Frequentist and Bayesian reasoning are two ways to handle Knightian uncertainty. Frequentism gives you statements that are outright true in the face of this uncertainty, which is fantastic. But this sets an incredibly high bar that is very difficult to work with.
For a classic example, let’s say you want have a possibly biased coin in front of you and you want to say something about its rate of heads. From frequentism, you can lock in a method of obtaining a confidence interval after, say, 100 flips and say “I’m about to flip this coin 100 times and give you a confidence interval for p_heads. The chance that the interval will contain p_heads is at least 99%, regardless of what the true value of p_heads is” There’s no Bayesian analogue.
Now let’s say I had a complex network of conditional probability distributions with a bunch of parameters which have Knightian uncertainty. Getting confidence regions will be extremely expensive, and they’ll probably be way too huge to be useful. So we put on a convenient prior and go.
ETA: Randomized complexity classes also feel fundamentally frequentist.
Frequentist and Bayesian reasoning are two ways to handle Knightian uncertainty. Frequentism gives you statements that are outright true in the face of this uncertainty, which is fantastic. But this sets an incredibly high bar that is very difficult to work with.
For a classic example, let’s say you want have a possibly biased coin in front of you and you want to say something about its rate of heads. From frequentism, you can lock in a method of obtaining a confidence interval after, say, 100 flips and say “I’m about to flip this coin 100 times and give you a confidence interval for p_heads. The chance that the interval will contain p_heads is at least 99%, regardless of what the true value of p_heads is” There’s no Bayesian analogue.
Now let’s say I had a complex network of conditional probability distributions with a bunch of parameters which have Knightian uncertainty. Getting confidence regions will be extremely expensive, and they’ll probably be way too huge to be useful. So we put on a convenient prior and go.
ETA: Randomized complexity classes also feel fundamentally frequentist.