In the last thread, all I had said about surprisingness (or all I meant to say) was that if our beginning expectations give the same probability distribution as a fair coin, then all sequences of a given length would be equally surprising to us, at least with respect to just this. However, in the real world, as I’ve noted, I don’t think anyone’s expectations are anything like the probability distribution of a fair coin, in that, in the real world, we all find 1001 H much more likely than 1000 H + 1 T, and such things. (I equate, in my Bayesian way, the extent to which we believe something, the probability we assign to it, the level to which we find it plausible, etc.)
I agree that the second result causes a feeling of surprise, or something like it, which the first one doesn’t, and that there is a justification for this. However, before I get to that, let me note that I actually think we view the second result as more likely than the first; after all, we think there is a possibility that a coin has been tampered with or is special in some way, that a special coin is more likely to be inclined towards the second result than towards the first, and that nonspecial coins are equally likely to go either way.
Now, then, as for the fact that we somehow, nevertheless, experience a feeling like surprise with the second one which we do not with the first, I think this is because the second one causes a sharp belief revision in a way which the first one does not. Upon viewing the second one, we find ourself switching over from a low probability to a high probability of the coin having some sort of weight upon it, or being magnetically controlled, or having heads on both sides, or simply being causally ordained by pure will of God to come up heads now and forevermore, or at least now and most of the time forevermore… We will also, as a result, find ourself switching to a very high probability of the next flips being heads. This updating of our beliefs/plausibility ratings is justified by the fact that our prior probability distribution/plausibility ratings, before viewing the data, was in fact not a uniform distribution, it was skewed in all the right ways to allow us to make these sorts of inductive leaps. Whereas, with the first sequence above, no drastic belief revision of the above sort happens; certainly, it is conceivable that a coin could be rigged in more complicated a manner as to come up in this fashion, but we consider such a rigging so implausible as to make the effect of viewing the first sequence almost negligible, in terms of revision of distant beliefs. [Of course, there are some beliefs which the first one causes us to revise too; it makes us go from thinking “It very likely won’t be HTHHTHHTHT” to thinking “It definitely was HTHHTHHTHT”, but that sort of thing doesn’t stir any feelings in us; feelings are stirred when evidence causes us to revise beliefs which are a little further out and of more importance or use to us in deriving new conclusions].
I agree that what’s unlikely, and what causes our feelings to stir, is a “meaningful” result coming up, such as would trigger us to begin carrying out inductive inferences. However, even though we find P(Something “meaningful” happens) to be much less than P(Nothing “meaningful” happens), I actually think, for our reasoning to work, it must also be the case that we find particular meaningful sequences rather more likely than particular nonmeaningful sequences, as I explained above.
