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1. How to Win at the Airport, for Heels First. I'm spending a lot of time in airports this week and need to keep this advice in mind.
2. Does the Surveillance and Security State Make Us Safer?, for Forbes.com. Fun with Bayes' Theorem and a festival of false positives.
3. What Does Good Feedback Get You?, for Kosmos Online.
When the revelations from Snowden first came out, I read a blog post from a scientist who explained via Bayes Theorem why the NSA's PRISM system was such a bad idea. A quick Google search showed that when allegations of NSA abuse surfaced during the Bush administration, a number of people used the Bayes Theorem critique to show why mass data collection was a bad idea. One such person who used this argument was economist Matt Yglesias. But now that it's a different regime in the White House, Yglesias has not bothered to trot out this critique again.
I looked at your Baseball edition of Better Living Through Statistics (http://www.forbes.com/sites/artcarden/2013/08/22/better-living-through-statistics/) – and although your statement of Bayes’ Theorem is fine, the example you’ve provided is not a good application of it.
In particular, the statement “Bob is correct 55% of the time when he predicts a victory” is not equivalent to “the probability that Bob predicts a victory, given that the Yankees subsequently win that game, is 55%”.
Perhaps this is best clarified with an example: suppose Bob is a terrible homer, and he predicts the Yankees to win 40 out of the first 40 games. It turns out that the Yankees start the season with a 22-18 Win-Loss record… not bad, but nowhere close to the 40-0 record that Bob predicted, so Bob gives up on the Yankees and predicts that they will lose the next 60 games. Then the Yankees go on an incredible 50-10 Win-Loss record for their next 60 games, bringing their record to 72-28. Now Bob is emboldened, and decides to get back on the “win” bandwagon for the 101st game. In this situation, going in to that 101st game, all of the following things are all true:
- Bob is correct 55% of the time when he predicts a victory.
- But in total, Bob is only correct 32% of the time (considering all his predictions for wins and losses).
- The probability that Bob predicts a victory, given that the Yankees win, is much worse than 55%: 22 / 72, or about 30.56%. This is the proper value of the “p(B | A1)” term.
- And note that the probability that Bob predicts a victory, given that the Yankees lose, is 18 / 28, or about 64.29%. This is the proper value of the “p(B | A2)” term
The net effect (borne out by both an application of Bayes’ theorem, and by actually looking at the Yankee’s Win-Loss record in games that Bob predicted they would lose) is that the Yankee’s in fact do a lot better than 72% when Bob predicts them to lose, and a lot worse than 72% when Bob predicts them to win!
@Tim: Thanks; I agree that it's awkwardly stated. Let me think about it and see if it needs fixing.