hood ratio is thus a ratio of normal densities determined from the respective predictive distributions, evaluated at the

observed value of m.

I n t h e f i r s t p a r t o f t h i s s e c t i o n , r e f e r e n c e was made t o t h e n o t i o n o f c o n j u g a t e d i s t r i b u t i o n s . I n t h e above example, f

( u )

and f 2 ( p ) were c o n j u g a t e d i s t r i b u t i o n s . For v a r i o u s

1

d a t a - g e n e r a t i n g p r o c e s s e s , i n c l u d i n g many o f t h e p r o c e s s e s commonly assumed i n a p p l i c a t i o n s ( e . g . t h e normal p r o c e s s , t h e ~ e r n o u l l i p r o c e s s , t h e P o i s s o n p r o c e s s , t h e normal r e - g r e s s i o n p r o c e s s , e t c . ) , t h e form o f t h e p r e d i c t i v e d i s t r i b u - t i o n h a s b e e n d e v e l o p e d u n d e r t h e a s s u m p t i o n t h a t f i ( 0 ) i s a c o n j u g a t e d i s t r i b u t i o n ( e . g . R a i f f a and S c h l a i f e r , [39] ) T h e r e f o r e , i f t h e h y p o t h e s e s of i n t e r e s t c a n b e e x p r e s s e d i n t e r m s o f c o n j u g a t e d i s t r i b u t i o n s , t h e a p p r o p r i a t e p r e d i c - t i v e d i s t r i b u t i o n c a n be found i n t h e B a y e s i a n l i t e r a t u r e and t h e d e t e r m i n a t i o n of t h e l i k e l i h o o d r a t i o i s m e r e l y a

m a t t e r o f c a l c u l a t i n g t h e a p p r o p r i a t e p r o b a b i l i t i e s ( d e n s i t i e s ) . Once a l i k e l i h o o d r a t i o i s d e t e r m i n e d , i t c a n be m u l t i - p l i e d by t h e p r i o r odds r a t i o t o a r r i v e a t t h e p o s t e r i o r o d d s r a t i o . F o r r e p o r t i n g p u r p o s e s , t h e e x p e r i m e n t e r may want t o c o n s i d e r v a r i o u s p o s s i b l e p r i o r odds r a t i o s . O f c o u r s e , i f t h e l i k e l i h o o d r a t i o i s g i v e n , i t i s e a s y f o r any r e a d e r t o i n s e r t a p r i o r odds r a t i o i n o r d e r t o d e t e r m i n e a p e r s o n a l p o s t e r i o r odds r a t i o .

It s h o u l d b e o b v i o u s by now t h a t i n t h e B a y e s i a n a p p r o a c h t o h y p o t h e s i s t e s t i n g , a g r e a t d e a l o f c a r e must be t a k e n i n t h e s p e c i f i c a t i o n o f h y p o t h e s e s . An e x a c t h y p o t h e s i s c a n o n l y be e n t e r t a i n e d i f one i s w i l l i n g t o p l a c e a n o n z e r o p r i o r p r o b a b i l i t y o n t h e s i n g l e v a l u e r e p r e s e n t e d by t h e e x a c t hypo- t h e s i s . F o r i n s t a n c e , a B a y e s i a n g e n e r a l i z a t i o n o f t h e n o t i o n

of testing a sharp null hypothesis is to consider

a

"spike"

of probability at the value specified by the sharp null hypo- thesis and an alternative hypothesis that is represented by

a distribution over the parameter space (e .g. see Jeffreys, [20] )

.

The alternative hypothesis might be taken to be a diffuse distribution, for example. If a "spikeI1 at a single point seems unreasonable, a further generalization is to let both fl(B) and f2(B) be centered at the exact value corresponding to the classical statistician's sharp null hypothesis but to make fl(0) a much tighter distribution than f2(B).

In general, the primary concern in Bayesian inference is the combination of prior information and sample informa- tion to form a poste~ior distribution. In many cases a Bayesian analysis of experimental data need not involve hypothesis testing at all. In this section, however, an attempt has been made to indicate how the Bayesian approach can be structured in terms of hypothesis testing if the experi- menter so desires.

5.

Discussion

In summary, there is an increasing interest in Bayesian procedures, although much of this interest is decision-oriented rather than inference-oriented and is concerned with development of theory rather than

with

the actual use of these procedures in practice. In the analysis of experimental results, the main concern is generally inference rather than decision,

and t h e b u l k o f c u r r e n t s t a t i s t i c a l p r a c t i c e i n t h i s

area

l e a v e s much t o be d e s i r e d , as i n d i c a t e d i n S e c t i o n 2 . Many f a c t o r s , i n c l u d i n g t r a d i t i o n , s t a t i s t i c a l t r a i n i n g , computa- t i o n a l d i f f i c u l t i e s , and r e p o r t i n g d i f f i c u l t i e s c o n t r i b u t e t o poor s t a t i s t i c a l p r a c t i c e . A s n o t e d a t t h e end o f S e c t i o n 3, an e x p e r i m e n t e r h a s l i t t l e i n c e n t i v e t o i n v e s t a g r e a t d e a l of t i m e a n d e f f o r t i n a c a r e f u l , a p p r o p r i a t e a n a l y s i s when i t a p p e a r s t h a t a s i m p l e s i g n i f i c a n c e l e v e l f o r a t e s t o f a s h a r p n u l l h y p o t h e s i s w i l l s e r v e t h e same p u r p o s e q u i t e w e l l i n t e r m s o f y i e l d i n g p u b l i s h a b l e r e s u l t s t h a t a r e a c c e p t a b l e p r o f e s s i o n a l l y .

How, t h e n , might t h e w e a k n e s s e s i n c u r r e n t s t a t i s t i c a l p r a c t i c e b e r e m e d i e d ? W i t h i n t h e c l a s s i c a l framework, improve- ments i n s t a t i s t i c a l t r a i n i n g t h a t p l a c e emphasis on meaning r a t h e r t h a n mechanics would be most u s e f u l , a s would a w i l l - i n g n e s s on t h e p a r t o f j o u r n a l e d i t o r s and r e f e r e e s t o demand c l e a r , m e a n i n g f u l s t a t i s t i c a l a n a l y s e s . The d i s c u s s i o n o f s c i e n t i f i c r e p o r t i n g i n S e c t i o n

4

i s r e l e v a n t h e r e . F u r t h e r - more, s i n c e t h i s p a p e r i s w r i t t e n from t h e B a y e s i a n s t a n d p o i n t , t h e view t a k e n h e r e i s t h a t t h e u s e o f B a y e s i a n t e c h n i q u e s would l e a d t o g r e a t improvements i n s t a t i s t i c a l p r a c t i c e , p r o v i d e d t h a t t h e s e t e c h n i q u e s a r e used c a r e f u l l y and a p p r o - p r i a t e l y . B a y e s i a n p r o c e d u r e s g e n e r a l l y p r o v i d e a n s w e r s t o t h e q u e s t i o n s o f i n t e r e s t t o t h e e x p e r i m e n t e r r a t h e r t h a n a n s w e r s t o r e l a t e d b u t d i f f e r e n t q u e s t i o n s . For example,

p r o b a b i l i t y s t a t e m e n t s c a n be made d i r e c t l y a b o u t t h e p a r a m e t e r s o f i n t e r e s t i n s t e a d of i n d i r e c t l y i n t e r m s o f p r o b a b i l i t i e s

of sample outcomes c o n d i t i o n a l upon t h e p a r a m e t e r s .

I n o r d e r t o i n c r e a s e t h e u s e o f B a y e s i a n i n f e r e n t i a l p r o c e d u r e s i n p r a c t i c e , i t i s n e c e s s a r y t o narrow t h e " t h e o r y - p r a c t i c e gap" by making B a y e s i a n p r o c e d u r e s more " a v a i l a b l e n t o e x p e r i m e n t e r s , A t t h e most b a s i c l e v e l , t h i s e f f o r t i n v o l v e s t h e u s e o f i n t r o d u c t o r y - l e v e l , i n f e r e n c e - o r i e n t e d B a y e s i a n t e x t s . M a t e r i a l on B a y e s i a n i n f e r e n c e above t h e e l e m e n t a r y i n t r o d u c t o r y l e v e l i s a v a i l a b l e i n books s u c h a s R a i f f a and S c h l a i f e r

[39],

J e f f r e y s [20], L i n d l e y [24], [25], P r a t t , R a i f f a , and S c h l a i f e r C37]

,

Good

[17] ,

^{DeGroot [lo],}

L a V a l l e [ 2 2 ] , Z e l l n e r [51], and Box and T i a o

151

^: many o f t h e s e r e f e r e n c e s a l s o c o n t a i n m a t e r i a l on d e c i s i o n - m a k i n g p r o c e d u r e s . Most i n t r o d u c t o r y t e x t s t h a t a r e B a y e s i a n i n n a t u r e a r e s t r o n g l y d e c i s i o n - o r i e n t e d ( e . g . R a i f f a , [38]

,

L i n d l e y , [ 2 6 ]

,

Moore, 2

,

and Brown, Kahr, and P e t e r s o n ,

[ 6 ] ) .

Some o t h e r i n t r o d u c t o r y B a y e s i a n t e x t s c o n t a i n a m i x t u r e o f i n f e r e n t i a l m a t e r i a l and d e c i s i o n - t h e o r e t i c m a t e r i a l . F o r example, S c h l a i f e r [43] was t h e p i o n e e r i n g i n t r o d u c t o r y - l e v e l book i n t h i s a r e a ( a l s o , s e e S c h l a i f e r , L44] ; S c h m i t t [463 p l a c e s some s t r e s s on i n f e r e n c e ; Winkler

[49]

i n c l u d e s q u i t e a b i t o f i n f e r e n t i a l m a t e r i a l ; a r e c e n t book by P h i l l i p s [31]

i s i n t e n d e d t o " f i l l t h e gap" somewhat i n t e r m s o f B a y e s i a n i n f e r e n c e ; and o t h e r books may b e i n p r e p a r a t i o n ( e . g .

Pitz,

p5]). More books e m p h a s i z i n g B a y e s i a n i n f e r e n c e a t t h e i n t r o - d u c t o r y l e v e l a r e needed.

Moving f r a m t h e t r a i n i n g Level t o t h e l e v e l o f a c t u a l

a p p l i c a t i o n o f t h e t e c h n i q u e s , f u r t h e r e f f o r t s h o u l d be expended

on expressing Bayesian procedures in forms that make them more accessible to users. This involves such steps as expres- sing the procedures in simplified form (e.g. simplifying

formulas for likelihood ratios as much as possible for situa- tions that are widely-encountered) and developing computer programs. Some individuals have worked on the first step

(e.g. Pitz, [34]) and on the second step (e.g. Schlaifer,

[45],

Novick, [ 3 0 ] ) . Furt.hermore, at the level of application, perhaps the most useful step in terms of the advancement of Bayesian inference would be the publication of more actual Bayesian analyses of experimental data in journals in the areas of application.

An

example of a particularly detailed analysis that might be useful for researchers to look at is a disputed- authorship problem studied in Mosteller and Wallace

[29]

; some applications in the area of medicine are presented in Cornfield [7]; and an application in the area of education is given in Novick

1301.

For an interesting (and somewhat con- troversial) application of Bayesian hypothesis testing, see Good [18] and Efron

[13.

Another area of interest is that of scientific reporting.

Research in this area might concentrate on the development of different lfpackagesll of items to be reported in different situations and on attempts to simplify these packages without a considerable loss in terms of the information content of the packages. For example, Dickey [ll] develops graphical techniques for relating parameters of prior distributions to

parameters of posterior distributions and considers bounds on odds ratios for various situations (also, see Dickey and

Freeman, [12]). More work along these lines would be valuable.

In addition to the need to make simple Bayesian procedures more available to users, further theoretical work would be useful. Such work might involve the development of Bayesian procedures for various types of models that have not been studied extensively from the Bayesian standpoint to date and the development of approximations that might simplify

a

Bayesian analysis. For instance many different situations are reviewed in Lindley [25], and various models have been considered in recent work in Bayesian econometrics (e.g. see Zellner, [51]

,

and Fienberg and Zellner, [16]).

In this paper, some weaknesses in current statistical practice have been discussed, and suggestions for remedying these weaknesses have been presented. The Bayesian approach, which has received much attention in recent years, particularly in terms of decision making, provides a useful framework for the analysis of experimental data. Efforts are needed to make Bayesian procedures more readily available to researchers dealing with experimental data, and some suggestions for the direction of such efforts have been given in this concluding section.

F o o t n o t e s

' ~ r o c e d u r e s t h a t do n o t i n v o l v e p r o b a b i l i t y r e v i s i o n a r e f r e q u e n t l y i n c l u d e d u n d e r t h e h e a d i n g " B a y e s i a n s t a t i s t i c s . ' ' I n p a r t i c u l a r , b e c a u s e B a y e s i a n methods u s e s u b j e c t i v e p r o b a - b i l i t i e s a s i n p u t s , i t i s o f t e n e r r o n e o u s l y assumed t h a t

" s u b j e c t i v e " and B a y e s i a n s ' a r e synonymous. A l s o , b e c a u s e B a y e s i a n methods a r e f r e q u e n t l y u s e d i n d e c i s i o n - m a k i n g p r o b l e m s , i t i s o f t e n e r r o n e o u s l y assumed t h a t t h e a d j e c t i v e

"Bayesian" i s always used i n c o n j u n c t i o n w i t h " d e c i s i o n t h e o r y . 2 ~ n d e r c e r t a i n c i r c u m s t a n c e s , B a y e s i a n and c l a s s i c a l

p r o c e d u r e s may y i e l d s i m i l a r n u m e r i c a l r e s u l t s . Even i n s u c h i n s t a n c e s , however, t h e i n t e r p r e t a t i o n s a t t a c h e d t o t h e numeri- c a l r e s u l t s by t h e two s c h o o l s o f t h o u g h t a r e q u i t e d i f f e r e n t .

tatis is tical

a n a l y s e s a r e p r e p a r e d f o r many d i f f e r e n t p u r p o s e s . I f t h e e x p e r i m e n t e r o n l y w a n t s t o u s e t h e a n a l y s i s f o r p e r s o n a l p u r p o s e s , t h e n i t i s a p p r o p r i a t e t o c o n s i d e r o n l y t h e e x p e r i m e n t e r ' s p r i o r d i s t r i b u t i o n . I f t h e a n a l y s i s i s b e i n g p r e p a r e d f o r a p a r t i c u l a r c l i e n t , t h e n t h e c l i e n t ' s p r i o r d i s t r i b u t i o n would b e t h e r e l e v a n t d i s t r i b u t i o n t o c o n s i d e r . T h i s p a p e r i s p r i m a r i l y c o n c e r n e d w i t h r e p o r t i n g t o t h e g e n e r a l s c i e n t i f i c community. F o r t h i s a u d i e n c e , t h e p o s t e r i o r d i s t r i b u t i o n f o l l o w i n g t h e e x p e r i m e n t e r ' s p e r s o n a l p r i o r d i s t r i b u t i o n might be o f some i n t e r e s t b e c a u s e o f t h e f a c t t h a t t h e e x p e r i m e n t e r p r e s u m a b l y h a s g i v e n t h e p r o b l e m a t hand. s e r i o u s t h o u g h t . However, o t h e r s may have d i f f e r e n t p r i o r d i s t r i b u t i o n s , and i t i s g e n e r a l l y i n a p p r o p r i a t e t o con- f i n e t h e a n a l y s i s t o t h e e x p e r i m e n t e r ' s own p o s t e r i o r d i s t r i - b u t i o n .

[I]

Bakan, D . "The T e s t o f S i g n i f i c a n c e i n P s y c h o l o g i c a l

Edwards,

W.,

Lindman,H., and Savage, L.J. "Bayesian

[28] Moore, P . G . R i s k i n B u s i n e s s D e c i s i o n . New York, W i l e y ,

Savage, L.J.,

et s.

The Foundations of Statistical- Tnference. London, Methuen, 1962.

Schlaifer, R e Probability and Statistics for Business Decisiong. New York, McGraw-Hi'll, 1959.

Schlaifer,

R.

Analysis of Decisions Under Uncertainty.

New York, McGraw-Hill, 1969 Schlaifer,

R.

Analysis.

Administration, Harvard University, 1971.

S chmi Measuring Uncert

ass., Slovic, P., and Lichtenstein,

S.

'Toomparison of Bayesian

and Regression App~oaches to the study of ~nfoknation Processing in Judgment,"

Human Performance,

5

⁽¹⁹⁷

Wilson, W., Miller, W.L., and Lower, J . S . "Much Ado About the Null Hypothesis," Psychological Bulletin, 67 (1967), - 188-196.

Winkler,

R.L. An

Introd~cti~on to Bayesian Inference and Decision. New York, Wolt, Rinehart and Winston, 1972.

Winkler, R.L. "Statist'ical Analysis: Theory Versus

practice

,"

Theory and ~ractice of ~ e a s b r i n ~ Subjective Probability. Dordrecht, D. Reidel, 1974, in press.

Zellner,

A. An

z,ntrod.u.ction, to Bsesian Inference

in

Econometrics. New York, Wiiey, 1971.

Im Dokument Bayesian Statistical Analysis of Experimental Data (Seite 27-38)