And looking at how he used up his time much sooner, he was more cautious today. He still lost and probably also took a psychological hit, so now my estimate of chances of Lee Sedol winning the whole match went down to ~5%.
Ignoring psychology and just looking at the results:
Delta-function prior at p=1/2 -- i.e., completely ignore the first two games and assume they’re equally matched. Lee Sedol wins 12.5% of the time.
Laplace’s law of succession gives a point estimate of 1⁄4 for Lee Sedol’s win probability now. That means Lee Sedol wins about 1.6% of the time. [EDITED to add:] Er, no, actually if you’re using the rule of succession you should apply it afresh after each game, and then the result is the same as with a uniform prior on [0,1] as in #3 below. Thanks to Unnamed for catching my error.
Uniform-on-[0,1] prior for Lee Sedol’s win probability means posterior density is f(p)=3(1-p)^2, which means he wins the match exactly 5% of the time.
I think most people expected it to be pretty close. Take a prior density f(p)=4p(1-p), which favours middling probabilities but not too outrageously; then he wins the match about 7.1% of the time.
So ~5% seems reasonable without bringing psychological factors into it.
Laplace’s law of succession gives Lee Sedol a 5% chance of winning the match (and AlphaGo a 50% chance of a 5-0 sweep). It gives him a 1⁄4 chance of winning game 3, a 2⁄5 chance of winning game 4 conditional on winning game 3, and a 1⁄2 chance of winning game 5 conditional on winning games 3&4. It’s important to keep updating the probability after each game, because 1⁄4 is just a point estimate for a distribution of true win probabilities and the cases where he wins game 3 tend to come from the part of the distribution where his true win probability is larger than 1⁄4. It is not a coincidence that Laplace’s law (with updating) gives the same result as #3 - Laplace’s law can be derived from assuming a uniform prior.
Hmm, I explicitly considered whether using LLS we should update after each new game and decided it was a mistake, but on reflection you’re right. (Of course what’s really right is to have an actual prior and do Bayesian updates, which is one reason why I didn’t consider at greater length and maybe get the right answer :-).)
And looking at how he used up his time much sooner, he was more cautious today. He still lost and probably also took a psychological hit, so now my estimate of chances of Lee Sedol winning the whole match went down to ~5%.
Ignoring psychology and just looking at the results:
Delta-function prior at p=1/2 -- i.e., completely ignore the first two games and assume they’re equally matched. Lee Sedol wins 12.5% of the time.
Laplace’s law of succession gives a point estimate of 1⁄4 for Lee Sedol’s win probability now. That means Lee Sedol wins about 1.6% of the time. [EDITED to add:] Er, no, actually if you’re using the rule of succession you should apply it afresh after each game, and then the result is the same as with a uniform prior on [0,1] as in #3 below. Thanks to Unnamed for catching my error.
Uniform-on-[0,1] prior for Lee Sedol’s win probability means posterior density is f(p)=3(1-p)^2, which means he wins the match exactly 5% of the time.
I think most people expected it to be pretty close. Take a prior density f(p)=4p(1-p), which favours middling probabilities but not too outrageously; then he wins the match about 7.1% of the time.
So ~5% seems reasonable without bringing psychological factors into it.
Laplace’s law of succession gives Lee Sedol a 5% chance of winning the match (and AlphaGo a 50% chance of a 5-0 sweep). It gives him a 1⁄4 chance of winning game 3, a 2⁄5 chance of winning game 4 conditional on winning game 3, and a 1⁄2 chance of winning game 5 conditional on winning games 3&4. It’s important to keep updating the probability after each game, because 1⁄4 is just a point estimate for a distribution of true win probabilities and the cases where he wins game 3 tend to come from the part of the distribution where his true win probability is larger than 1⁄4. It is not a coincidence that Laplace’s law (with updating) gives the same result as #3 - Laplace’s law can be derived from assuming a uniform prior.
Hmm, I explicitly considered whether using LLS we should update after each new game and decided it was a mistake, but on reflection you’re right. (Of course what’s really right is to have an actual prior and do Bayesian updates, which is one reason why I didn’t consider at greater length and maybe get the right answer :-).)
Sorry about that.