David Brenner/D.A.S. Fraser

On Foundations for Conditional Probabilit Y with Statistical Models - When is a Class of Functions a Function

I. Introduction

We are concerned here with the foundations of statistical inference. For this the starting point is a statistical model together with data. We do not examine the requirements for and the validation of a statistical model. These pre-in q ference questions are examined elsewhere.

A statistical model and data provide two aspects of the in- vestigation of a stable system: the statistical model records the background information as made relevant to the investig- ation; and the data are the results of the performances of the system in the investigation.

For statistical inference the traditional model is a selected class of random variables and the objective is insight con- cerning the random variable that most accurately describes the stable system under investigation. In its raw form this standard model is simply an indexed class of random variables which may be denoted~1 = {y(8) : 8e~} where'y(8) = (~,~,Ps) is a probability space with 8 in the index space Q of un- specified structure; the objective is insight concerning the @ value that yields the most accurate description of the system.

For some stable systems, however, a basic or intrinsic vari- ation can be identified and the observable response is a transformation or presentation of that variation; the model is a selected class of transformations operating on a proba- bility space and the objective is insight concerning the transformation (presentation) that most accurately describes the system.

This transformation model can be denoted~ 2 = (z,9) where z = (~,2,P) is a probability space and % is a class of measur- able transformations ~ : ~ +~; note that ~ is also used as the index space ~ = #.

For many applications the identification of the basic vari- ation is primary. For example randomization in the design of experiments is used to generate a clearly defined variation or error.

The ~enera!ized transformation model allows for uncertainty concerning the distribution for variation:~63=({z(1) :leA},~) where the index space is now q = A x e.

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The basic analysis of a transformation model with data can involve (Fraser, 1968) a deductive step that is necessary- determined by the model and data alone, and not dependent on external reduction criteria or principles. This occurs if the class of transformations is a group with respect to com-

position; the model correspondingly is called a structural model or G-model.

The transformation structure is an explicit component of the transformation model. Nevertheless it is easy to misconstrue such a model as just a standard model. Certainly a trans- formation model by contraction generates a standard model via the identification

y(~) = r

however in the contraction the transformation structure is lost as an explicit component.

In a mathematical sense the reverse process is always pos- sible: the standard model J41 can be represented as a con- traction from a transformation model. For this let z be the independent product | and ~ be the coordinate ~rojection of z on y(8); the transformation model is~& 2 = (z,{~:~s This is an existence statement and says nothing about unique- ness. For example the contraction from a G-model can also be presented as the preceding contraction from an independent product model. However, two distinct transformation models that contract to the same standard model are just that: distinct. In a given application the presented structure is specific and elemental (Fraser, 1967).

We mention that not every standard model can be represented as a contraction from a G-model. For example if (Yl,Y2) is bivariate normal with standard normal marginals and correl- ation p then a differentiable group structure is not avail- able.

2. The Canonical Model

Consider briefly the basic analysis of the G-model. There is a direct factorization of the space ~ into a cross section space and an orbit space; and correspondingly there is a factorization of the variation z into a marginal distribution on the cross section and a conditiona.l distribution on the orbit space. If G acts freely (without isotopy) then the con- ditional distribution is directly on the group as a space giving coordinates on the orbit space.

In scheme, the analysis depends only on simple properties of transformation groups. For a G-model with data y, the equali- ty Gz = Gy is an event for the variation and the orbit of z is observed; there is no information concerning the position of z on its orbit and the position of z is not observed. If G acts freely the position is given directly by the group.

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A central concern for transformation models is to establish conditions under which the available information from a data response takes the form of a genuine event for the variation, with the result that the distribution P is replaced by the conditional distribution given the observed event. We are led to the solution by determing degrees of resolution of trans- formation models.

3. Categorical Information

Conditional probability may well be the most important and fundamental concept in statistics. To obtain conditional probabilities for a variable z from available information, it seems necessary that the information identify an observed set C from a partition ~ of ~ and provide nothing further con- cerning the realized value in C. More specifically the in- formation source gives an observed set C in a partition and admits a value in C has occured, each value in C could have occured, and each value in C would have given the same information. In such cases we will speak of categorical in- formation. For this we must not only have information but also know how it was produced.

The requirements are satisfied in a simple and familiar manner if we have a measurable function f : ~5 + ~ and obtain just the observed value say y of the function. A value y for the function just identifies the antecedent z as a point in the

set f-l(y) of the pre-image partition ~ =~/f. For this two functions are equivalent if they display the same information concerning the antecedent value. This of course is ordinary equivalence of functions: f and g are equivalent if ~/f=2e/g.

Now more generally consider an information producing process and a data space ~. Let S(y) be the set of possible antecedent values z on ~ that are consistent with the data y and the in- formation process; we assume ~ = im Sc~ and we call S(y) the antecedent set for the data y. Also let H(y) be the totality of additional information from the data y with the information process; let ~ = im H. Then the information process with data y gives (S(y),H(y)) in ~ x ~; note that we have presented the categorical type information as a first component. The re- quirements then give the following definition.

Definition 3.1. The information process (S,H) is categorical if im S is a partition of ~ and if im (S,H) is the graph of ~-function ~ + ~.

Note that the second part of the definition asserts that H can be written as a function of S. For the first part of the definition we see that an observed y identifies a set S(y) in a partition; and for the second part we see that the same in- formation is presented whenever a particular antecedent set S = S(y) is obtained.

For an information process (S,H) we will call S the apparent information process. Note that an apparent information pro- cess is categorical if im S is a partition.

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We now develop some notation that allows us to characterize information processes that are categorical. For a given y in

the set S(y) records the set of all possible antecedent values in~ . Each of these values could in accord with the information process have produced various possible alternative data values; let T(y) be the alternatives set consisting of all these values alternative to the given y; and let ~ = im T. Note that T is a function of S, say T(y) = fS(y).

Proposition 3.2. An apparent information process S is cate- gorical if and only if S(y) is constant valued on each alter- natives set T .

o

Proof. Suppose that S is categorical, that is, ~ is a par- tition. Consider S(y) = S O in the partition. The alternatives set T(y) consists of data values with an antecedent value in S(y) ; but ~ is a partition so that S(y)~S(y ') # ~ implies S(y) = S(y'), thus T(y) = S-IS(y). The inverse function shows that ~is a partition in one-one correspondence with ~ and that f : ~ ~ ~ is a bijection. Hence S(y) = f-IT o for y in T O and S is constant valued on sets T o .

Conversely, suppose that S is constant valued on each alter- natives set T o : for y in To,S(y) = S o say. For y in To, T(y) consists of data values with an antecedent value in S(y) = So; hence for y in T O , T(y) is determined by S O and thus T(y) = To; accordingly T is a projection map with partition ~. Also we obtain that S is a function of T; thus f : ~ + ~ is a bi-

jection. Consider two sets S I and S 2 that intersect (an ante- cedent value in common); then fSINfS 2 @ @ and thus fS I = fS 2 and S I = S 2. It follows that ~ is a partition.

Proposition 3.3. An information process (S,H) is categorical if and only if (S(y) ,H(y)) is constant valued on each alter- natives set T O .

Proof. Suppose that (S,H) is categorical. By Proposition 3.2., S is constant valued on each alternative set To; by Definition 3.1., H is a function of S and thus also is constant valued on each alternatives set T o .

Now suppose (S,H) is constant valued on each alternatives set T o . By Proposition 3.2. ~ is a partition; and for y in T o ,

-1To S(y) = f and H is constant valued; thus H is a function of S.

4. Resolutions of an Information Process

An information process may not be categorical and yet have a derived process that is categorical; we call such a derived process a resolution of the initial process.

A covering of 2~ is a collection of subsets whose union is all ofZ . We will say that a covering ?z is finer than% w (~ > 7) if for any u in%6 there is a v inT/ , and vice versa, such that uCv; and in the same circumstances we can say that 7/ is coarser than ~z(7/<~) .

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Consider an information process (S,H). Then associated with any partition ~ coarser than ~, there is a well defined map R : ~ + ~ that takes y to the unique element of ~ that con- tains S(y). In fact if we let p denote the canonical pro- jection on ~, we can write R(y) = pS(y). Clearly R is cate- gorical and we call it a resolution of the process (S,H). Of course such things exist, for the trivial map R(y) = ~ is a resolution. We will say that one resolution is greater than another if its image partition is finer than the others.

A resolution has a simple property associated with the alter- natives sets T(y).

ProDos{tion 4.1. A resolution R ~s constant valued on any alternatives set T in ~.

o

Proof. Let T o = T(y o) and consider any y in T o . From the de- finition of T we have a value z antecedent to Yo and y is a possible data value from z; accordingly S(y)NS(y o) is nonempty and hence R(y) = R(Yo).

Proposition 4.2. For any information process (S,H) there is a unique maximum resolution.

Proof. Let H denote the full collection of partitions ~or equivalently of canonical projections p that are coarser than ~. We define a map pj fromX to~ by p$(z) = Np(z) with peH

and we let ~= im p; we will show that P~ is a partition

satisfying ~>~ > ~ for any ~ in H, and thus have that

R~ = p~S is the unique maximum resolution.

For any z in ~ we have zep~(z). Let zep~(z o) for a given z O.

Then zep(z O) or p(z) = p(z O) for each p in ~. Accordingly we have p~(z) = Np(z) = Np(z O) = p~(zo); hence p2~= p~. Thus p/

is a projection map and ~ is a partition.

The relation ~ > ~ is obvious from the definition of~. The

relation ~ > ~ follows easily: for any set S in ~ and z in S

we have SCp(z) for each p in H and hence SCp~(z) ; this with~j being a partition gives the relation.

From now on when we speak of the resolution of an information process we will mean the unique maximum resolution R~ , typically designated R.

For examples we turn to the transformation models from Section 1. The information process given by the transformations or presentations is easily examined to isolate the categorical- type information S: for the response value y we have the fol- lowing antecedent set.

S(y) = {z : 4)(z) = y for some 4) in ~} = U ~-ly = ur

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The alternative set T(y) can also be recorded explicity in terms of the class r

-I T(y) = u ~S(y) = ur = ur162 y.

ExamDle 4.3. Consider a probability distribution for (Zl,Z 2)

on ~2 , and as an information process suppose that (yl,Y2) =

(z I + 9, z 2 + 9) is reported for some 9 in [O,1] . The set

S(Yl,Y2) is a closed interval with unit slope in ~2, length

~, and upper end point (yl,Y2) . The maximum resolution has

R(Yl,y 2) equal to the full affine line containing the pre-

ceding interval.

Example 4.4. Consider a probability distribution for (Zl,Z 2)

on 2 = ~ • (~\O) , and as an information process suppose

(yl,Y2) = (zi-9, z2/@) is reported for some @ in (O,1). The

set S(Yl,Y2) is an open interval with end points (Y1,O) ,

(Y1+1,Y2) . The maximum resolution ~(yl,Y2 ) is however the

entire space ~; the resolution is trivial.

5. Classification of Transformation Models

We now consider a classification of transformation models 2 based on the amount of categorical information, or equivalent. ly on the degree of resolution.

For a model with R(y) H ~ it is reasonable to say that the model has null resolution and is of resolution class ~o- The model in Example 4.4. is of resolution class ~o- As a second example it is straight forward to see that the independent product model derived from a standard model is of class ~o-

Now consider a model for which the resolution is non trivial R(y) ~ ~ and yet its partition is strictly coarser than~ . We will say that such a model has resolution class ~ I" The model in Example 4.3. is of resolution class~ I"

For a model of resolution class~ I we could perhaps consider the conditional distribution zlp(z) = R(y) but we would lack the full requirements recorded in Section 3; indeed we know that the variable z is constrained to a greater degree than expressed by the conditioning relation. On the positive side however we do have that p: ~ + ~ is a nontrivial function on the probability space and is observable.

Suppose we think of the preceding case but modified to have a I dependent distribution on the variation space: z = z(1) = (~,~,Pl) with i in <. This gives a generalized transformation model. With a reported response y we can of

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course determine the likelihood function re ~ based on the observed p(z) = R(y). This provides an extension of the marginal likelihood (as in Fraser 1968, 1979) to more general models.

Now consider a model for which the resolution R coincides with the apparent information S; that is R(y) { S(y) or equivalently the image J of the apparent information is a partition on the probability space. We will say that such a model is of resolution classl2.

From the remarks following'Definition 3.1. we have that a model is of resolution classY2 if the apparent information is categorical, and from Proposition 3.2. we have that a model is of resolution classY2 if and only if S is a constant valued on any alternatives set T o .

Example 5.1. Consider a probability distribution for ~ on ~n

and let k project z onto the orthogonal complement of

I = (I .... ,I) by kz = z - zl. Define <U,o> z = [~,a]kz =

I + o(z - z I) and let ~ = {<~,~> : !~! < I, ~ > o}. Then

(z,~ ~) is~ 2. For we have T(~) = {b ~ + c(~-y) :Ib!<1,c>o} and

S(y) = f-IT(y) = {a I + cy : - ~ < a < , c > o} is an entire

half plane (orZ (I)) and these partition ~n.

For a model of resolution classY2 we could consider with somewhat more comfort than for~ I the conditional distri- bution zlp(z) = R(y). For this we do have the partition re- quirement from Section 3: the variable z is not constrained to a greater degree than expressed by the conditioning re- lation. However the information from the observed y may not be categorical.

Now consider a generalized transformation model with the present resolutionS2 for the transformations #. With a re- ported response y we have that p(z) = S(y) is the full cate- gorical observation on the variation space and we are in a somewhat stronger position than with 4 1 when we calculate the likelihood function re y based on the observed p(z)=S(y) ; this is the marginal likelihood for I.

Now consider a model for which the information process itself is categorical; we will say that such a model is of resolution class~ 3. This class~ 3 is a special case of resolution~ 2 and it fully satisfies the criteria for conditional probability in Section 3. We examine and characterize this resolution class in the next section. We then define in Section 7 a re- solution classY4 which represents a simple pure form of~3.

6. Models with Categorical Information

We first develop some notation that allows the explicit ex- pression of the information process with a transformation

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model. In turn this allows the characterization of models with categorical information.

For a response value y with ~ unknown in ~ we can contemplate in turn each set r of antecedent values and record with it the associated transformation ~. We formalize this as an in- formation display concerning the possible antecedent z and

write D(#,y) = {(~,r : ~s Note that a display is a function from # to~ . For given spaces~ and~ , let ~ denote the collection of all displays D(~,y), that is, ~ = im D(.,.); of course, this collection is extremely large.

We now eliminate the choice of labelling for the space ~ in the way that the partition ~ was obtained from a function f at the beginning of Section 3. Let ~(~,~') denote the set of all bijections of the space ~ to another ~', and for re- lebelling of y we consider the group B(~,_ ~) . Consider ssB(~,~) = ~ ; the model (z,~) as given initially becomes the model (z,s%) as relabelled. The information display from the relabelled sy is D(sy,s~)=[(sr162162162 CE#}. We can now eliminate the arbitrariness of notation by forming the equivalence class over relabelling; the in- formation from # and y is

I(~,y) =~D(#,y) = {D(s~,sy) : se~}.

We now make explicit the definition at the end of Section 5. A model is of resolution classY3 if the information I(~,.) is categorical.

From Proposition 3.3. we have that a model is of resolution classY3 if and only if the information I(~,.) is constant valued on the sets T(y). The details of the proof of Pro- position 3.3. then give immediately the following:

Proposition 6.1. A model is of resolution class ~3 if and only if the information function I(r is constant valued on each alternatives set T in ~.

o --

We can obtain a more incisive characterization of models that are of resolution class ~ 3. For this we introduce invariant ro~ G = G# of ~:

G~ = {t : ts t# = #}.

We denote the corresponding orbit partitions on ~ and �9 by ~/G and r respectively. Of course G~ is a group.

proposition 6.2. The preimage of the information function is

Proof. Let y and y' having I(~,y) = I(r Then D(sr = D(~,y') for some s in~ ; hence y' = sy and r = sr which gives seG. Now let y' = sy for some s in G; then ~ = s~ and D(s%,sy) = D(~,y') which give I(~,y) = I(r

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Theorem 6.3. A model is of resolution~ 3 if and only if ~/G : x.

Corollary 6.4. A model is of resolution~ 3 ~f and only if G is transitive on the alternative sets T(y).

Proqf. A direct composition of Propositions 6.1, 6.2.

We now have the basic structure of ~3 models. However before some more general comments we record a corollary that exhibits the information as a direct function on the probability space for z.

Corollaru 6.5. ~ = Gy gives a well defined map ~ § ~ whose inverse bartition isY if and only if the model is

3"

Proof. If the model is~3, then for any z in S(y) and any in r #zsT(y) and GCz = Gy; thus G%z is independent of ~ in ~; hence Gy = Gt z = Gr z = %z. Conversely the condition gives that an orbit of G is obtained as the ~ images of any antecedent z; thus ~ /G = ~.

An~3 model has substantial structure. G acts on @ by com- position and r is the orbit structure; let A be a cross section. Then ~ is the disjoint union of orbits GI with l in A. Let w = (z,A) by the standard model defined by w(l)=lz for z in A. Then (w,G) gives a partly contracted version of the initial model and has a group as the class of transform- ations. This is the canonical G-model but with a parameter l on the space for variation.

7. Repeated Sampling

With repeated samples, ~ acts coordinate by coordinate from

~n toTn and the group G with G# = % also acts coordinate by

�9 oerdinate on~n. As our final resolution we will say that a model has resolution class~ 4 if it is ~3 and remains ~3 under repeated sampling.

We can characterize an ~4 model rather easily provided we introduced an assumption that assumes that the group G is not too large.

Assumption 7.1. There is a sample size n, such that the G- action has an effective poin t [o : if g[o = ~o then g = I.

We call this condition EP.

Theorem 7.2. A model (z,~) satisfying EP is of class R 4 i_~f and only if ~ = G#o for some ~o in @. The relation ~ = g~o is_a correspondence.

Proof. Let ~o = ~o~o be an effective point in~ n Then

t~o = g~o~o for some g in G. Let z be any point in ~ and con-

sider (~o,Z) as a sample point for~n+1. Then

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~(Zo,Z)~ = gz~o(Zo,Z)% for some gz in G. Thus g ~ z =~z =g~ z z o o o 0%0

which gives gz = g independent of z. It follows that @z=g@oZ

for all z and thus that ~ = gr with g uniquely determined.

Corollar~ 7.3. Under the condition of Theorem 7.2., if the function are all onto then @ = G@ for any ~ in @.

An~4 model is effectively a G-model. For 4o in @ let W=@oZ; this relabelling presents (z,~) in the form (w,G) which is the canonical G-model.

References

I. Brenner, David (1977). When is a class of functions a function, Ph.D. Thesis, University of Toronto.

2. Fraser, D.A.S. (1967). Data transformations and the linear model, Ann. Math. Stat. 38, 1456-65.

3. Fraser, D.A.S. (1968). The Structure of Inference, Huntington, N.Y.: Krieger Publishing Company.

4. Fraser, D.A.S. (1971). Events, information processing, and the structural model, Proceedings of the Symposium on the Foundations of Statistical Inference, Ed. V.P. Godambe and D. Sprott, Toronto: Holt, Rinehart and Winston.

5. Fraser, D.A.S. (1979). Inference and Linear Models, New York: MacGraw Hill.

Zusammenfassunq

D. Brenner/D.A.S. Fraser: Uber die Grundlegung der bedingten Wahrscheinlichkeit in statistischen Modellen -Wann ist eine Klasse von Funktionen eine Funktion?

In Diskussionen Hber logische Strukturen bei Inferenz hat Fraser (1967, 1968, 1970, 1979) gezeigt, dab die Datenanalyse mit einem Transformationsgruppenmodell einzig und allein be- stimmt wird durch die exakte Zerlegung des Fehlers oder der Variation in einen beobachtbaren (G-Orbit) und einen nicht beoabachtbaren Teil (Position auf einem G-Orbit) : die Infor- mation, die der Reaktionswert liefert, ist ~quivalent zu ei- nem Ereignis f~r die Variation und bildet die Basis fHr die direkte Anwendung bedingter Wahrscheinlichkeiten. Solche Mo- delle sind - bezogen auf diese Eigenschaft - in einer noch zu beschreibenden Weise kanonisch.

In diesem Artikel untersuchen wit f~r diese kanonische Tren- nung von beobachtbaren und unbeobachtbaren Ereignissen und f~r die Erzeugung reiner Ereignisse ein allgemeines statisti- sches Modell. Es wird eine Klassifizierung statistischer Mo- delle nach dem Grad, in dem die Information die Gestalt eines

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Ereignisses anni~mt, entwickelt. Gruppentheorie wird auf fun- damentaler Ebene in die statistischen Betrachtungen einbezo- gen.

Stichw6rter: Inferenz, stabiles System, auserw~hlte Klasse, Standardmodell, Transformationsmodell, Variation, Randomisie- rung, Kontraktion, Presentation, Representation, Ereignis, Aufl~sung, vorhergehend, Alternative, Darstellung, Informa- tion, invariante Gruppe, wiederholtes Stichprobenziehen, marginale Likelihood, Strukturanalyse.

Summary

D. Brenner/D.A.S. Fraser: On Foundations for Conditional Probability with Statistical Models - When is a Class of Functions a Function

In discussions of logical structures in inference Fraser (1967, 1968, 1970, 1979) has shown how the analysis of data with a transformation-group model is uniquely and intrinsic- ally determined by the exact factorization of the error or variation into what is observable (a G-orbit) and what is un- observable (position on a G-orbit): the information supplied by a response value is equivalent to an event for the vari- ation and provides the basis for the direct use of conditional probability. Such models are canonical, in a sense to be des- cribed, with respect to this property.

In this paper we examine a general statistical model for this canonical separation of observable and unobservable and for the production of pure events. A classification of statistical models is developed in terms of the degree to which information takes the form of an event. Group theory enters the statistical considerations at a fundamental level.

Keywords: inference, stable system, selected class, standard model, transformation model, variation, randomization, con- traction, presentation, representation, event, resolution, antecedent, alternative, display, information, invariant group, repeated sampling, marginal likelihood, structural analysis.

R~sum~

D. Brenner/D.A.S. Fraser: Sur des bases pour une probabilit~ conditionelle avec des modules de statistique - Quand est une groupe de fonctions une fonction

Dans les discussions sur les structures logiques en inference, Fraser (1967, 1968, 1970, 1979) a montr~ comme l'analyse des dates avec un module de transformation de groupe est d~ter- min~ seulement par la factorisation exacte d'erreurs ou une variation dans ce qui peut ~tre observ@ et dans ce qui ne peut pas ~tre observe. L'information fournie par les donn~es d'enqu@tes est ~quivalente ~ un ~v~nement pour cette variation

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et fournie la base pour l'usage directe des probabilit~s con- ditionelles. De tels mod~les sent canoniques, dans un certain sense, avec respect ~ cette proprietY.

ans cet article nous examinons un mod~le g~n~ral de statistique pour cette s~paration canonique qui peut ~tre observ~ et qui ne peut pas ~tre obsery~ pour une production d'~v~nements pQrs. Une classification des mod~les de statistiques est d~velopp~e de telle fagon que l'information prend la forme d'un ~v~nement. La th~orie de groupe entre les considerations statistiques sur un niveau fondamental.

Inference, syst~me stable, classes, mod~le standardis~, variation, mod~le de transformation, presentation, ~v~nement, r~solution, alternative, information, groupe in- variant, ~chantillonnage r~p~t~, analyse structurelle.

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