network EditorialEditorial fter 20 years of outstanding ser- Sir [3]. A recent ar- vice, Graham Hoare has retired as ticle by Matthews [4] endorses the Amer- A Letters Editor for Mathematics To- ican Statistical Association’s criticisms of day. He brought much expertise and pro- p-values for similar reasons, particularly in fessionalism to this honorary position, at- connection with the current replication cri-

tending meetings regularly and responding Library Science Photo sis. Let us consider the problems in more | to letters in a characteristically erudite and detail. jocular style. The Editorial Board is most When a teacher tests a simple hypoth- grateful and wishes him well for the future. esis about a parameter by collecting data Graham celebrated this retirement with a Terry © Sheila and generating conclusions such as reject substantial donation to the IMA coffers. H0 at the 5% level of significance be- Christopher Hollings (University of cause a 95% confidence interval excludes Oxford) has kindly agreed to take over the test value or because the p-value is the role of Letters Editor forthwith. Else- less than 0.05, many students are con- where, Ron Knott (University of Surrey) fused by the arbitrary terminology and was recently elected to succeed Edmund illogical reasoning. Chadwick as North West Branch Chair. As probability is the agreed measure of Chris and Ron are well known to IMA uncertainty, what roles do significance and Pierre Simon de Laplace (1749–1827) members for their valued contributions. confidence play? Why choose 5%, 95% Congratulations to Dr Priya Subramanian (University of and 0.05 thresholds to make binary decisions? How can some Leeds), who was awarded a L’Oréal-UNESCO Fellowship for null and alternative hypotheses not partition the set of possible Women in Science this May. At our request, she agreed to values? Why specify a power of 80% to determine a suitable write an article for Mathematics Today, which you can find on sample size? Even more perplexing, why perform any statistical pages 140–141. We also have items about this year’s winners of analysis when H0 is often clearly false anyway? the Royal Society’s Copley Medal (Sir Andrew Wiles) and the For example, if H : π =0.5 where π (0, 1) is the prob- 0 ∈ Shaw Prize in Mathematical Sciences (Professors János Kollár ability that a coin toss results in a head, the teacher might toss and Claire Voisin). a coin n times and count the number x of heads. Suppose that In October, we shall publish a special issue on the theme n = 10 and x =9. We estimate π 0.9 with 95% confidence ≈ of Space, as requested in last year’s survey, with guest editor interval (0.55, 1.00) and p-value about 0.02, so we reject H0 at Professor Jörg Fliege (University of Southampton). Meanwhile, the 5% level of significance. However, slight asymmetries in de- August’s issue includes articles on algorithms, further education, sign, manufacture and wear mean that all coins exhibit some bias, personnel management, centres of gravity and d’Alembert, so so we already know that π =0.5. The test is ill defined and the  there should be something to please everybody. conclusion is irrelevant. From September, new AS and A-level syllabi in maths and What we really wish to determine is whether the coin is ap- further maths will be delivered in England, Wales and Northern proximately unbiased as defined by the parameter π A, where ∈ Ireland. Regrettably, these courses teach significance tests and A = (0.4, 0.6) or some other interval of acceptability, given the confidence intervals as the predominant forms of statistical infer- observed data D = 9 heads, 1 tail . Elementary probability { } ence, despite their substantial deficiencies. These methods sim- theory does exactly this: ilarly feature in the Scottish advanced higher statistics qualifica- tion and in many UK university degree programmes. P (π A D)= f(π D) dπ (1) ∈ | | The concepts behind significance tests and confidence inter- A vals contributed much to the development of mathematical statis- with, from Bayes’ theorem, tics in the 20th century. However, O’Hagan and Forster [1] note that they have some philosophical flaws, whereas the method- p(D π)f(π) f(π D)= | (π; D)f(π) (2) ology published by Thomas Bayes in 1763 and Pierre-Simon | p(D) ∝L Laplace in 1814 is fundamentally sound. where (π; D)=p(D π) as a function of π. For simplicity, Dennis Lindley remarked that significance tests and confi- L | dence intervals are statements about data given parameter, rather probability mass functions p and probability density functions f than parameter given data. They interpret the conditional prob- are distinguishable only by their arguments here. Relation (2) is cited as ‘posterior likelihood prior’ and ability P (data parameter) as though it were P (parameter data). ∝ × | | is the key to . The function (π; D) is the This misinterpretation can be devastating: the probability of di- L agnosis given disease is different from that of disease given di- likelihood of parameter π given data D, which is the binomial agnosis. Similarly, the probability of evidence given innocence probability (π; D) π9(1 π) is different from that of innocence given evidence. If these prob- L ∝ − abilities were equal, we should prosecute all lottery winners for for our illustration. The function f(π) is a prior density that re- fraud! flects existing knowledge about the coin. My prior beliefs are Other critics of significance tests and confidence intervals in- expressed by the beta density clude Chair of the Council for the Mathematical Sciences Sir Adrian Smith [2] and President of the Royal Statistical Society f(π) π25(1 π)25 ∝ −

Mathematics TODAY AUGUST 2017 138 for π (0, 1) based on tertiles of 0.47 and 0.53, though you could Evaluating these limits involves numerical solution of nonlinear ∈ use a different formula to express your prior beliefs. We need only equations with nested quadrature. Such algorithms are readily specify functions that are proportional to the likelihood and prior. available in standard packages, though teachers could use a sim- Substituting these two functions of π into relation (2) gives pler method. As posterior densities are asymptotically normal, the posterior density approximate limits are given by µ 1.96σ where µ = E(π D) ± | ≈ 0.56 and σ2 = var(π D) 0.0039 here. 34 26 f(π D) π (1 π) | ≈ | ∝ − for π (0, 1). This necessarily integrates to one over the unit f ∈ interval, so the constant of proportionality is  1 4 1018.  1 34 26 ≈ × 0 π (1 π) dπ −  Finally, equation (1) gives prior posterior 0.6  P π (0.4, 0.6) D = f(π D) dπ 0.71 { ∈ | } | ≈ 0.4  and this probability measures my belief that the coin is fair. Un- like a significance test, this analysis involves nothing arbitrary or  illogical and generates precisely the answer that we sought! My prior and posterior densities for π are displayed in           π Figure 1 and the probability of 0.71 corresponds to the Figure 1: Prior and posterior densities. area shaded green. Teachers could use freely available math- ematics software to plot graphs and evaluate integrals. Typing All these techniques can be simplified for practical applica- (integral p∧34(1-p)∧26 from 0.4 to 0.6)/(integral tion and have been developed substantially for advanced research. p∧34(1-p)∧26 from 0 to 1) into WolframAlpha returns 0.71 With today’s powerful computers, perhaps it is time to stop mis- almost immediately. leading pupils with significance tests and confidence intervals, An alternative presentation of this analysis is useful when and return to the sound methods of Bayes and Laplace. there is no obvious interval of acceptable values. This involves finding the limits of a 95% posterior probability interval (or union DavidDavid F. F. Percy Percy CMathCMath CSciCSci FIMA University of Salford of intervals) B for π and checking whether the test value lies in- University of Salford side B. These intervals are usually defined by R References 1 O’Hagan, A. and Forster, J. (2004) Kendall’s Advanced Theory of 1 O’Hagan, A. and Forster, J. (2004) Kendall’s Advanced Theory of f(π D) dπ =0.95 Statistics: Bayesian Inference, Edward Arnold, UK. B | Statistics: Bayesian Inference, Edward Arnold, UK.  2 Bernardo, J.M. and Smith, A.F.M. (2006) Bayesian Theory, John Wi- 2 Bernardo, J.M. and Smith, A.F.M. (2006) Bayesian Theory, John ley & Sons, Canada. and Wiley & Sons, Canada. f(π D) f(π D) 1| ≥ 2| 33 Lunn,Lunn, D.,D., Jackson, C., Best, N., Thomas, A. and Spiegelhalter, D. Spiegelhalter, D. for all π B and π / B. They might resemble confidence (2012)(2012) TheThe BUGS BUGS Book: Book: A PracticalA Practical Introduction Introduction to Bayesian to Bayesian Anal- 1 ∈ 2 ∈ intervals but are their antitheses. ysisAnalysis, CRC, CRC Press, Press, USA. USA. For our example, B (0.44, 0.69) includes the test value 44 Matthews,Matthews, R.A.J.R.A.J. (2017)(2017) The ASA’s pp-value-value statement, one year on on ≈ π =0.5, and so supports the hypothesis that the coin is fair. (with(with discussion),discussion),Significance Significance,, vol. vol. 14, 14, pp. pp. 38–41. 38–41.

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