Tim McCarver is a Bayesian with very strong priors.... 

 The Red Sox beat the Indians last night in Game 5 of the ALCS, sending the series back to Fenway and enabling the majority of us at Harvard who are (at least fair-weather) Sox fans to, as Kevin Youkilis said last night, come down off the bridge for a few more days.  Why do I bring this up?  Well, after Boston's loss in Game 4, a commenter on this blog asked the following question: 

 In the disastrous inning of the Red Sox game tonight, the announcer (maybe Tim McCarver?) said “One would think that a lead-off walk would lead to more runs than a lead-off home-run, but it’s not true. We’ve researched it and this year a lead-off home-run has led to more multi-run innings than have lead-off walks.”

 I must not be "one", b/c I think a lead-off home-run is much more likely to lead to multiple-run innings, b/c after the home-run, you have a run and need only 1 more to have multiple, and the actions after the first batter are mostly independent of the results of the first batter.  So, I think he has it totally backwards. I was a fair stats student, so I need confirmation. He was backwards, right?  

 The short answer is that it was Tim McCarver, and as an empirical matter he was wrong to be surprised.  I don't have access to full inning-by-inning statistics over a long period of time, but the most convincing analysis I found in a quick search ( here ) suggests that between 1974 and 2002, the probability of a multi-run inning conditional on a leadoff walk is .242 and the probability of a multirun inning after a leadoff home run is .276.   

  

  
  


 The blogosphere has had a lot of fun at McCarver's expense (not that it takes much to provoke such a reaction, granted):   It's Math! ,  Zero > One ,  Tim McCarver Does Research , etc.  His observation, though, is a good example of Bayesian updating at work: while I doubt that most baseball observers "would think that a lead-off walk would lead to more runs than a lead-off home-run," it is very clear that  Tim McCarver  thought that at some point.  As evidence, in a 2006 game he made the  following comment : 

 "There is nothing that opens up big innings any more than a leadoff walk. Leadoff home runs don't do it. Leadoff singles, maybe. But a leadoff walk. It changes the mindset of a pitcher. Since he walked the first hitter, now all of a sudden he wants to find the fatter part of the plate with the succeeding hitters. And that could make for a big inning." 

 In 2004, he said during the Yankees-Red Sox ALCS that " a walk is as good as a home run ."  And back in 2002, he made a similar comment during the playoffs; in fact, it was that comment that prompted the analysis that I linked to above!  Clearly, he had a strong prior belief (from where, I don't know) that leadoff walks somehow get in the pitcher's head and produce more big innings.  Now that he's been confronted by data, those belief are updating, but since his posterior has shifted so much from his prior it's not surprising that he thinks this is some great discovery.  In a couple of years, he'll probably think that he always knew a leadoff home run was better.  

 As for the intuition, it looks like the commenter is also correct.  Using the data cited above, the probability of scoring zero runs in an inning is approx. .723, while the probability of scoring no additional runs after a leadoff homer is approx. .724; the rest of distribution is similar as well.