Bruno LecoutreBruno Lecoutre
C.N.R.S. et Université de Rouen C.N.R.S. et Université de Rouen
E-mail: [email protected]
Internet: http://www.univ-rouen.fr/LMRS/Persopage/Lecoutre/Eris
EEquipe quipe RRaisonnement aisonnement IInduction nduction SStatistiquetatistique
MaxEnt 2006 Twenty sixth International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and EngineeringCNRS, Paris, France, July 9, 2006
And if you were a Bayesian without And if you were a Bayesian without knowing it?knowing it?
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)
ProbabilityProbability
andandStatistical InferenceStatistical Inference
Many statistical users misinterpret the p-values of significance tests as “inverse” probabilities:
1-p is “the probability that the alternative hypothesis is true”
(Mis)intepretations of (Mis)intepretations of pp-values-valuesin Bayesian termsin 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
(Mis)intepretations of confidence levels(Mis)intepretations of confidence levelsin Bayesian termsin Bayesian terms
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
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
(Mis)intepretations of frequentist procedures(Mis)intepretations of frequentist proceduresin Bayesian termsin Bayesian terms
All the attempts to rectify these interpretations have been
a loosing battle
We ask themselves:
“And if you were a Bayesian without knowing it?”
Virtually all users interpret frequentist confidence intervals
in a Bayesian fashion
Two main definitions of probabilityTwo main definitions of probability(already in Bernoulli, 17th century)(already in Bernoulli, 17th century)
“Frequentist (“classical”, “orthodox”, “sampling theory”) conception
A measure of the degree of belief (or confidence) in the occurrence of an event or more generally in a proposition
The long-run frequency of occurrence of an event, either in a sequence of repeated trials or in an ensemble of “identically” prepared systems
Seems to make probability an objective property, existing in the nature independently of us, that should be based on empirical frequencies
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
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
Frequentist approachSelf-proclaimed “objective” contrary to the “Bayesian” inference that should be necessary “subjective”
Bayesian approachThe 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
Statistical InferenceStatistical 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
Bayesian inference
Parameters can also be probabilized
Frequentist inference
All probabilities (in fact frequencies) are conditional on [unknown] parameters
Significance tests (parameter value fixed by hypothesis)
Confidence intervals
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
Known data0 0 0 1 0
f = 1/5
Unknownparameter
= ?
A finite population of size N=20With a dichotomous variable
1 (success) – 0 (failure)Proportion of success
A sample of size n=5from this populationhas been observed
A simple illustrative situationA simple illustrative situation
Inductive reasoning: generalisation from known to unknown
Known data0 0 0 1 0
f = 1/5
Unknownparameter
= ?
In the frequentist framework: no probabilities no solution
Frequentist inference: from unknown to known
Known data0 1 0 0 0
f = 1/5
Unknownparameter
= ?
no more solution
Frequentist inference
Imaginary repetitions of the observations
f = 0/5 : 0.00006f = 1/5 : 0.005f = 2/5 : 0.068f = 3/5 : 0.293f = 4/5 : 0.440f = 5/5 : 0.194
One sample have been observed out 15 503 possible samples
Sampling probabilities = frequencies
ParameterFixed value
Example:
= 15/20 = 0.75
Data: 0 0 0 1 0 (f = 1/5 = 0.20)
If = 0.75 one find in 99.5% of the repetitionsa value f > 1/5 (greater than the observation f=0.20)
The null hypothesis = 0.75 is rejected (Significant: p = 0.00506)
Imaginary repetitions of the observations
f = 0/5 : 0.00006f = 1/5 : 0.005f = 2/5 : 0.068f = 3/5 : 0.293f = 4/5 : 0.440f = 5/5 : 0.194
Null hypothesis
Example 1: = 0.75 (15/20)
Level: = 0.050.995
Data0 0 0 1 0 f = 0.20
Frequentist significance test
However, this conclusion is based on the probabilityof 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 theactual value is nearly always negligible.
What the use of P implies, therefore, is that a hypothesis that may be truemay be rejected because it has not predicted observable results that havenot occurred.
This seems a remarkable procedure.”’Jeffreys, 1961
Null hypothesis
Example 2: = 0.50 (10/20)
Level: = 0.05
Data0 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 : 0.293f = 4/5 : 0.440f = 5/5 : 0.194
0.848
If = 0.50 one find in 84.8% of the repetitionsA value f > 1/5 (greater than than the observation f =0.20)
The null hypothesis = 0.50 is not rejected (Non significant: p = 0.152)
Obviously this does not prove that = 0.50!
Set of possible values for that are not rejected at level
Data0 0 0 1 0 f = 0.20
Frequentist confidence interval
Example = 0.05One get the “95% confidence” interval
[0.05 , 0.60]
How to interpret the “95% 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 computedincludes this value”
Interpretation of frequentist confidence?
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 calculatingConfidence 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
Set of all possible valuesof the unknown parameter
= 0/20, 1/20, 2/20… 20/20
Bayesian inference
Probabilities that express our uncertainty
(in addition to sampling probabilities)Known data0 0 0 1 0
f = 1/5
Return to the inductive reasoning:Generalisation from known to unknown
“As long as we are uncertain about values of parameters,we will fall into the Bayesian camp”
Iversen, 2000
Bayesian inference
All the frequentist probabilities associated with the data
Pr(f = 1/5 | )
Likelihood function
= 0/20 0 = 10/20 0.135= 1/20 0.250 = 11/20 0.089= 2/20 0.395 = 12/20 0.054= 3/20 0.461 = 13/20 0.029= 4/20 0.470 = 14/20 0.014= 5/20 0.440 = 15/20 0.005= 6/20 0.387 = 16/20 0.001= 7/20 0.323 = 17/20 0= 8/20 0.255 = 18/20 0= 9/20 0.192 = 19/20 0
= 20/20 0
Data0 0 0 1 0
f = 1/5
joint probabilities (a simple product):
Pr( and f=1/5) = Pr(f=1/5 |) × Pr()
likelihood × prior probability
Bayesian inference
Predictive probabilities (sum of the joint probabilities)
Pr(f=1/5)
A weighted average of the likelihood function
We assume prior probabilities Pr() (before observation)
posterior probabilities (A simple application of the definition of conditional probabilities)
Pr( | f=1/5) = Pr( and f=1/5) / Pr(f)
The normalized product of the prior and the likelihood
“Bayesian statistics is difficult in the sense that thinking is difficult”
Berry, 1997
We can conclude with Berry
Considerable difficultiesConsiderable difficulties
with the frequentist approachwith the frequentist approach
The mysterious and unrealistic useThe mysterious and unrealistic useof the sampling distributionof 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”
Experts in statistics are not immuneExperts in statistics are not immunefrom conceptual confusionsfrom conceptual confusionsAbout confidence intervalsAbout confidence intervals
A methodological paper by Rosnow and Rosenthal (1996)
They take the example of an observed difference between two means d=+0.266
They consider the interval [0,+532] whose bounds are the “null hypothesis” (0) and what they call the “counternul value” (2d=+0.532), the symmetrical value of 0 with regard to d
They interpret this specific interval [0,+532] as “a 77% confidence interval”
(0.77=1-2×0.115, 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
Experimental research and statistical inference:Experimental research and statistical inference:A paradoxical situationA paradoxical situation
Null Hypothesis Significance Testing (NHST)
An unavoidable norm in most scientific publications
BUT
Innumerable misinterpretations and misuses
Often appears as a label of scientificness
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
Today is a crucial timeToday 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 estimatesin addition to or in place of the results of NHST
Common misinterpretations of NHSTCommon misinterpretations of NHST
Emphasized by empirical studiesRosenthal & 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 instructorsHaller & Krauss, 2001
Professional applied statisticians are not immune to misinterpretationsM.-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
Why these misinterpretations?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
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
A more or less “naïve” mixture of NHST resultsA more or less “naïve” mixture of NHST resultsand other informationand other information
BUT
this is not an easy task!
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, nullhypothesis, 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
A set of recipes and ritualsA 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 do not supply real statistical thinking
“We need statistical thinking, not rituals”
Gigerenzer, 1998
They are both partially technically redundant and conceptually incoherent
Confidence intervals could quickly become a compulsory Confidence intervals could quickly become a compulsory norm in experimental publicationsnorm 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)
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”
The ambivalence of statistical instructorsThe 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 usethis heretic interpretation
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 1 à estimer a une probabilité 0.95 de se trouver dansla fourchette: [0.58,0.64]”
“If in a public opinion poll of size 1000, one find P [the observed proportion]
= 0.613, the proportion 1 to be estimated has a 0.95 probability to be in the
range: [0.58,0.64]''
Claudine Robert, 1995, page 221
The ambivalence of statistical instructorsThe 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
The ambivalence of statistical instructorsThe ambivalence of statistical instructors
The ambivalence of statistical instructorsThe 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
Other authors claim that the “correct” frequentist interpretationthey advocate can be expressed as :
Hard to imagine that readers can understand that “confident” refers here to a frequentist view of probability!
“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
We will distinguish between probability as frequency, termed probability, and probability as information/uncertainty, termed confidence”
Schweder & Hjort (2002)
Teaching the frequentist interpretation:
a losing battle
“we are fighting a losing battle”Freeman, 1993
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
The Bayesian therapyThe Bayesian therapy
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)
A better understanding of frequentist procedures
“Students [exposed to a Bayesian approach] come to understand the frequentist conceptsof confidence intervals and P values better than do students
exposed only to a frequentist approachBerry, 1997
Combining descriptive statistics and significance tests
A basic situation: the inference about the difference 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 is ageneralized (or scaled) t distribution (with the same degrees of freedomAs the t test), centered on d and with scale factor the ratio e=d/t
(see e.g. Lecoutre, 2006)
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 has the opposite sign of the observeddifference
Bayesian interpretation of the confidence interval
It becomes correct to say that “there is a 95% [for instance] probability of
being included between the fixed bounds of the interval” (conditionally on the data)
If d>0, 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
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
Bayesian procedures are no more arbitrarythan frequentist ones
Many potential users of Bayesian methods continue to think that they are too subjective to be scientifically acceptable
BUT
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)
ExampleFor instance, let us consider the usual Binomial one-tailed test for the null hypothesis =0 against the alternative <0
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
But, for discrete data, it depends on whether the observed data are included or not
The choice of a noninformative prior distribution cannot avoid conventions
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 hypothesisThey 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 is greater than
0respectively 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)
But the particular choice of such a prior is an exact counterpart of the arbitrariness involved within the frequentist approach
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 propertiesIts coverage probability is very close to the nominal level, even for small-size samplesIt 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
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)
The preceding results can be generalized to more general situations of comparisons between proportions
(see for the case of a 2×2 contingency table Lecoutre & Charron, 2000)
Similar results are obtained for negative-Binomial (or Pascal) sampling
This suggests that the intermediate Beta distribution of parameters 0 and ½ is an objective procedureIt is precisely the Jeffreys prior
This result concerns a very important issue related to the “likelihood principle”
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
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
“An essential aspect of the process of evaluating design strategies is the ability to calculate predictive probabilities of potential results”
Berry, 1991
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
Consequently, they regret the fact that the Bayesian methods, unlike the frequentist practice, generallyignore 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 experimentationshould 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”?
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 thatthe 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 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(| i, e0 ,d) p(| e0 ,d) p(i |,e0,d)
It becomes evident that the prior depends on d
p(| i, e0 ,d) p(| e0 ,d) p(i |,e0,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 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)
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)
ConclusionConclusion
In actual fact I suggest that such a theory is by no means a speculativeviewpoint 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
“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
Why scientists really appear to want a different kind Why scientists really appear to want a different kind of inference but seem reluctant to use Bayesian of inference but seem reluctant to use Bayesian
inferential procedures in practice? 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”
WinklerWinkler, 1974
The times we are living in at the moment appear to be crucial
“we [statisticians] will all be Bayesians in 2020,
and then we can be a united profession”Lindley in Smith, 1995, page 317
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
“It is their straightforward, natural approach to inference
that makes them [Bayesian methods] so attractive”
Schmitt, 1969
Text and references available upon requestMail to : [email protected]