Could you make it work using a conditional probability? i.e. P(heads | do(“bet tails”)) = 0, P(tails | do(“bet heads”)) = 1 . I saw in the post that you said that there was no distribution that could express this, but I am unsure why conditionals cannot be used.
Conditional distributions are perfectly valid beliefs in my sense (see Definition 8), but conditional distributions are not actually probability distributions about anything — rather, a conditional distribution of given is an -indexed family of distributions about , . There is no way to lift this to a probability distribution (except, under the assumption of Haar measure, MaxEnt — under which the d20 bet is strictly dominated by both the Heads bet and the Tails bet).
This paucity of probability alone for epistemology is also what inspired Judea Pearl to invent Structural Causal Models, the source of the do-notation you used. Pearl’s Causal Hierarchy also makes the point that ordinary probability distributions are inadequate to express causal beliefs.
Could you make it work using a conditional probability? i.e. P(heads | do(“bet tails”)) = 0, P(tails | do(“bet heads”)) = 1 . I saw in the post that you said that there was no distribution that could express this, but I am unsure why conditionals cannot be used.
Conditional distributions are perfectly valid beliefs in my sense (see Definition 8), but conditional distributions are not actually probability distributions about anything — rather, a conditional distribution of
This paucity of probability alone for epistemology is also what inspired Judea Pearl to invent Structural Causal Models, the source of the do-notation you used. Pearl’s Causal Hierarchy also makes the point that ordinary probability distributions are inadequate to express causal beliefs.
Thanks, this explains it.