Transcript Beyond the significance test controversy: Prime time for Bayes?

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: [email protected]
Internet: http://www.univ-rouen.fr/LMRS/Persopage/Lecoutre/Eris
E
quipe
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I
aisonnement nduction
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tatistique
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)
Probability
and
Statistical Inference
(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”
(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
(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?”
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
 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 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
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
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
A simple illustrative situation
A finite population of size N=20
With a dichotomous variable
1 (success) – 0 (failure)
Proportion j of success
Known data
00010
f = 1/5
A sample of size n=5
from this population
has been observed
Unknown
parameter
j=?
Inductive reasoning: generalisation from known to unknown
Unknown
parameter
j=?
Known data
00010
f = 1/5
In the frequentist framework:
no probabilities
no solution
Frequentist inference: from unknown to known
Known data
01000
f = 1/5
no more solution
Unknown
parameter
j=?
Frequentist inference
Data: 0 0 0 1 0 (f = 1/5 = 0.20)
Parameter
Fixed value
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
 Example:
Sampling probabilities = frequencies
j = 15/20 = 0.75
Frequentist significance test
Null hypothesis
Example 1:
j = 0.75 (15/20)
Level: a = 0.05
Data
00010
f = 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
0.995
f = 4/5 : 0.440
f = 5/5 : 0.194
If j = 0.75 one find in 99.5% of the repetitions
a value f > 1/5 (greater than the observation f=0.20)
 The null hypothesis j = 0.75 is rejected (Significant: p = 0.00506)
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
Frequentist significance test
Null hypothesis
Example 2:
j = 0.50 (10/20)
Level: a = 0.05
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
Data
00010
f = 0.20
0.848
If j = 0.50 one find in 84.8% of the repetitions
A value f > 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!
Frequentist confidence interval
Data
00010
f = 0.20
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”?
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”
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
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
Known data
00010
f = 1/5
(in addition to sampling probabilities)
“As long as we are uncertain about values of parameters,
we will fall into the Bayesian camp”
Iversen, 2000
Bayesian inference
Data
00010
f = 1/5
All the frequentist probabilities associated with the data
Pr(f = 1/5 | j)
 Likelihood function
j = 0/20  0
j = 1/20  0.250
j = 2/20  0.395
j = 3/20  0.461
j = 4/20  0.470
j = 5/20  0.440
j = 6/20  0.387
j = 7/20  0.323
j = 8/20  0.255
j = 9/20  0.192
j = 10/20 
j = 11/20 
j = 12/20 
j = 13/20 
j = 14/20 
j = 15/20 
j = 16/20 
j = 17/20 
j = 18/20 
j = 19/20 
j = 20/20 
0.135
0.089
0.054
0.029
0.014
0.005
0.001
0
0
0
0
Bayesian inference
We assume prior probabilities Pr(j) (before observation)
 joint probabilities (a simple product):
Pr(j and f=1/5) = Pr(f=1/5 | j) × Pr(j)
likelihood × prior probability
 Predictive probabilities (sum of the joint probabilities)
Pr(f=1/5)
A weighted average of the likelihood function
 posterior probabilities (A simple application of the definition
of conditional probabilities)
Pr(j | f=1/5) = Pr(j and f=1/5) / Pr(f)
The normalized product of the prior and the likelihood
We can conclude with Berry
“Bayesian statistics is difficult in the sense that thinking is difficult”
Berry, 1997
Considerable difficulties
with the frequentist approach
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”
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
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:
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
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
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
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 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!
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
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)
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 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
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
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
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
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)
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 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 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 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 d has the opposite sign of the observed
difference
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
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)
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
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
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 j=j0 against the alternative j<j0
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
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)
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 smallsize 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
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 2×2 contingency table Lecoutre & Charron, 2000)
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
“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
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, 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”?
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 
 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)
p( | i, e0 ,d)  p( | e0 ,d) p(i |  ,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 
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)
Conclusion
“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
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
“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
Text and references available upon request
Mail to : [email protected]
“It is their straightforward, natural approach to inference
that makes them [Bayesian methods] so attractive”
Schmitt, 1969