Maths for the Modern Economy

Mathematics Today Vol. 53 No. 5 October 2017

Extracts concerning uncertainty

Maths for the Modern Economy at the Royal Society (pg 209)

Both talks described the role of a mathematician in understanding the assumptions that have gone into a decision , subsequently challenging and testing these, and made a compelling case for employing [mathematicians].

Optimising Resilience: at the Edge of Computability (pg 233)

A number of authors argued that the classical Bayesian approaches can fail and a different type of is required to capture partial or conflicting sources of information.

The value of p-values (Letters, pg 241)

[As] one digs more deeply into any school of inference, difficulties arise. This should not be surprising: we are trying to make inferences about the real world, not some mathematical idealisation. Modern thought recognises that different schools of inference are helpful in shedding light on different kinds of question: there is, and indeed can be, no universally best method.

… Bayesian methodology is coherent – meaning that it is internally consistent. [This] is important, but our ultimate objective is to make a statement about the real world, so the key question is how well the data and our theories match. … [The] old criticism [is]that the Bayesian approach leads to the tail of mathematical coherence wagging the dog of the scientific question.

Comments

The first quote implies that uncertainty is important and widespread, and mathematicians have a role in uncovering and dealing with it. The second quote implies that even those who work on ‘quantifying uncertainty’ recognize that this ultimately impossible. The third quote is part of a suggestion that – done properly – the statistician’s p-values may be a useful adjunct to probabilities. I agree. But I do not think that this is always enough.

It seems to me that one can often derive a reasonable probability based on assumptions that are generally accepted by some community, or possibly derive different probabilities for different communities. The next most significant stage is not to supplement the probability/probabilities by some number, but to explicate the assumptions and to develop credible scenarios under which they may fail.

For example, if asked for the probability of a run of Heads when tossing a coin, we might calculate the answer for the usual assumptions and then point out how these could fail (e.g. by using a carefully engineered machine to toss).

Dave Marsay

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