being good Bayesian reasoners. They had a prior: "The Cubs are a good team, and the Mets cannot sweep them." In other words, the Mets cannot win four straight games against a team as good as the Cubs.

But they were failing to update this prior given new data. First of all, having won the first two games, the odds of the Mets "winning four straight games" became irrelevant: the only relevant odds were of their now winning two straight games.

Secondly, the Mets' decisive wins in games one and two should have altered their evaluation of the relative strength of the two teams: the Mets were playing particularly well, while the Cubs were not, and thus the odds of the Mets winning two (more) straight games ought to have increased in their minds.

1. You may well be correct, but it depends on the strength of their prior. If we take a simple prior like "for a given game, the Cubs have a 20% chance of winning", then we would believe that the Mets have only a 41% chance to sweep. If we lower the prior for an individual Cubs win to 15%, then we would expect the Mets to sweep.

On the second point, I'd argue that changing your prior while the series is in progress is doing something other than - or in addition to - purely statistical reasoning. Which is not necessarily invalid, of course.

1. No, Matt, no change of prior involved: Sorry, I meant to write "Update to new odds, given their prior and the results so far."

If you think the odds of the Cubs winning a game are 1 in 2, you think the odds of the Mets winning four straight are 1 in 16. If the Mets win two straight, now the odds of a sweep are 1 in 4, since now a sweep only means winning two straight.

2. Aided by the clear thinking of a morning cup of coffee, I realize that my first comment was incorrect with respect to your first point about the odds of the Cubs winning the remaining two games given that they lost the first two. The prior would have to be higher to expect the Cubs to win a game: around 30%.

I find the second point more interesting. It suggests that to the extent that our intuitive reasoning is Bayesian, our priors are more nuanced and multifaceted than we would put in a model. For example, instead of holding a prior like "for a given game, the Cubs have X chance of winning", we would hold many detailed prior beliefs about the team's strengths and weaknesses and the relationships between them. This allows us to update said priors with data other than just "win" or "loss".

1. Of course, there might be an element of confidence or demoralization occurring after the first two results!