Philip Pilkington, author of the buyable The Reformation in Economics: A Deconstruction and Reconstruction of Economic Theory, asks a follow-up question to the material on the precautionary principle.
If probability is not decision then, I hope you’ll grant, that decision is not probability. (Where’s my Nobel Prize!?). But it seems to me that many probability enthusiasts/fetishists do indeed think that decisions should be made in line with probability theory.
If Px = 0.49, then “naw”. If Px = 0.51, then “yeah, boi”. If Px = 0.5, then crushing indifference, the absurdity of all life and meaning, and so on.
I think what Taleb is getting at (in characteristically cack-handed fashion) is that, as the far more interesting GLS Shackle would say, probability and possibility are two entirely different fields. Probability can only be applied to situations with a known, limited number of outcomes – dice rolls, n = 6. While possibility is relevant when the number of situations is unbounded and subjected to the potential infinities of our imaginations – n = 1… ?. (We mere mortals are only able to conceive of approaching the ? rather than realising the ? in our imaginations – the latter being the province of the Other Being).
Decisions are almost always made in a sphere of possibilities rather than probabilities. Probabilitists often do not realise this. And, in not doing so, they assumed that their models act like dice rather than being the products of their mind’s possibility-generating function which, naturally, is constrained by n = 1… ?.
There is another distinction in British and American English between the words possibility and probability. In Brit, possibility means of small chance, while probability means not unlikely (and not a plain likely). In Yank, possibility means it isn’t impossible, while probability means some kind of quantification or statement of a chance, regardless of size, large or small.
But I do not quite agree with this distinction in Pilkington’s question. Probability is the study of propositions in relation to evidence, and nothing more. Thus C = “The god Zeus had at least three wives” given P = “The god Zeus had at least four wives” has probability 1, and this is so even though if the god Zeus existed or not, i.e. whether we speak of the myth or the superman.
In Uncertainty, I liked to use the example of interocitors, fictional devices from the planet Metaluna, featured in This Island Earth. If we accept P = “This interocitor can and must be in states 1, 2, or 3 only”, then the probability C = “This interocitor is not in state 1″ is 2/3. But if we modify P to “This interocitor will be in some state”, then the probability of C is the uncapped unit interval. There is no information in P to say anything about what the states are except that some states exist. Tacit in that is the possibility of state 1; state 1 is certainly not impossible give P. So there is no fixed number.
What is C = “This interocitor is not in state 1 or 2″? We’re asking more of P. But P has nothing more to give. The probability doesn’t change. We don’t have certainty of C’s falsity or truth. We only have its possibility.
So there is a distinction. Possible is not impossible. In interocitors were real, we would say C is contingent.
Is it possible the number of states of the interocitor are infinite? There are no real-life infinities, which we could say is tacit in P, but the interocitor is not real life, and, anyway, in mathematics there are plenty of infinities (to say the least). If we’re to move to infinities, as folks do when used boxed models, nearly all of which rely on the infinity of the continuum, then we need to be especially careful about the infinities we invoke. People are usually not so careful.
If in some real-life situation we do not know the possible number of states (or situations), we have to assume some boundaries, else probability measures won’t be forthcoming, nor decisions based on them.
The bad but true pun is that infinity is a large subject, which I cover in detail in Uncertainty. There is something else here that I think of greater potential interest, and is also a subject I cover in Uncertainty, but not well enough, I think. And that is weight of evidence, related to comments above about probabilities of 0.49, 0.50, and 0.51. More to come…
(I only just realized I typed all the quotation marks as if I were in LaTeX. Occupational hazard.)