suppes - models of data

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Institute for athematical StudiesAPPLIED MATHEMATICS AND STATISTICS LABORATORIES

STANFORD UNIVERSITYReprint No 7

Models of DataPATRICKUPPES

Reprinted f r o mLogic, Metho dology and Philosophy of Science: Proceedingsof the 1960 International Congress1962

E. Nagel, P. Suppes, & A. Tarski (Eds.), Logic, Methodology, and Philosophy

of Science: Proceedings of the 1960 International Congress. Stanford:

Stanford University Press, 1962. pp. 252-261.


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MODELS O F DATAPATRICK SUPPES

S tan fordUnivers i t y , Stanford C a l i f o r n i a , U.S.A1. Introduction

To nearly all the members of this Congress the logical notion of a modelof a theory is too familiar to need detailed review here. Roughly speaking,a model of a theory may be defined as a possible realization in which allvalid sentences of the theory aresatisfied, and a possible realization of thetheory sanentity of theappropriateset-theoreticalstructure.For n-stance, we may characterizeapossible ealization of themathematicaltheory of groups as an ordered couplewhose first member is a non-empty setand whose second member is a binary operation on this set. possible reali-zation of the theoryof groups is a model of the theory if the axioms of thetheory are satisfied in the realization, for in this case (as well as in manyothers ) the validsentences of the theor y are defined as hosesentenceswhich are logical consequences of th e axioms. To provide complete mathe-matical flexibility I shall speak of theories axiomatized within general settheoryby defining anappropriate et-theoreticalpredicate e.g., isagroup) rather thanof theories axiomatized directly within first-order logicas a formal language. Fo r the purposes of this paper, this difference is notcritical. In he set-theoretical case it isconvenientsometimes tospeakof the appropriate predicates being satisfied by a possible realization. Butwhichever sense of formalization is used, essentially the same logical notionof model app1ies.lIt is my opinion that this notion of model is the fundamenta l one for theempirical sciences as well as mathematics. T o assert this is not to deny aplace for variant uses of the word model by empirical scientists, as, forexample, when a physicist talks about a physical model, or a psychologistrefers to a quantitative theory of behavior as a mathematical model. Onthis occasion I do not want to argue for this fundamental character of thelogical notion of model, for I tried to make out a detailed case at a collo-quium in Utrecht last January, also sponsored by the In ternational Unionof History and Philosophy f Science [3] . Perhaps the most persuasive argu-ment which might be singled out for mention here is that the notion ofmodel used in any serious statistical t reatment of a theory and its relationto experiment does not differ in any essential way from the logical notionof model.The focus of th e presentpaper is closely connected to hestatisticalThe writing of this paper was partially supported by the Group Psychology Branchof the Office of naval Research.lForadetailed discussion of axiomatization of theories within set heory, seeSuppes [2, Chap. 121.

282


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SUPPE: Models of Data 253analysis of the empiricaladequacies o theories. What I want to try toshow is that exact analys is of the relation between empirical theories andrelevant data calls for ahierarchy of models of different logical type. Gener-allyspeaking, npuremathematics hecomparison of models nvolvescomparison of two models of the same logical type, as in the assertion ofrepresentation theorems. A radically different situation often obtains in thecomparison of theory and experiment. Theoretical notions are used in thetheory which have no direct observable analogue in the experimental data.I n addition, it iscommonformodels of a theory o contain continuousfunctionsor nfinite sequences although he confirming dataare highlydiscreteandfinitistic ncharacter.Perhaps I may adequately describe the kind of ideas in which I am in-terested in the following way. Corresponding to possible realizations of th etheory I introduce possible realizations of th e dat a. Models of the da ta ofan experiment are then defined in the customary manner in termsof possi-ble realizations of the da ta . As should be apparent, from a logical stand-point possiblerealizations of data are defined n ust hesamewayaspossiblerealizations of the theory being tes ted by the experimentfromwhich the da ta come. The precise definition of models of th e da ta for anygiven experiment requires that there be a theory of the data in the senseof the experimental procedure, as ell as in the ordinaryense of the empir-ical theory of the phenomena being studied.Before analyzing some of the consequences 2nd problems of this view-point, it may be useful t o give the ideas more definiteness by consideringan example.2. Example from learning theory

I have deliberately chosen an example from learning theory because it isconceptually imple,mathematicallynon-trivialand horoughlyproba-bilistic. More particularly, I consider inear response theory as developedby Estes and myself [ l ] . To simplify the presentation of the theory in aninessential way, let us assume ha t on every trial the organism in thexperi-mental situation can make exactly one of two responses, A or A,, andaf te r each response it receives a reinforcement, E, or E,, of one of the twopossibleresponses. A possible experimentaloutcome n hesense of th etheory is an nfinitesequence of orderedpairs,where the erm of thesequence represents th e observed response -the firs t member of the pairand the actual reinforcement he second member of the pair n tria ln of theexperiment.A possible realization of the theory is an ordered triple < X , P , Oof the following sort. The set X is the set of all sequences of ordered pairssuch that the first member of each pair is an element of some set A and thesecond member an element of some set B , where A and B each have twoelements. The set A represents the t w o possibleresponses and the set Bthe two possible reinforcements. The function P is a probability measure


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54 METHODOLOGY AND PHILOSOPHY OF SCIENCEon the smallest Borel field containing the field of cylinder sets of X; nd 6,a real number in the interval < 6 1 is the learning parameter. (Ad-mittedly, for theories whose models have a rather complicatedset-theo-retical structure the definition of possible realization is a t points arbitrary,but this is not an issue which affects in any way the development of ideascentral t o this paper.)Thereare woobviousrespects nwhicha possible realization of thetheory cannot be a possibe realization of experimental da ta . The first isthat no actual experiment can include an infinite number of discrete trials.The second is that the parameter 6 is not directly observable and is notpart of the recorded data.To pursue further relations between theory and experiment, it is nec-essary to sta te the axioms of the theory, .e., to define modelsof the theory.For this purpose a certain amount of notation is needed. Let Ai, e theevent of response A i on trial n, Ej, heevent of reinforcement Ei ontrial n, where i 1,2, and for x in X let x, be the equivalence class of allsequences in X which are identical with x through trial n. A possible reali-zation of the linear response theory is then a model of the theory if the fol-lowing two axioms are satisfied in the realization:

Axiom 1. If .P(Ei, Ait,,x,-l)> O themP(Ai,,+l IEi,,Ai,,,X,-l) (1 ) P W , -l) + 6 .P(&,n+l Ei,nAi,nXn-l) (1 W i , n x,-1).

Axiom 2 If P(Ei,nAi,,n~,-l)> O and i j , thenThe first axiom asserts ha t when a response is reinforced, the probability

of making that response on the next trial is increased by a simple lineartransformation. The second axiom asserts that when a different response isreinforced, the probability of making the response is decreased by a secondlinear transformation. T o those who are concerned about the psychologicalbasis of this theory t maybe remarked th at it s derivablefrom a muchmorecomplicated theory that assumes processes of stimulus sampling and con-ditioning. The linear response theory is the limiting case of the stimulussampling theory as the numberf stimuli approaches infinity.For still greater definiteness it will be expedient t o consider a particularclass of experiments to which the linear response theory has been applied,namely, those experiments with simple contingent reinforcement schedules.On every trial,if an A , response is made, the probability of an E , reinforce-ment is q , ndependent of the trial number and other preceding events. Ifan A , response is made, the probability of an E , reinforcement is z,. Thus,in summary for every mw , , , 4 % ) n 1 w , , , iAl,,),

PF IA2,n) n 1 P(E1,n lA2,n).This characterization of simple contingent reinforcement schedules hasbeen made in the language f the theory, as s necessary in order o compute


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SUPPES: Models of Data 255theoretical predictions. This is not possible forhe finer detailso the experi-ment. Letus suppose he experimenterdecides on600 trials for each subject.A brief description (cf. [4 p. 81-83j) of theexperimentalapparatusmightrun as follows.

Thesubjectsits at a table of standardheight . Mounted verticaliy n rontof thesubject s a large opaquepanel.Twosilentoperatingkeys A, and A,responses) are mounted at the base of the panel 20 cm. apart. Three milk-glasspanel lights are mounted on the panel. One of these lights, which serves as thesignalfor the subj ect to respond, is centered between he keys at the subjectseye level. Each of the other two lights, the einforcing events E, and E,, is mounteddirectlyabove one of the keys. On all rials he signal light is on for 3.5 sec.;the time between successive signal exposures is 10 sec. A reinforcing light comeson 1.5 sec. af te r the cessation of the signal light and remains on for 2 sec.I t is not surprising tha t this description o the apparatus is not incorpo-

rated in any direct wayt all into the theory. The important points to takethe linear response theory and this description as two extremes betweenwhich a hierarchy of theories and their models is to be fitted in a detailedanalysis.In th e class of experiments we are considering, the experimenter recordsonly the response made andreinforcement given on each trial. This suggeststhe definition of the possible realizations of the theory that s the first stepdown from the abst ract evel of the linear response theory itself. This theoryI shall call the theory o/ the experiment, which term must not be taken torefer to what statis ticians call the theory of experimental design-a topicto be mentioned later. A ossible realization of the theoryof th e experimentis an ordered couple < Y , P > where (i) Y is a finite set consistingof ail possible finite sequences of length 600 with, as previously, the termsof th e sequences being ordered pairs, the first member of each pair beingdrawn from some pair set A and correspondingly for t he second members,and (ii) the function P s a probability measure on the set of all subsets ofY .A possible realization Y < Y , P > of the theory of the experimentis a model of the theory if the probability measure P atisfies the definingcondition orasimplecontingent einforcementschedule. Models of th eexperiment thus defined are entities still far removed from the actual data.The finitesequences that are elements of Y may indeed be used o representany possible experimentaloutcome,but nanexperimentwith,say, 40subjects, heobserved 40 sequencesare an insignificant part of the 4600sequences n Y . Consequently,amodel closer to heactualsituation isneeded to represent the actual conditional relative frequencies of reinforce-mentused.The appropriate realization for this purpose seems to be an N-tuple Zof elements from Y , where N is the number of subjects in the experiment .An N-tuple rather than a subset of Y is selected for two reasons. The fi rs tis that if a subset is selected there is no direct way of indicating th at two


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256 METHODOLOGYAND PHILOSOPHY O F SCIENCEdistinct subjects had exactly the same equence of responses and reinforce-ments dmittedly a highly improbable event. The second and more im-por tant reason is that the N-tuple may be used t o represent the time se-quence in which subjects were run , a point of some concern in consideringcertain detailed questions of experimental design. It may be noted that inusing an N-tuple as realization of the data rathe r than more complicatedenti ty that couldbeused to express the actual times at which subjectswere run in the experiment,we have taken yet another step of abstractionand simplification away from th e bewilderingly complex complete experi-mental phenomena.2The next question is, Whens a possible realization of th e dat aa model ofthe d ata ? The complete answer, as I see it, requires a detailed statisticaltheory of goodness of fit. Roughly speaking, an N-tuple realization is amodel of the da ta if th e conditional relative frequencies of E , and E , rein-forcements fit closely enough the probability measure F of the model of theexperiment. T o examine in detail statistical tests for this goodness of fitwould be inappropriate here, but it will be instructive of the complexitiesof the issues nvolved to outlinesome of th e mainconsiderations. Thefirst thing to note is that no single simple goodness of f it test will guaranteethat a possible realizationZ of the datas an adequatemodel of the data. Thekinds of problems that arise are these: (i) (Homogenei ty ) Are the conditionalrelative frequencies (C.R.F.)of reinforcements approximately nior l - qas thecase may be, for eachsubject?To answer thiswe must compare mem-bers of the N-tup le Z. ii) (S ta t ionar i ty ) Are the C.R.F. of reinforcementsconstant over trials? To answer this practical ly we sum over subjects, i.e.,over members of Z o obtain sufficient data for a test. (iii) (Order) Are theC.R.F. of reinforcements independent of preceding reinforcements and re-sponses? To answer thiswe need to show that the C.R.F.define a zero orderprocess ha t serial correlations of all order arezero. Note, of course, th atthe zero order is with respect to the conditional events i iven A,, for i ,

1 2 . These three questions are by no means exhaustive; they do reflectcentralconsiderations.To ndicate heiressentially ormalcharacter tmay be helpful to sketch their formulation in a relativelylassical statisticalframework. Roughly speaking, the approach is asfollows. For each possiblerealization Z of theda ta we define a statistic T ( Z ) foreachquestion.This statistic is a random variable with a probability distribxtion -pref-erablyadistribution hat s(asymptotically) ndependent of theactualC.R.F. under the null hypothesis that is a model of the data. In statisticalterminology, we accept the null hypothesis if the obtained value of thestatistic T ( Z ) as a probability equal to o r greater than some significancelevel M on the assumption that indeed the null hypothesiss true.

LThe exact character of a model Y of the experiment anda model Z f the data 1snot determined uniquely by the experiment. I t would be possible,for instance, todefine iy in erms of X-tuples.


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SLJ PPES Models of Data 267For the questions of homogeneity , stationarity, and order stated above,maximum likelihood or chi-square sta tis tic s would be appropriate. There isnot adequate space to iscuss details, but these statistics are standard in theliterature. For he purposes of this paper t is not mportan t hat some

subjectivists ike L. J . Savage might be critical of the unfet tered use ofsuch classical tests. A more pertinen t caveat is that joint satisfact ion ofthree statistical tests (by satisfaction I mean acceptance of the null hy-pothesiswitha evel of significance .05) corresponding to he hreequestions does not ntuitively seem completelysufficient or a possiblerealization Z to be a model of the data.3 Yo claim for completeness wasmade in listing these three, but it mightlso be queried as to what realisticpossibility thereis of drawing up a finite listf statistical testswhich may beregarded as jointly sufficient for Z o be a model of t he da ta . A skepticalnon-formalistic experimenter might claim that given any usable set of testshe could produce a conditional reinforcement scheduie th at would satisfythe test s and yet be intuitively unsatisfactory. For example, suppose thestat istical tests for order were constructed to look at no more than fourth-ordereffects, theskepticalexperimenter could then construct a possiblerealization Zwith a non-random fifth-order pattern . Actually the procedureused in well-constructed experiments makes such a dodge rather difficult.The practice is to obtain the C.R.F. from some published table of randomnumbers whose propertieshave been thoroughly nvestigated by a widebatte ry of sta tistical tests. From the systematic methodological standpointit is not important that the experimenter himself perform the tests on Z.

On the other hand, in the experimental literature relevant to this exampleit is actually the case that greater care needs to be taken to guarantee thata possible realization Z of the da ta s indeed a model of the da tafor the ex-periment at hand. A typical instance is the practice of restricted randomi-3For use a t this point, a more explicit definition of models of th e da ta would runas fol ows. Z s an X-fold model of t he da ta for experiment ?V if an d on ly if there is aset Y and a p robability measure on su bsets f Y such th at < Y , P> is a model ofthe theory of the experiment, Z s an N-tu ple of elements of Y , and Z atisfies thestatistical tests of homogene ity, stationarity, and order. fully formal definition ould

spell out the statistical tests in exact mathematical detail. For example, a chi-squaretest of homogeneity for E , reinforcements follow-ingA esponses would be formulatedas follows. Let N, be the numberof A esponses (excluding the last trial) for subject,i.e ., as ecorded in Z , he j t member of the N-tup leZ , and let be the numberof E ,reinforcements following A esponses forsubject j . Then

and this f has N degrees of freedom. If the value ; z l ( Z ) as probability greater thanO5 the null hypothesis is accepted, i.e., with respect to homogene ity Z is satisfactory.


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258 METHODOLOGY A N D PHILOSOPHY O F SCIENCEzation. To illustrate, if l '(E,, , A l , n .B then some experimenters wouldarrange that in every block of 10 A , responses exactly 6 are followed byE , reinforcements, a result that should have a probabilityof approximatelyzero for a large number of trials.*The most important objection of the skeptical experimenter to the im-portance of models of the data has not yet been examined. The objectionis that the precise analysis of these models includes only a small portion ofthe manyproblems of experimental design. For example, y most canons ofexperimental design the assignment of A , to the left (or to the right) forevery subjectwould bea mistake. More generally, the use of an experimentalroom in which there was considerably more light on the left sidef subjectsthan on the rightwould be considered mistaken. There is a difference, how-ever, in these two examples. The assignment of A , to the left or right foreach subject is information th at can easily be incorporated into models ofthe da ta nd requirements of randomization can be tat ed. Detailed infor-mationabout hedistribution of physicalparameterscharacterizing heexperimental environment is not a simple matter to incorporate in modelsof data and s usually not reported in the literature; roughly speaking,omegeneral ceteris par ibus conditions are assumed to hold.The characterization of models of da ta is not really determined,however,by relevant nformationaboutexperimental designwhichcaneasilybeformalized. I n one sense there is scarcely any limit to information of thiskind; itcan range from phasesof the moon to I.Q. data on subjects.

The central idea, corresponding well, I think, to a rough but generallycleardistinctionmade by experimentersandstatisticians, is to estrictmodels of the da ta to those aspects of the experiment which have a para-metric analogue in he theory.A model of the da ta s designed to incorporateall the information about the experiment which can be used in statisticaltests of the adequacy of the theory. The point I want to make is not assimple or as easily made precise as I could wish. Table 1 is meant to in-dicate a possible hierarchy of theories, models, and problems th at arise ateach level to harass the scient ist. At the lowest level I have placed ceterispar ibus conditions. Here is placed every intuitive consideration of experi-mental design that involves no formal statistics. ontrol of loud noises, badodors, wrong times of da y or season go here. At thenext levelformalprob-lems of experimental design enter,but of thesort hatfar exceed thelimits of the particular theory being tested. Randomization of A s the eftor right response is a problem for this level, as is random assignment ofsubjects to different experimental groups. All the considerations that enter

4To emphasize t ha t conceptually here is nothing special abo ut hisparticularexample chosen from earning heory, i t is per tin ent oremark hatmuchmoreelaborateanalyses of sources of experimentalerrorarecustomary incomplicatedphysical experiments. I n the literature of learning the ory it is as yet uncommon t oreport the kind of statistical tests described above which play a role analogous to thephysicists' summary of experimentalerrors.


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SUPPES: Models of DataTABLE 1

HIERARCHYF THEORIES, ODELS,A X D PROBLEMSTheory of Typicalroblems

259

Linear responsemodels Estimation of 8, goodness of f i t omodels of dataModels of experiment Number of trials, choice of experimentalparameters

Models of data Homogeneity,tationarity,it ofexperimental parametersExperimental design Left-right andomization,assignmentof subjects

Cetevis pavibus conditions Xoises, lighting, odors, phases of themoon

at this level can be formalized,nd their relationo models of the da ta ,whichare at the next level, can be made explicitn contrast to theeemingly end-less number of unstated ceteris par ibus conditions.At the next level,models of the experiment enter. They bear the relationt o models of the data already outlined. Finally at the topf the hierarchy arethe linear response models, relatively far removed from th e concrete ex-perimental experience. It is to be noted that linear esponse modelsare relat-ed directly to models of the data, without explicit consideration of modelsof the experiment. Also worth emphasizing once again is that the cr ite riafor deciding if a possible realization of the data s a model of the data in owaydependupon tsrelation to a inear responsemodel.These criteriaare to determine if the experiment was well run , not to decide if the linearresponse theory has merit.The dependence is actually the other way round. Given a model of th edata we ask if there is a linear response model to which it bears a satis-factory goodness of fi t relation. The rationaleof a maximum likelihood esti-mate of 0 is easily stated in this context:iven the experimental parametersq and z we seek that linear response model, i.e ., the linearesponse modelwith learning parameter d which will maximize the probability of th e ob-served data, asgiven in the model of the data .It isnecessary at this point o break off rather sharply discussion ofthis example from learning theory, but there is one central point that hasnot been sufficiently mentioned. The analysis f th e relation between theoryandexperimentmust proceed a t every level of thehierarchyshown nTable 1 . Difficulties encountered at all bu t the topevel reflect weaknesses inthe experiment, not in the fundamental learning theory. It is unfortunatethat it s not possible to give here citations from the experimental literatureof badly conceived or poorly executed experiments that are taken to invali-date the theory they resume to test, but in fact do not.


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260 X E T H O D O L O G YA S D PHILOSOPHY O F SCIENCE3. The theory of models in the empirical sciences

I began by saying thatI wanted to try tohow that exact analysisof therelation between empirical theories and relevant data calls for a hierarchyof models of different logical type. The examination of th e example fromlearning heory was meant o exhibitsomeaspects of thishierarchy. Iwould like to conclude with some more general remarks that are partiallysuggested by this example.One point of concern on my part has been to show tha t in moving fromth e level of theory to the level of experiment we do not need to abandonformal methods of analysis. From a conceptual standpoint the distinctionbetween pure and applied mathematics is spurious oth deal with set-theoretical entities, and the same is true of theory and experiment.It is a fundamental contribution of modern mathematical statistics tohave recognized the explicit need of a model in analyzing the significanceofexperimental data.It is a paradox of scientific method that the branchesofempirical science that have the least substantial theoretical developmentsoften have the most sophisticated methods of evaluating evidence. In suchhighlyempiricalbranches of science a argehierarchy of models isnotnecessary, for the theory being tested is not a theory with a genuine logicalstructure but a collection of heuristic ideas. The only models needed aresomething like th e models of the experiment and models of the data dis-cussed in connection with the example from learning theory.Present statistical methodology is ess adequate when a genuine theory isat stake. The hierarchy f models outlined in our example corresponds in avery rough way to statist icians concepts of a sample space, a population,and a sample. It is my own opinion that the explicit and exact use of thelogical conceptof model will tu rn out toe a highly useful device in clarifyingthe theory of experimental design, which many statist icians still thinkf asan art rather thanscience.Limitations of space have prevented workingout the formal relations between the theory of experimental design and thetheory of models of the data , as I conceive it.However, my ambitions for the theory of models in the empiricalsciencesare not entirely such practicalnes. One of the besetting sinsof philosophersof science is to overly simplify the structure of science. Philosophers whowrite about the representationof scientific theories as logical calculi then goon to say that a theory is given empirical meaning by providing interpre-tations or coordinatingdefinitions orsome of th e primitiveordefinedterms of t he calculus. What I have attempted to argue is that a wholehierarchy of models stands between the model of the basic theory and thecomplete experimental experience. Moreover, for each level of the hierarchythere is a theory in its own right. Theory at one level is given empiricalmeaning by making formal connections with theory a t a lower level. Sta-tistical or logical investigation of the relations between heories at thesedifferent levels can proceed in a purely formal, set-theoretical manner. The


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SUPPES: Rlodels of Data Z6 1more explicit he analysis the less place there is for non-formal considerations.Once the empirical data are put incanonical form (at the level of models ofdata in Table l , every question of systematic evaluation th at arises is aformal one. It is important to notice that the questions to be answered areformal but not mathematical ot mathematical in the sense that thei ranswers do not in general follow from the axioms of set theory (or someother standard ramework fcr mathematics).It is precisely the fundamentalproblem of scientific methodt o sta te the rinciples of scientific methodologythat are tobe used to answer these questions uestions of measurement,of goodness of fit, of parameter estimation, of identifiability, and the like.The principles needed are entirely formal in character in the sensehat theyhave as their subject matter set-theoretical models and their comparison.Indeed, the line of argument I have tried to follow in this paper leads tothe conclusion that the only systematic results possible in the theory ofscientificmethodology arepurely ormal, bu t ageneral defense of thisconclusion cannot be madehere.

REFEREXCES[ l ] ESTES, . K., and P. SUPPES. oundations of linear models. Chap. 8 in S t u d i e s in

M a t h e m a t i c a lL e a r n i n g T h eo r y , R R. Bush and W K. Estes,eds.,Stanford,Calif., StanfordUniv.Press, 1959,432 pp.[2] SUPPES,P. I d r o d u c i i o n o o g i c , Princeton, Van Nostrand, 1957,312 pp.[3] SUPPES, . A comparison of th e meaning and uses of models in mathematics and[4]SUPPES, . , andR . A4^^^^^^^. M a r k o v L e a r n i n g M o d e l s for a Zztltiperson Imte r-the empirical sciences, S y n t h e s e , Vol. 12 (1960), 287-301.

act ions, Stanford, Calif., StanfordUniv. Press, 1960, xiit-296pp.

suppes - models of data

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