Bruno Lecoutre

1
MaxEnt 2006 Twenty sixth International Workshop
on Bayesian Inference and Maximum Entropy
Methods in Science and Engineering CNRS, Paris,
France, July 9, 2006
And if you were a Bayesian without knowing it?

• Bruno Lecoutre
• C.N.R.S. et Université de Rouen
• E-mail bruno.lecoutre_at_univ-rouen.fr
• Internet http//www.univ-rouen.fr/LMRS/Persopage/
  Lecoutre/Eris
• Equipe Raisonnement Induction Statistique

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Bayes, Thomas (b. 1702, London - d. 1761,
Tunbridge Wells, Kent), mathematician who first
used probability inductively and established a
mathematical basis for probability inference (a
means of calculating, from the number of times
an event has not occured, the probability that
it will occur in future trials)
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(No Transcript)
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Probability and Statistical Inference
5
(Mis)intepretations of p-values in Bayesian terms
  ? Many statistical users misinterpret the
p-values of significance tests as inverse
probabilities 1-p is the probability
that the alternative hypothesis is true
6
(Mis)intepretations of confidence levels in
Bayesian terms
Frequentist interpretation of a 95 confidence
interval ? In the long run 95 of computed
confidence intervals will contain the true
value of the parameter ? Each interval in
isolation has either a 0 or 100 probability
of containing it
This correct interpretation does not make sense
for most users! ? It is the interpretation in
(Bayesian) terms of a fixed interval having a
95 chance of including the true value of
interest which is the appealing feature of
confidence intervals
7
(Mis)intepretations of frequentist procedures in
Bayesian terms

• ? Even experienced users and experts in
  statistics are not
• immune from conceptual confusions
• In these conditions a p-value of 1/15, the
  odds of 14 to 1
• that this loss was caused by seeding of clouds
• do not appear negligible to us
• Neyman et al., 1969

? All the attempts to rectify these
interpretations have been a loosing battle
? Virtually all users interpret frequentist
confidence intervals in a Bayesian fashion
We ask themselves And if you were a Bayesian
without knowing it?
8
Two main definitions of probability (already in
Bernoulli, 17th century)
? The long-run frequency of occurrence of an
event, either in a sequence of repeated trials or
in an ensemble of identically prepared systems
? Frequentist (classical, orthodox,
sampling theory) conception

• Seems to make probability an objective property,
  existing in the nature independently of us, that
  should be based on empirical frequencies

? A measure of the degree of belief (or
confidence) in the occurrence of an event or more
generally in a proposition

• ? The Bayesian conception
• A much more general definition Ramsey, 1931
  Savage, 1954 de Finetti, 1974

Jaynes, E.T. (2003) Probability Theory The
Logic of Science (Edited by G.L.
Bretthorst) Cambridge, England Cambridge
University Press
9
  ? The Bayesian definition fits the meaning of
the term probability in everyday language
  ? The Bayesian probability theory appears to be
much more closely related to how people
intuitively reason in the presence of
uncertainty

•  
• It is beyond any reasonable doubt that for most
  people,
• probabilities about single events do make sense
• even though this sense may be naïve and fall
  short from numerical accuracy
• Rouanet, in Rouanet et al., 2000, page 26

10

• Frequentist approach
• Self-proclaimed objective contrary to the
  Bayesian inference that should be necessary
  subjective

• Bayesian approach
• The Bayesian definition can serve to describe
  objective knowledge,
• in particular based on symmetry arguments or
  on frequency data

? Bayesian statistical inference is no less
objective than frequentist inference It is even
the contrary in many contexts
11

• Statistical Inference
• Statistical inference is typically concerned with
  both known quantities - the observed data - and
  unknown quantities - the parameters and the data
  that have not been observed.

• The raw material of a statistical investigation
  is a set of observations these are the values
  taken on by random variables X whose distribution
  P? is at least partly unknown.
• Lehmann, 1959

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• Frequentist inference
• ? All probabilities (in fact frequencies) are
  conditional on unknown parameters

• Significance tests (parameter value fixed by
  hypothesis)
• Confidence intervals

• Bayesian inference
• ? Parameters can also be probabilized

• Distributions of probabilities that express our
  uncertainty
• before observations (does nor depend on data)
  prior probabilities
• after observations (conditional on data)
  posterior (or revised) probabilities
• also about future data predictive probabilities

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A simple illustrative situation
A finite population of size N20 With a
dichotomous variable 1 (success) 0
(failure) Proportion j of success
Unknown parameter j ?
Known data 0 0 0 1 0 f 1/5
A sample of size n5 from this population has
been observed
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Inductive reasoning generalisation from known to
unknown
Unknown parameter j ?

• In the frequentist framework
• no probabilities
• no solution

Known data 0 0 0 1 0 f 1/5
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Frequentist inference from unknown to known
Known data 0 1 0 0 0 f 1/5

• no more solution

Unknown parameter j ?
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Frequentist inference
Data 0 0 0 1 0 (f 1/5 0.20)
Imaginary repetitions of the
observations f 0/5 0.00006 f 1/5
0.005 f 2/5 0.068 f 3/5 0.293 f 4/5
0.440 f 5/5 0.194 One sample have been
observed out 15 503 possible samples

• Parameter
• Fixed value
• ? Example
• j 15/20 0.75

Sampling probabilities frequencies
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Data 0 0 0 1 0 f 0.20
Frequentist significance test
Imaginary repetitions of the observations f
0/5 0.00006 f 1/5 0.005 f 2/5
0.068 f 3/5 0.293 f 4/5 0.440 f 5/5
0.194

• Null hypothesis
• Example 1
• j 0.75 (15/20)
• Level a 0.05

0.995
If j 0.75 one find in 99.5 of the
repetitions a value f gt 1/5 (greater than the
observation f0.20) ? The null hypothesis j
0.75 is rejected (Significant p 0.00506)
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However, this conclusion is based on the
probability of the samples that have not been
observed!
If P is small, that means that there have been
unexpectedly large departures from prediction.
But why should these be stated in terms of
P? The latter gives the probability of
departures, measured in a particular way, equal
to or greater than the observed set, and the
contribution from the actual value is nearly
always negligible. What the use of P implies,
therefore, is that a hypothesis that may be
true may be rejected because it has not predicted
observable results that have not occurred. This
seems a remarkable procedure. Jeffreys, 1961
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Data 0 0 0 1 0 f 0.20
Frequentist significance test
Imaginary repetitions of the observations
f 0/5 0.00006 f 1/5 0.005 f
2/5 0.068 f 3/5 0.293 f 4/5
0.440 f 5/5 0.194

• Null hypothesis
• Example 2
• j 0.50 (10/20)
• Level a 0.05

0.848
If j 0.50 one find in 84.8 of the
repetitions A value f gt 1/5 (greater than than
the observation f 0.20) ? The null hypothesis j
0.50 is not rejected (Non significant p
0.152)
Obviously this does not prove that j 0.50!
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Data 0 0 0 1 0 f 0.20
Frequentist confidence interval
Set of possible values for j that are not
rejected at level a

• Example a 0.05
• One get the 95 confidence interval
• 0.05 , 0.60

How to interpret the 95 confidence?
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Interpretation of frequentist confidence?
The frequentist interpretation is based on the
universal statement Whatever the fixed value
of the parameter is, in 95 (at least) of the
repetitions the interval that should be
computed includes this value
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A very strange interpretation it does not
involve the data in hand!
It is at least unrealistic Objection has
sometimes been made that the method of
calculating Confidence Limits by setting an
assigned value such as 1 on the frequency of
observing 3 or less (or at the other end of
observing 3 or more) is unrealistic in treating
the values less than 3, which have not
been observed, in exactly the same manner as the
value 3, which is the one that has been
observed. This feature is indeed not very
defensible save as an approximation. Fisher,
1990/1973, page 71
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Return to the inductive reasoning Generalisation
from known to unknown
Set of all possible values of the unknown
parameter j 0/20, 1/20, 2/20 20/20

• Bayesian inference
• Probabilities that express our uncertainty
• (in addition to sampling probabilities)

Known data 0 0 0 1 0 f 1/5
As long as we are uncertain about values of
parameters, we will fall into the Bayesian
camp Iversen, 2000
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Data 0 0 0 1 0 f 1/5
Bayesian inference

• All the frequentist probabilities associated with
  the data
• Pr(f 1/5 j)
• ? Likelihood function
• j 0/20 ? 0 j 10/20 ? 0.135
• j 1/20 ? 0.250 j 11/20 ? 0.089
• j 2/20 ? 0.395 j 12/20 ? 0.054
• j 3/20 ? 0.461 j 13/20 ? 0.029
• j 4/20 ? 0.470 j 14/20 ? 0.014
• j 5/20 ? 0.440 j 15/20 ? 0.005
• j 6/20 ? 0.387 j 16/20 ? 0.001
• j 7/20 ? 0.323 j 17/20 ? 0
• j 8/20 ? 0.255 j 18/20 ? 0
• j 9/20 ? 0.192 j 19/20 ? 0
• j 20/20 ? 0

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Bayesian inference
We assume prior probabilities Pr(j) (before
observation)

• joint probabilities (a simple product)
• Pr(j and f1/5) Pr(f1/5 j) Pr(j)
• likelihood prior probability

• Predictive probabilities (sum of the joint
  probabilities)
• Pr(f1/5)
• A weighted average of the likelihood function

• posterior probabilities (A simple application of
  the definition
• of conditional probabilities)
• Pr(j f1/5) Pr(j and f1/5) / Pr(f)
• The normalized product of the prior and the
  likelihood

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Bayesian statistics is difficult in the sense
that thinking is difficultBerry, 1997
We can conclude with Berry
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Considerable difficulties with the frequentist
approach
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The mysterious and unrealistic use of the
sampling distribution

• Frequent questions asked by students and
  statistical users
• why one considers the probability of samples
  outcomes that are more extreme than the one
  observed?
• why must one calculate the probability of
  samples that have not been observed?
• etc.

• No such difficulties with the Bayesian inference
• Involves the sampling probability of the data ,
  via the likelihood function
• that writes the sampling distribution in the
  natural order
• from unknown to known

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Experts in statistics are not immune from
conceptual confusions About confidence intervals

• A methodological paper by Rosnow and Rosenthal
  (1996)
• They take the example of an observed difference
  between two means d0.266

They consider the interval 0,532 whose bounds
are the null hypothesis (0) and what they call
the counternul value (2d0.532), the
symmetrical value of 0 with regard to d
They interpret this specific interval 0,532
as a 77 confidence interval (0.771-20.115,
where 0.115 is the one-sided p-value for the
usual t test)
Clearly, 0.77 is here a data dependent
probability, which needs a Bayesian approach to
be correctly interpreted
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Experimental research and statistical
inference A paradoxical situation

• Null Hypothesis Significance Testing (NHST)
• An unavoidable norm in most scientific
  publications

• Often appears as a label of scientificness

BUT

• Innumerable misinterpretations and misuses

• Use explicitly denounced by the most eminent and
  most experienced scientists

The test provides neither the necessary nor the
sufficient scope or type of knowledge that basic
scientific social research requires Morrison
Henkel, 1969
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Today is a crucial time

• Users' uneasiness is ever growing

• In all fields necessity of changes in reporting
  experimental results
• ? routinely report effect size indicators
• ? and their interval estimates
• in addition to or in place of the results of
  NHST

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Common misinterpretations of NHST

• Emphasized by empirical studies
• Rosenthal Gaito, 1963 Nelson, Rosenthal
  Rosnow, 1986
• Oakes, 1986 Zuckerman, Hodgins, Zuckerman
  Rosenthal, 1993
• Falk Greenbaum, 1995 Mittag Thompson,
  2000 Gordon, 2001
• M.-P. Lecoutre (2000), B. Lecoutre, M.-P.
  Lecoutre Poitevineau, 2001

• Shared by most methodology instructors
• Haller Krauss, 2001

• Professional applied statisticians are not immune
  to misinterpretations
• M.-P. Lecoutre, Poitevineau B.Lecoutre (2003)
  - Even statisticians are not immune to
  misinterpretations of Null Hypothesis
  Significance Tests. International Journal of
  Psychology, 38, 37-45

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Why these misinterpretations?

• An individual's lack of mastery?
• This explanation is hardly applicable to
  professional statisticians

• Judgmental adjustments or adaptative
  distorsions'
• (M.-P. Lecoutre, in Rouanet et al., 2000, page
  74)
• designed to make an ill-suited tool fit their
  true needs

• Examples
• - Confusion between statistical significance
  and scientific significance
• - Improper uses of nonsignificant results as
  proof of the null hypothesis
• - Incorrect (non frequentist)
  interpretations of p-values as inverse
  probabilities

34
? NHST does not address questions that are of
primary interest for the scientific research

• This suggests that
• users really want to make a different kind of
  inference
• Robinson Wainer, 2002, page 270

35
A more or less naïve mixture of NHST
results and other information

• The task of statisticians in pharmaceutical
  companies

• Actually, what an experienced statistician does
  when looking at
• p-values is to combine them with information on
  sample size, null
• hypothesis, test statistic, and so forth to form
  in his mind something
• that is pretty much like a Confidence interval to
  be able to interpret
• the p-values in a reasonable way
• Schmidt, 1995, page 490

BUT
this is not an easy task!
36
A set of recipes and rituals

• Many attempts to remedy the inadequacy of usual
  significance tests
• See for instance the Task Force of the
  American Psychological
• Association (Wilkinson et al. 1999)

• They are both partially technically redundant
  and conceptually incoherent

• They do not supply real statistical thinking

We need statistical thinking, not rituals
Gigerenzer, 1998
37
Confidence intervals could quickly become a
compulsory norm in experimental publications
In practice two probabilities can be routinely
associated with a specific interval estimate
computed from a particular sample

• The first probability is the proportion of
  repeated intervals that contain the parameter
• It is usually termed the coverage probability

• The second is the Bayesian posterior
  probability that this interval contains the
  parameter (given the data in hand), assuming a
  noninformative prior distribution

In the frequentist approach, it is forbidden to
use the second probability In the Bayesian
approach, the two probabilities are valid
Moreover, an objective Bayes interval is often
a great frequentist procedure (Berger, 2004)
38
The debates can be expressed on these
terms whether the probabilities should only
refer to data and be based on frequency or
whether they should also apply to parameters and
be regarded as measures of beliefs
39
The ambivalence of statistical instructors
It is undoubtedly the natural (Bayesian)
interpretation a fixed interval having a
95 chance of including the true value of
interest that is the appealing feature of
confidence intervals
Most statistical instructors tolerate and even
use this heretic interpretation
40
The ambivalence of statistical instructors

• In a popular statistical textbook (whose
  objective is to allow the reader accessing the
  deep intuitions in the field), one can found the
  following interpretation of the confidence
  interval for a proportion

Si dans un sondage de taille 1000, on trouve P
la proportion observée 0.613, la proportion
p1 à estimer a une probabilité 0.95 de se trouver
dans la fourchette 0.58,0.64 If in a public
opinion poll of size 1000, one find P the
observed proportion 0.613, the proportion p1
to be estimated has a 0.95 probability to be in
the range 0.58,0.64'' Claudine Robert, 1995,
page 221
41
The ambivalence of statistical instructors

• In an other book that claims the goal of
  understanding statistics, a 95 confidence
  interval is described as

an interval such that the probability is 0.95
that the interval contains the population
value Pagano, 1990, page 228
42
The ambivalence of statistical instructors
It would not be scientifically sound to justify
a procedure by frequentist arguments and to
interpret it in Bayesian terms Rouanet, 2000
43
  Other authors claim that the correct
frequentist interpretation they advocate can be
expressed as
  We can be 95 confident that the population
mean is between 114.06 and 119.94 Kirk, 1982
  We may claim 95 confidence that the
population value of multiple R2 is no lower than
0.266 Smithson, 2001
  ? Hard to imagine that readers can understand
that confident refers here to a frequentist
view of probability!
We will distinguish between probability as
frequency, termed probability, and probability
as information/uncertainty, termed confidence
Schweder Hjort (2002)
44
  Teaching the frequentist interpretation a
losing battle
we are fighting a losing battle Freeman, 1993  
45
Most statistical users are likely to be
Bayesian without knowing it!
It could be argued that since most
physicians use statement A the probability the
true mean value is in the interval is 95 to
describe confidence intervals, what they really
want are probability intervals. Since to get
them they must use Bayesian methods, then they
are really Bayesians at heart! Grunkemeier
Payne, 2002
46
The Bayesian therapy
47
It is not acceptable that that future
statistical inference methods users will continue
using non appropriate procedures because they
know no other alternative
Since most people use inverse probability
statements to interpret NHST and confidence
intervals, the Bayesian definition of
probability, conditional probabilities and Bayes
formula are already - at least implicitly -
involved in the use of frequentist methods
Which is simply required by the Bayesian
approach is a very natural shift of emphasis
about these concepts, showing that they can be
used consistently and appropriately in
statistical analysis (Lecoutre, 2006)
48
A better understanding of frequentist
procedures Students exposed to a Bayesian
approach come to understand the frequentist
concepts of confidence intervals and P values
better than do students exposed only to a
frequentist approach Berry, 1997

• Combining descriptive statistics and
  significance tests
• A basic situation
• the inference about the difference d between two
  normal means

• Let us denote by d (assuming d?0) the observed
  difference and by t the value of the Student's
  test statistic

• Assuming the usual non informative prior, the
  posterior for d is a
• generalized (or scaled) t distribution (with the
  same degrees of freedom
• As the t test), centered on d and with scale
  factor the ratio ed/t
• (see e.g. Lecoutre, 2006)

49
Conceptual links

• Bayesian interpretation of the p-value
• The one-sided p-value of the t test is exactly
  the posterior Bayesian probability that the
  difference d has the opposite sign of the
  observed
• difference

If dgt0, there is a p posterior probability of a
negative difference and a 1-p complementary
probability of a positive difference
In the Bayesian framework these statements are
statistically correct

• Bayesian interpretation of the confidence
  interval
• It becomes correct to say that there is a 95
  for instance probability of d being included
  between the fixed bounds of the interval
  (conditionally on the data)

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Some decisive advantages

• overcoming usual difficulties
• In this way, Bayesian methods allow users to
  overcome usual difficulties encountered with the
  frequentist approach

• public use statements
• The use of noninformative priors has a privileged
  status in order to gain public use statements

• Combining information
• when good prior information is available other
  Bayesian techniques also have an important role
  to play in experimental investigations

51
Many potential users of Bayesian methods continue
to think that they are too subjective to be
scientifically acceptable
BUT
Bayesian procedures are no more arbitrary than
frequentist ones
52

• frequentist methods are full of ad hoc
  conventions
• Thus the p-value is traditionally based on the
  samples that are more extreme than the observed
  data (under the null hypothesis)

But, for discrete data, it depends on whether the
observed data are included or not

• Example
• For instance, let us consider the usual Binomial
  one-tailed test for the null hypothesis jj0
  against the alternative jltj0

This test is conservative, but if the observed
data are excluded, it becomes liberal
A typical solution to overcome this problem
consists in considering a mid-p-value, but it
has only it ad hoc justifications
53

• The choice of a noninformative prior
  distribution cannot
• avoid conventions

But the particular choice of such a prior is an
exact counterpart of the arbitrariness involved
within the frequentist approach
For Binomial sampling, different priors have been
proposed for an objective Bayesian analysis (for
a discussion, see e.g. Lee, 1989, pages 86-90)
It exists two extreme noninformative priors that
are respectively the more unfavourable and the
more favourable priors with respect to the null
hypothesis They are respectively the Beta
distribution of parameters 1 and 0 and the Beta
distribution of parameters 0 and 1
The observed significance levels of the inclusive
and exclusive conventions are exactly the
posterior Bayesian probabilities that j is
greater than j0 respectively associated with
these two extreme priors
These two priors constitute an a priori
ignorance zone' (Bernard, 1996), which is
related to the notion of imprecise probability
(see Walley, 1996)
54
The usual criticism of frequentists towards the
divergence of Bayesians with respect to the
choice of a non informative prior can be easily
reversed
Furthermore, the Jeffreys prior, which is very
naturally the intermediate Beta distribution of
parameters ½ and ½ gives an intermediate value,
fully justified, close to the observed mid-p-value
The Jeffreys prior credible interval has
remarkable frequentist properties Its coverage
probability is very close to the nominal level,
even for small-size samples It is undoubtedly an
objective procedure that can be favourably
compared to most frequentist intervals
We revisit the problem of interval
estimation of a binomial proportion We begin by
showing that the chaotic coverage properties of
the Wald interval are far more persistent than is
appreciated... We recommend the Wilson interval
or the equal tailed Jeffreys prior interval for
small n Brown, Cai and DasGupta, 2001, page 101
55
Similar results are obtained for
negative-Binomial (or Pascal) sampling
In this case, the observed significance levels of
the inclusive and exclusive conventions are
exactly the posterior Bayesian probabilities
associated with the two respective priors
Beta(0,0) and Beta(0,1)
This suggests that the intermediate Beta
distribution of parameters 0 and ½ is an
objective procedure It is precisely the Jeffreys
prior
This result concerns a very important issue
related to the likelihood principle
The preceding results can be generalized to more
general situations of comparisons between
proportions (see for the case of a 22
contingency table Lecoutre Charron, 2000)
56
The predictive probabilities A very appealing
tool
The predictive idea is central in experimental
investigations The essence of science is
replication. A scientist should always be
concerned about what would happen if he or
another scientist were to repeat his
experiment Guttman, 1983
57
An essential aspect of the process of evaluating
design strategies is the ability to calculate
predictive probabilities of potential
results Berry, 1991
Bayesian predictive probabilities a very
appealing method to answer essential questions
such as
? Planning How many subjects? How big should
be the experiment to have a reasonable chance of
demonstrating a given conclusion?
? Monitoring When to stop? Given the current
data, what is the chance that the final result
will be in some sense conclusive, or on the
contrary inconclusive?
These questions are unconditional in that they
require consideration of all possible value of
parameters Whereas traditional frequentist
practice does not address these questions,
predictive probabilities give them direct and
natural answer
58
The stopping rule principle A need to rethink
Experimental designs often involve interim looks
at the data
Most experimental investigators feel that the
possibility of early stopping cannot be ignored,
since it may induce a bias on the inference that
must be explicitly corrected
59
Consequently, they regret the fact that the
Bayesian methods, unlike the frequentist
practice, generally ignore this specificity of
the design
Bayarri and Berger (2004) consider this
desideratum as an area of current disagreement
between the frequentist and Bayesian approaches
This is due to the compliance of most Bayesians
with the likelihood principle (a consequence of
Bayes' theorem), which implies the stopping rule
principle in interim analysis
Once the data have been obtained, the reasons
for stopping experimentation should have no
bearing on the evidence reported about unknown
model parameters Bayarri and Berger, 2004, page
81
Would the fact that people resist an idea so
patently right (Savage, 1954) be fatal to the
claim that they are Bayesian without knowing
it?
60
This is not so sure, experimental investigators
could well be right!
They feel that the experimental design
(incorporating the stopping rule) is prior to the
sampling information and that the information on
the design is one part of the evidence
It is precisely the point of view developed by de
Cristofaro (1996, 2004, 2006), who persuasively
argued that the correct version of Bayes' formula
must integrate ? the parameter q ? the
design d ? the initial evidence (prior to
designing) e0 ? the statistical information i
Consequently Bayes' formula must be written in
the following form
p(q i, e0 ,d) ? p(q e0 ,d) p(i q ,e0,d)
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p(q i, e0 ,d) ? p(q e0 ,d) p(i q ,e0,d)
? It becomes evident that the prior depends on d
? With this formulation, both the likelihood
principle and the stopping rule principle are no
longer an automatic consequence
? It is not true that, under the same likelihood,
the inference about q is the same, irrespective
of d
Box and Tiao (1973, pages 45-46), stated that the
Jeffreys priors are different for the Binomial
and Pascal sampling as the two sampling models
are also different
In both cases, the resulting posterior
distribution have remarkable frequentist
properties (i.e. coverage probabilities of
credible intervals)
62
? This result can be extended to general stopping
rules (Bunouf, 2006)
The basic principle is that the design
information, which is ignored in the likelihood
function, can be recovered in the Fisher
information (which is related to Shannon's notion
of entropy)
Within this framework, we can get a coherent and
fully justified Bayesian answer to the issue of
sequential analysis, which furthermore satisfy
the experimental investigators desideratum
(Bunouf and Lecoutre, 2006)
63
Conclusion
64
A widely accepted objective Bayes theory, which
fiducial inference was intended to be, would be
of immense theoretical and practical
importance. A successful objective Bayes theory
would have to provide good frequentist properties
in familiar situations, for instance, reasonable
coverage probabilities for whatever replaces
confidence intervals Efron, 1998, page 106
In actual fact I suggest that such a theory is by
no means a speculative viewpoint but on the
contrary is perfectly feasible (see especially,
Berger, 2004) It is better suited to the needs
of users than frequentist approach and provide
scientists with relevant answers to essential
questions raised by experimental data analysis
65
Why scientists really appear to want a different
kind of inference but seem reluctant to use
Bayesian inferential procedures in practice?

• This state of affairs appears to be due to a
  combination of factors including
• philosophical conviction,
• tradition,
• statistical training,
• lack of availability,
• computational difficulties,
• reporting difficulties,
• and perceived resistance by journal editors

Winkler
Winkler, 1974
66
we statisticians will all be Bayesians in
2020, and then we can be a united
profession Lindley in Smith, 1995, page 317
The times we are living in at the moment appear
to be crucial
One of the decisive factors could be the recent
draft guidance document of the US Fud and Drug
Administration (FDA, 2006)
This document reviews the least burdensome way
of addressing the relevant issues related to the
use of Bayesian statistics in medical device
clinical trials
It opens the possibility for experimental
investigators to really be Bayesian in practice
67
Text and references available upon request Mail
to bruno.lecoutre_at_univ-rouen.fr
It is their straightforward, natural approach to
inference that makes them Bayesian methods so
attractive Schmitt, 1969