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 :-).)
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.