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## Truth and probability

from Lars Syll

Truth exists, and so does uncertainty. Uncertainty acknowledges the existence of an underlying truth: you cannot be uncertain of nothing: nothing is the complete absence of anything. You are uncertain of something, and if there is some thing, there must be truth. At the very least, it is that this thing exists. Probability, which is the science of uncertainty, therefore aims at truth. Probability presupposes truth; it is a measure or characterization of truth. Probability is not necessarily the quantification of the uncertainty of truth, because not all uncertainty is quantifiable. Probability explains the limitations of our knowledge of truth, it never denies it. Probability is purely epistemological, a matter solely of individual understanding. Probability does not exist in things; it is not a substance. Without truth, there could be no probability.

William Briggs’ approach is — as he acknowledges in the preface of his interesting and thought-provoking book — “closely aligned to Keynes’s.”

Almost a hundred years after John Maynard Keynes wrote his seminal A Treatise on Probability (1921), it is still very difficult to find statistics textbooks that seriously try to incorporate his far-reaching and incisive analysis of induction and evidential weight.

The standard view in statistics — and the axiomatic probability theory underlying it — is to a large extent based on the rather simplistic idea that ‘more is better.’ But as Keynes argues – ‘more of the same’ is not what is important when making inductive inferences. It’s rather a question of ‘more but different.’

Variation, not replication, is at the core of induction. Finding that p(x|y) = p(x|y & w) doesn’t make w ‘irrelevant.’ Knowing that the probability is unchanged when w is present gives p(x|y & w) another evidential weight (‘weight of argument’). Running 10 replicative experiments do not make you as ‘sure’ of your inductions as when running 10 000 varied experiments – even if the probability values happen to be the same.

According to Keynes we live in a world permeated by unmeasurable uncertainty – not quantifiable stochastic risk – which often forces us to make decisions based on anything but ‘rational expectations.’ Keynes rather thinks that we base our expectations on the confidence or ‘weight’ we put on different events and alternatives. To Keynes, expectations are a question of weighing probabilities by ‘degrees of belief,’ beliefs that often have preciously little to do with the kind of stochastic probabilistic calculations made by the rational agents as modelled by “modern” social sciences. And often we ‘simply do not know.’ As Keynes writes in Treatise:

The kind of fundamental assumption about the character of material laws, on which scientists appear commonly to act, seems to me to be [that] the system of the material universe must consist of bodies … such that each of them exercises its own separate, independent, and invariable effect, a change of the total state being compounded of a number of separate changes each of which is solely due to a separate portion of the preceding state …  In my judgment, the practical usefulness of those modes of inference … on which the boasted knowledge of modern science depends, can only exist … if the universe of phenomena does in fact present those peculiar characteristics of atomism and limited variety which appears more and more clearly as the ultimate result to which material science is tending.

Science according to Keynes should help us penetrate to “the true process of causation lying behind current events” and disclose “the causal forces behind the apparent facts.” Models can never be more than a starting point in that endeavour. He further argued that it was inadmissible to project history onto the future. Consequently, we cannot presuppose that what has worked before, will continue to do so in the future. That statistical models can get hold of correlations between different ‘variables’ is not enough. If they cannot get at the causal structure that generated the data, they are not really ‘identified.’

How strange that writers of statistics textbook, as a rule, do not even touch upon these aspects of scientific methodology that seems to be so fundamental and important for anyone trying to understand how we learn and orient ourselves in an uncertain world. An educated guess on why this is a fact would be that Keynes concepts are not possible to squeeze into a single calculable numerical ‘probability.’ In the quest for quantities one puts a blind eye to qualities and looks the other way – but Keynes ideas keep creeping out from under the statistics carpet.

It’s high time that statistics textbooks give Keynes his due.

1. November 15, 2018 at 6:42 pm

Humans are driven to reduce ambiguity and uncertainty or they risk feeling overwhelming anxiety. Beliefs — even irrational ones — serve to reduce that anxiety. Give me ambiguity or give me something else is not a comfortable place psychologically. When we expect something to be a certain way and it isn’t, we experience stress and anxiety and/or conflict arises. Traditional statistics are often used to strengthen beliefs and reduce ambiguity but should not be confused with analysis using statistics as a tool to uncover truths or circumscribe the unknown to make it emotionally manageable.

2. November 15, 2018 at 7:48 pm

If I may be excused some pedantry, may I note that Briggs also says that probability is not a thing ( at https://wmbriggs.com/post/25749/ ). But if I (correctly) deny that (unconditional) probability is a thing and instead talk about the uncertainty of the thing to which the probability estimate relates people are rarely interested: they want a probability, not some poor substitute that fails to deliver clear-cut decisions in they way that ‘rational’ decision theory does.

Instead, I have found it expedient to say that I am ‘uncertain’ about the probability. Those claiming that a given probability estimate is in some sense dependable often respond by attempting to justify it, in the course of which the ‘true nature’ of the claim becomes more evident. It is rarely useful to attempt to discuss whether or not in the particular instance probability is actual ‘a thing’ or not: it is usually enough to cast doubt on any estimate and hence uncertainty about any decision that rests on it. Usefully, this can sometimes lead to some useful insights (e.g. about ‘causality’ and ‘optionality’) that straightforward probability estimates rarely do.

Hope this makes some sense. Regards.

3. November 16, 2018 at 9:44 am

Yet again the elephant in the room is totally ignored. By conflating the concepts of abstract and concrete relationships, economics ceases to have any claim to being scientific in any way. The curve fitting which underlies the above discussion can only provide concrete descriptions. There are no restrictions on the forms of the relationships which can be used. Read the literature and the multiplicities of form you will discover will be apparent. This is completely valid. However, if is then claimed that any of these relationships are theoretically valid, then those relationships MUST satisfy the REQUIREMENTS of the quantity calculus. NONE DO! How simple is that?

Keynes did not use the term quantity calculus, but he implicitly recognises it in the words “The kind of fundamental assumption about the character of material laws…”. That is exactly what the quantity calculus is. If “abstract”, “concrete” and “quantity calculus” were used consistently in economic discourse, then progress would be possible. Without them, discussion in the same old fruitless way will continue. Please, accept and use scientific concepts and terminology.

4. November 19, 2018 at 12:48 am

With respect to all of you dear colleagues, I am, once again, utterly lost! I simply fail to understand how details regarding potential, possible, probable, optional, partial acceptance or denials and the rest help us in putting right an already completely derailed discourse.

So, let us ask our readers, potential followers and students ONE key question tonight:
What is the probability that the survivors of the recent tragedy in the US will outnumber those who died on 9/11? Is it possible that life will ever be the same for either group or those close to them?

• November 19, 2018 at 8:38 am

We may be in total agreement that conventional analysis is a “completely derailed discourse”. But that allows an enormous number of alternative hypotheses to coexist. I continue to stress the necessity of appropriate analysis to ensure that discussion will lead to empirically valid conclusions.

The blog concludes “It’s high time that statistics textbooks give Keynes his due.” Lars Syll’s conclusion is a very narrow point. I pointed out that the lesson to be learned is not the narrow point but it is the universal truth, well understood in the physical sciences, that abstract relationships MUST satisfy the rules of the quality calculus. So Lars Syll continues to ignore this and as you say, and I agree with you, that “details [of] … an already completely derailed discourse” being reiterated does not facilitate progress.

I will continue to stress the scientific truths which economics need to take to the heart of its discourse. I would commend all of you to determine for yourselves what true scientific understanding is and to apply it relentlessly. It may not be simple but it is and will be worth while.

5. November 26, 2018 at 6:36 am

We need to cast the net much wider. Rationality, irrationality, uncertainty, risk, etc. are all cultural defined ways of life, processes. Simply put, humans create cultures and the cultures speak to humans about the nature of rationality, uncertainty, risk, etc. An example can illustrate better I think than a long obscure “theoretical” essay by me. When I speak with a research source about actions and reasons for actions, how do I respond when the source tells me, “What I did felt right,” “It was the only course of action that solved my problem,” or “I did it because I was certain of the results?” And then adds “This was my only rational course of action.” Do I over rule the subject, correct their views on rationality, problem solving, or uncertainty? If I do that, I have little chance of understanding the actions of this person. The same issues exist with small and large groups, of which my subject is a member. Interactions making groups are an element, often the major one is each of these actions and thoughts. My taking over the situation rubs out the groups the subject lives within every day. As a psychologist I do this to re-socialize the person with the goal of changing actions and thoughts in the direction of what I as psychologist believe are groups behaving and thinking “correctly.” As historian, anthropologist I must do just the opposite. Reject taking over in favor of allowing the persons/groups’ actions and beliefs of my sources to be as visible to me and everyone else as possible. I assume it is the same for economists. Is this not correct?