Psychology and Uncertainty

Rational Belief versus Logical Doubt

This note is about two kinds of psychology as they relate to two kinds of uncertainty. Contemporary behavioural psychology supposes that there is some normative rational decision method compared with which people have biases, presumed to be ‘a bad thing’. This normative rationality uses a normative type of uncertainty: numeric (Bayesian) probability. In contrast an older ‘scientific’ psychology has a broader view of uncertainty, that I seek to show here.

Preliminary Logic

To start with, consider mathematical probability theories. These considers situations where there is a map P( ) that assigns normalised measures to some statements, using the notation P(X) = p to mean that ‘for the given measure, the probability of the statement X is the number p’. From these are derived conditional probabilities, denoted P(X|Y) = p. The key result is then Bayes’ theorem. This is not controversial. But where do the probabilities come from?

Two views of Psychology

The point of departure between the behavioural and scientific versions is that in the behavioural version it is implicitly believed that there is only one true probability P( ), and anything else is biased.

William James’ scientific notion of psychology is quite different:

“What characterizes both consent and belief is the cessation of theoretic agitation, through the advent of an idea which is inwardly stable, and fills the mind solidly to the exclusion of contradictory ideas.”

Thus logic and mathematics, which often fail to find any theoretical justification for any definite probability, are allowed to withhold consent and belief from both particular precise probabilities and from the general proposition that such probabilities are always possible and even necessary to rationality.

On the other hand, James gives psychological reasons why at least some people do tend to form beliefs even when there is inadequate logical support for them. In particular, people tend to believe things that are instrumental, as Bayesian probability is. But, armed with such a description, we may doubt that such a result should be either normative or reliable.

A slight complication of James’ characterisation is that we can consider cessation of theoretic agitation on different time-frames. A current belief might be one in which we are not actively agitating. A firm belief might be one which we would not doubt, no matter what. And there may be intermediate cases to consider. Thus, far from being ‘biased’ and ‘irrational’ (bad), James leaves open the possibility that some people are rational with respect to their logics, and perhaps even better than common ‘rationality’ (good).

Intermediate Cases, Informed by Mathematics

Two extreme cases are people (like James’) who tend to settle on their beliefs quickly, and those who always agitate, perhaps as a matter of principle. More interesting is ‘theoretic agitation’ that depends on the circumstances, and to consider how conditional belief might be.

 

If we consider a schoolbook example involving a coin toss, then by convention we are supposed to believe that THE ‘probability’ of a ‘Head’ is a half (conventionally denoted “P(‘Heads’) = 1/2”). Logically, this must rely on some missing premises, but in this context (a school book) we have been taught what assumptions to make (e.g., that any given coin is fair). If faced with a real coin really being tossed, we would have to form our own premises. If we accept that real coins are just like text-book ones, we can be considered ‘rational’. If not, we might be able to determine the text book answer, or otherwise make a best guess, and yet still doubt it. So instead of considering probabilities P(X) as if they were normative, I want to condition them on the premises, P(X:premises). It is not controversial that this has a normative value for SOME premises. But the conventional claim seems to be that there is always a normative set of premises for which the probability has a normative value.

I do claim that one can always make bounds such as

p <= P(X:premises) <= q, where

but I do not claim that the bounds can always be precise (p = q). Besides probability, I then have the following potential causes of uncertainty:

  • Bounds that I think should be precise, but I doubt my estimate for their value (as in statistics).
  • Bounds that I doubt are precise.
  • Doubt about the premises.

For example, for a real coin:

  • I may doubt that it is exactly fair.
  • I may doubt that the probability of ‘Head’ is independent of how it is tossed.
  • I may doubt that it has been selected at random out of similar coins in circulation or that the person tossing it is not a magician – or a sneaky mathematician.

The view that uncertainty is ‘nothing but’ numeric probability implicitly holds that the above doubts are irrational. Various arguments have been given to this effect, but these severely limit the actions that those involved can take, forcing the subjects to act ‘as if’ they are rational in the behavioural psychologists’ sense. For example, some examples involve forced gambles. Implicitly, one ‘should’ think that to be rational one ‘should’ behave consistently with one’s actions, even when they were forced. In contrast, using James’ definition, we may think it reasonable not to be committed to be consistent with forced actions. Thus people might reasonably doubt the probability values that their actions imply if they were acting under duress.

Having been to a few public lectures on the public understanding of probability, it seems to me that the lecturers used to use a reasonably fair coin, but more recently have started using double-headed coins to illustrate the difference between a schoolbook coin and a real one. Thus supposedly ‘irrational’ theoretic agitation can be justified (in James’ sense) by attending such talks.

We can always reduce the doubt for a statement by making the dependence on assumptions and context explicit. For example “the probability of ‘Heads’ is 1/2, as long as the coin is fair and the tosser isn’t a cheat” is much less doubtful than the unconditional claim. As Boole noted, it is also useful to be able to identify the main source of doubtfulness: the conditioned probability or some specific premise. Typically, if one doubts an assumption made in particular case, one will tend to reconsider it. Sometimes identifying doubtful factors can also help avoid misleading claims. For example ‘an economic crash is extremely unlikely’ superficially sounds reassuring. ‘An economic crash is extremely unlikely, as long as the economy is stable’ much less so.

One should strive to make assumptions explicit, but the context is often hard to define. Instead one may be able to give some scenarios in which the context (whatever it is) would be changed, focussing on those scenarios that contribute most to the doubt over the context. For example, according to game theory, economic business as usual is more doubtful whenever a dominant economic power becomes challenged.

If doubt could be measured precisely one could derive a probability. Bayesians often implicitly assume that this can be done. But we only need to assume a partial order, such as derives from imprecise measures. One then wants to identify sources of doubt that are not dominated. It seems, then, that there are a range of circumstances in which ‘theoretic agitation’ might be more or less indicated, and that we could relate these to types of belief (and inversely, doubt) which – at least in some cases – might be compared. In economics, for example, it is usual to assume away a whole range of possibilities. Such theories may be regarded as more doubtful the more they assume away.

In the case of tossing a real coin, we might significantly reduce doubt by experimenting with the coin, using statistics to estimate the probability of ‘Heads’. We might go further still by challenging others to get biased results. We might then perhaps believe in a fairly precise probability of ‘Heads’, unless the coin tosser is clever than those who accepted our challenge.

There is also a problem that the actual context tends to change with time. If, years ago, I had gone around studying swans then the probability ‘for me’ that ‘all swans are white’ might well have increased to the point where I simply believed it. But as Captain Cook set sail it may nonetheless occurred to me to doubt it. Thus we may distinguish between a belief that it would be pointless for us to examine the subject further and a more absolute belief that it will always be pointless. For example, we might give up trying to get biased results from coin tossing yet remain open to the possibility that others might achieve it.

Induction

The above suggest to me that in the short-term we match what we sense to our mental constructs, and in the longer term we may update our constructs, but only when we admit to some need, which we will be loth to do if the sense-making habit is well established. In particular, there seems to be some psychological law of induction: the longer something has been established without any apparent snags, the firmer it is established and the less likely to be revised. Such induction has long been controversial. Perhaps we could merely say that the longer that something has been established, the firmer it is established within the current context. But this is not to say that the current context will endure for ever. For example, geometry was firmly established as ‘the’ theory of space until Einstein devised an alternative and Eddington conducted the crucial experiment.

Classification

In the behavourist’s extreme we can classify things arbitrarily and then assign probabilities, so that P(‘X is a dingbat’)=p is always treated as if it is meaningful, no matter how arbitrary the class ‘dingbat’ may be. Many Bayesians are not so extreme: they simply assume that meaningful probabilities exist for practically useful classes. In this they follow James. Yet there seems to be no way to be sure that any such classification will remain sound in all contexts. For example, ‘smart phones’, ‘smart TVs’ and Internet-enabled computers blur the old hard distinctions between telephones, televisions and computers. Thus, for me, reliance on a particular classification depends on some premises, which may matter.

For example, suppose that X is a dingbat and 80% of dingbats are whatnots. Then, conventionally, the probability that X is a whatnot is 80%. But the implicit premise here is that ‘dingbat’ is an appropriate class to judge whatnot-ness. For example, if 80% of Muslims named in news headlines are terrorists and X is a Muslim named in the headlines, then – conventionally – X is probably a terrorist. There seems no logical reason why this should be so, or at least why ‘X is probably a terrorist’ should have its usual implications.

James notes that the scientific method relies on classification. This seems reasonable in so far as the classes of the ‘hard’ sciences (e.g. physics) tend to be well established and continually checked as being both meaningful and relevant by an appropriate scientific process. It may be less reasonable in some of the softer ‘sciences’, such as economics, where classifications often seem more ideological than empirical. It is often noted that some minds and organisations give undue weight to what they think is measureable (e.g. probability). Perhaps they also give undue weight to what they think is classifiable, as an essential precursor to measurement.

 

Discrimination

Treating some person or thing as a member of an inappropriate classification is ‘discrimination’ in the bad sense. But some discrimination is always necessary.

If our habitual classifications fail to make a particular discrimination then we may wrangle with reality as much as we will and yet find no reason to update our classifications, perhaps attributing variability and unpredictability to randomness.

Where we have appropriate classes it is conventional to attach precise probabilities. The root of ‘bad’ discrimination, it seems to me, is to attach such precise probabilities to the discriminated against class and claim it to be ‘rational’. James opens up the possibility that we might agitate against this, perhaps insisting on imprecise probabilities and the identification of premises and appropriate sub-classes. Thus in walking through a rough area at night or meeting someone at work I might class people as ‘foreign-looking’ but avoid serious discrimination by not taking undue account of this.

Interpretation

If some asks for a probability, expecting a number, they are implicitly asking us to determine the ‘true’ premises. If this seems inappropriate then we should be able give them as caveats, and in some cases give imprecise probabilities. Often we would also want to give scenarios where the probabilities are significantly different from those obtained with the generally accepted premises, particularly where we are doubtful about them.

Similarly, we might challenge the views of others (including ‘experts’) by looking for premises, scenarios and discriminations that they may have overlooked. Even in the conventional view this makes sense, because their premises, no matter how rational, may not be ours.

Updating

Bayesians use Bayes’ rule, which some psychologists regard as rational. In brief, they suppose that there are precise, fixed, conditional probabilities, so that the probability is modified whenever one gets extra data. For example, for a coin we may think that a coin is definitely has “P(‘Heads’) = 1/2” exactly. Then no matter how many Heads in a row we observe, “P(‘Heads’) = 1/2” remains true ‘for us’, according to Bayes’ rule. (Because we have assign probability 0 to every other possibility, and Bayes’ rule never promotes the impossible to being probable.) But such a sequence may lead to us doubting our premises, in which case “P(‘Heads’) = 1” will eventually become more probable.

More generally:

  • If we have no more reason to doubt the premises:
    1. If we have precise probabilities, Bayes’ rule applies.
    2. If the probabilities are imprecise, we can adapt Bayes’ rule.
  • If we doubt the premises more, we may study them in the light of the data.

In terms of statistics and science we can proceed routinely until we have cause to doubt the premises. We might then use the data to form new premises, but we can’t use the same data to estimate likely errors. We need more data, and also theory beyond the scope of this post.

Engineers typically apply science ‘as if’ they believed it absolutely. Yet at the same time their professional institutions require them to look out for and report potential signs of errors, including in the science. They are thus required to act ‘irrationally’ as if they had doubt, but experience shows this to be a ‘good thing’.

Collaboration

If we have two similar sources that give different estimates for some P(X) we can only form some kind of average. Using imprecise probabilities we might form a range. But often we should take a different approach: try to identify the premises behind the probabilities. It may be that one set of premises is far from our own, and can be discarded. Perhaps one estimate is based on experience of a context that is much closer to our view. Or perhaps the sources have different assumptions that might be rationalised, and our sources asked for conditional probabilities that we might then use. Similarly, we might well ask our sources for scenarios that they think improbable but which would make a significant difference to their estimates. (This can be quite fruitful.)

Dave Marsay

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