Something Bayesian. Despite my best effort I just do not get Bayesian probability, it more or less just does not make sense to me. Can you convince me otherwise? What is your best example of something with a probability that can not be analyzed in terms of frequencies or other proportions? And your Bayesian account of it must make sense, I am 90 % certain that P != NP and that is why I would take bets based on those odds does not cut it.

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glial 34 minutes ago | parent | next [-]

Someone walks out of a magic store holding a coin.

They propose a bet. If they flip it 100 times and the proportion of heads is within [0.4, 0.6], you win $100. If it's not, you pay $100. Do you take that bet?

Explanation: absent the magic store scenario, a `rational' person would take the bet. Your prior belief is that most coins are roughly unbiased. Given that they walked out of a magic store, you now have additional information. Maybe the coin is a trick coin. In that case, your belief that the coin is unbiased should be weaker, even if you don't know which direction the coin is biased in.

This illustrates two things: one, additional information (magic store) can update your beliefs. Two, a strong prior and a weak prior, in this case about the coin's bias, can lead to materially different decisions.

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jonahx an hour ago | parent | prev | next [-]

Any one off event is an example. But I assume you know that, so can you clarify what you mean by "a probability that can not be analyzed in terms of frequencies or other proportions"?

	▲	danbruc 27 minutes ago \| parent [-]
		Let us start with coin flips. You repeatedly flip a coin and the number of heads will come out to be about half the number of trials. Where does that come from? It is not some intrinsic property of the coin, it comes from varying initial conditions. If you had enough precision when controlling your hand movements, you could in principle force an outcome with high probability. But assuming you can not or at least do not do that, there is a certain set of initial states, some will lead to heads, some to tails, and each toss will start from a randomly selected initial state. So given my ignorance of the exact initial state, the coin will land heads with a probability equal to the number of initial states leading to heads divided by the number of initial states compatible with my observations of the initial state. [1] Repeatedly tossing a coin will sample the set of initial states and the result will match the proportion of the number of states. At least as long as I am not wrong about the set of initial states. The same applies to something like an election. I have imperfect knowledge about the state of the world but there is a set of states compatible with my knowledge about the world and certain subsets of them will lead to certain candidates to win. [1] Maybe adjusted by some probability distribution over the initial states if they are not equally likely to be picked.

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kgwgk an hour ago | parent | prev [-]

What's the probability that the sinking of the USS Maine in 1898 was accidental?

	▲	danbruc 5 minutes ago \| parent [-]
		One could look at the likelihood of spontaneous coal fires and their effects, gather evidence about the activities surrounding the event, essentially trying to narrow down the set of possible states that would lead to the incident and then see what proportion of those states cause the sinking by fire, military action, or something else. But then attaching a number to that seems a dauting task.