P[ : C]: Applications
Possible probabilities, P<A|B:C> are a generalisation of the more conventional precise probabilities, P(A|B). They include the full range of possible solutions consistent with the given constraints and the context, C, including assumptions and heuristics. The idea can either be applied directly, or indirectly.
- To apply possible probabilities directly and fully one can proceed as normal, but preserving the full range of solutions rather than collapsing them to a precise value. This can then inform decision-making, e.g. using an appropriate attitude to risk.
- One could use interval representations of probability to yield a potentially vaguer result, refining the assumptions or the hypotheses to give the required precision, if necessary.
- One can also apply the theory in full but introduce explicit assumptions and heuristics to get the conventional result.
- Indirectly, one can often take an existing conventional probability estimate and assess whether or not it is likely to be reasonable.
In the last case, one needs a general theory that identifies when the conventional (precise) approach is or is not reliable. We have already identified the following issues that Ramsey, von Neumann et al highlighted:
- Conventional methods are extrapolations, and hence only valid in the short-run. In particular, they assume stability and hence cannot be used to draw conclusions about it.
- Conditional probabilities P(A|B) only apply to random samples from B, and only apply to sub-sets of B if B is assumed to be homogenous. (This is often not the case.)
- When combining evidences Ej against hypotheses Hi the conventional definition of likelihoods, P(Ej|Hi), can give systematically biased results when some Hi are imprecise, due to their dependence on priors.
Prior to the 2007/8 financial crisis the notion of ‘risk’ was based on variability, and it was supposed that a crash of the type that happened was highly improbable. But with the above points in mind we see, like Keynes and von Neumann and Morgenstern, that:
- The notion of risk was simply the chance of the current system failing due to random variability. It took no account of any possible systemic crash.
- To inform effective action it is not enough to think in broad categories: the details can matter. (Are we in another slump, where we need to apply the remedies that worked in the past? Or do we need to be more forensic?)
- It may not always be the case that honest people drawing on a huge evidence base will necessarily converge on their views. (Some ideologies may so shape priors that they are virtually unfalsifiable. Again, one needs to be more forensic.)
Of course, mathematics could be part of any forensic investigation. It is a pity that, as in economics, it has often been relegated to mere extrapolation.