TEACHING BAYESIAN STATISTICS IN THE SCIENTIFIC CURRICULA

  Giulio D'Agostini 

  Physics Department, University of Rome "La Sapienza", Rome, Italy
  Email: giulio.dagostini@roma1.infn.it
  URL: http://roma1.infn.it/~dagos
    

  It is well known that the best way to learn something is to teach it. 
  When I had to give the Laboratory of Physics course to Chemistry students 
  and introduce elements of probability and statistics applied to data 
  analysis, I did as most new teachers do: I started repeating what 
  I had learned  years before, more or less using the same lecture notes. 
  This worked well for explaining the experiments, but when I moved to 
  probability, the situation was quite embarrassing. In the very first 
  lecture I realized that I was not convinced of what I was saying. 
  I introduced probability as the ratio between favourable and possible cases, 
  but I had no courage to add `if the cases are equally probable'.  
  I cheated by saying `if the cases are equally possible' and moved rapidly 
  to examples. The students had no time to react, the examples were well 
  chosen, and I was able to survive that lesson and the following weeks.
    
  The problem returned when we came to the evaluation of measurement 
  uncertainty, a typical application of statistics in scientific disciplines. 
  I had to acknowledge that the reasoning used in practice by physicists 
  was quite in contradiction with the statistics theory we learn and teach. 
  The result was that I had started the semester saying that subjective 
  probability was not scientific, and ended it teaching probability inversion 
  applied to physics quantities. 

  In the years since, I have clarified my ideas and, thanks also to the 
  freedom allowed by the Italian university system, I have set up an 
  introductory mini-course based on Bayesian concepts. Recently I have 
  published an article (Am. J. Phys. 67, December 1999, pp.1260-1269) that 
  gives a short account of the way I introduce subjective probability
  and Bayesian inference. The paper is also available on my web page,
  together with other applications of Bayesian inference and teaching
  material. Here I would like to touch very briefly on some points in 
  which other teachers might be interested.

  University students have already developed, from high school, some 
  negative reaction towards the words `belief' and `subjective' ("Science 
  is objective!"). It is important, therefore, to clarify immediately the 
  difference between belief and `imagination', between subjective and 
  `arbitrary'. For the first pair of concepts I find Hume's analysis 
  particularly convincing. For the second one I find crucial the role of 
  de Finetti's coherent bet, without which probability statements are, in my 
  opinion, empty statements. I also find important to separate assessments 
  of probability from decision issues (belief versus `convenience'). 

  The interplay of subjective probability with combinatorial and 
  frequency-based evaluations is also an important point to be clarified soon.
  In particular, it is important to distinguish observed frequencies from 
  expected frequencies. The former are statistical data and can be used to 
  assess beliefs under well-specified conditions, whereas the latter are 
  just a way to express the beliefs. Many scientists think they are 
  frequentists because they are used to expressing their beliefs in terms of 
  frequencies (but they also apply the same reasoning to probability of 
  hypotheses and give a probabilistic interpretation of frequentistic 
  coverage). 

  As far as the inferential aspects are concerned, I find it unavoidable 
  to speak about probability of true values, i.e. of objects which are not 
  directly observable and therefore, strictly speaking, not verifiable. 
  I understand that it is possible to skip the intermediate state of the 
  true values, and only speak about probability of future observations
  conditioned by past observations. But in practice it is not easy to follow 
  strictly this approach. The scientific method is based on models for the 
  real world and on making statements about the parameters of the model: 
  Who has never `seen' a mass, an electric charge or an angular momentum? 
  What is important is to be consistent, and to always make a distinction 
  between the things which are really observed and the `metaphysical' objects.
  For example, it is important to clarify that it is possible to make a 
  frequency distribution of an experimental observable (such as the reading 
  of a scale) under apparently similar experimental conditions and use it 
  to evaluate the probability distribution that we associate with  the 
  likelihood. In contrast, it is impossible to evaluate the probability 
  distribution of true values by extrapolating from frequency distribution. 
  The only way to assess their probability is to use Bayesian probability 
  inversion. 
     
  Once the Bayesian inference is presented, it is important to show that 
  many conventional methods can be recovered as limit cases of the Bayesian 
  ones, under certain well-understood restricting conditions. The best example 
  is the least square fit. The advantage is that the interpretation of the 
  result corresponds to what scientists have anyway, and students are aware 
  of the hidden hypotheses behind the methods. It is also important to show 
  what are the implicit assumptions that make the conventional scheme of 
  hypothesis test `often to work', though it is not logically grounded. 
  So, even p-values can be used to spot a possible problem, but certainly 
  not to draw scientific conclusions or to take decisions.