P[ :C]: Change
This develops the notion of ‘possible probabilities’where instead of representing an uncertainty situation by a single, precise, ‘Bayesian’ probability distribution, P( ), one considers (after Boole) sets of possible probabilities, conditioned on the context, including any assumptions. These are denoted P<A|B:C>.
Merging short-run and long-run uncertainty
The situation of interest is where the short and long-run probabilities differ. Typically the long-run probabilities will be less precise than the short.
Suppose, for now, that we expect a series of epochs within a single supra-epoch, each epoch having its own probabilistic characteristics. Suppose that we have good estimates of
P< :Long> for the supra-epoch.
P< :Short> for the current epoch
and we want to estimate P(Heads) for the next turn, without knowing whether the epoch will continue. We can say:
- P< :Same> = P< :Short>
- P< :Change> = P< :Long>
Now, for any λ∈(0,1],
P< :C> = (1-λ).P< :Short> + λ.P< :Long>
is a valid set of probability functions, and is the appropriate set for a probability, λ, of the epoch changing on the next turn. In this situation the probability of the epoch having changed by the n-th turn is 1-(1-λ)n. If we use Pn< :C> to denote the state on the n-th turn, then
Pn< :C> = (1-λ)n.P< :Short> + 1-(1-λ)n.P< :Long>.
Pn< :C> → P< :Long> as n → ∞.
This is a kind of ‘reversion to the long-run probabilistic characteristics’. If we are only interested in the probability at some specific time then then the uncertainty associated with the possibility of change is of the same type as the more conventional probability function: one only needs to adjust the parameters. Moreover, the adjustments to be made increase gradually with n.
Now, suppose that instead of a change of epoch being imposed randomly from outside (by the supra-epoch), it comes about as a result of the state of the current epoch. As a simple example, suppose that the epoch can satisfy some condition ‘x’, and when it is satisfied ‘x’ m times in succession, the epoch changes. Suppose that P(‘x’) = λ, a constant, and that ‘x’ has already been met k times (k < m). Then:
Pn< :C> = 1 for n ≤ m-k, otherwise
(1-λm-k).P< :Short> + λm-k.P< :Long>.
This is qualitatively different from the previous case, of independent random changes. Instead of there always being the same gradually increasing profile of uncertainty out to the future, unless λ is almost 1 there is a modest effect due to the possibility of change until one is close to meeting the condition (k ≈ m), when the effect suddenly ‘blows up’.
Notice that in this model the timing of a change to the supra-epoch is determined by events in the epoch, but the supra-epoch determines the nature of the change, as in a trial-and-improvement process. This is a simple form of ‘Holism‘.
An uncertainty principle
According to the Heisenberg uncertainty principle, the variance in location measurement is inversely proportion to the variance in momentum measurement. The equivalent here is that if we have statistical measurements of the mean within the epoch and supra-epoch, then – for enough measurements – the variance of the first is proportional to 1/λ, while the variance of the second is proportional to λ, a similar result.
More generally, estimation will add errors proportional to 1/λ to the variability inherent in P< :Short> and proportional to λ in P< :Long>. Hence for each n there is a λ for which estimates of Pn< :C> are most accurate. But the difference may not be very significant if Pn< :C> is already ‘muddled’.
Draft: More to come.