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Home , Confidence distribution, Fiducial inference, Pivotal quantity

Feudalism

Rader T 1971 The Economics of Feudalism. Gordon & Breach, 1. A Simple Example New York Roth P 1863 FeudalitaW t und Unterthan!erband. Bo$ hlau, Weimar, Consider a variable y that is directly available or has Germany arisen by some preliminary reduction process and Sander P 1906 Feudalstaat und BuW rgerliche Verfassung. Ein suppose that y measures θ in an unbiased manner and Versuch uW ber das Grundproblem der deutschen Verfassungs- has error that is normal with known σ#. Then geschichte. Bath, Berlin, Germany ! we can say for example that P(y ! θj1.64σ!; θ) l 95 Schluchter W 1979 Die Entwicklung des okzidentalen Ration- percent. alismus. Eine Analyse !on Max Webers Gesellschaftsge- schichte. Mohr (Paul Siebeck), Tu$ bingen, Germany Parenthetically, we note that the preliminary re- Schreiner K 1983 ‘Grundherrschaft’. Entstehung und Bede- duction could have occurred as part of the underlying utungswandel eines geschichtswissenschaftlichen Ordnungs- investigation or as part of some subsequent simplifi- und Erkla$ rungsbegriffs. In: Patze H (ed.) Die Grundherrschaft cation of the by one of the common im Mittelalter. Thorbecke, Sigmaringen, Germany pp. 11–74 statistical reduction methods, sufficiency or con- Sidney P 1961 Feudalism and Liberty: Articles and Addresses. ditionality. If the reduction is to a sufficient , Johans Hopkins Press, Baltimore then the conditional distribution describing possible Stephenson C 1948 Medie!al Feudalism. Cornell University antecedent has no dependence on the Press, Ithaca, NY and the model for the sufficient statistic is used in place Strayer J R 1965 Feudalism. Van Nostrand, Princeton Sweezy P u. a. 1976 The Transition from Feudalism to Capitalism. of the original model. If the reduction is by con- Verso, London ditionality then there is typically an ancillary variable Sydney H 1921\1969 The Fall of Feudalism in France. Barnes & with a distribution free of the parameter and the given Noble, New York model for the possible original data is replaced by the van Horst Bartel 1969 SachwoW rterbuch der Geschichte Deut- conditional model given the ancillary (supportive) schlands und der deutschen Arbeiterbewegung Dietz, Berlin, variable; see also Statistical Sufficiency. Germany Vol. 1, pp. 582–6 With a data value y! the fiducial methodology Von Gierke O 1868\1954 Das Deutsche Genossenschaftsrecht. would take the above probability expression and Graz Akademische Druck-und Verlangs-Anstadt, Graz, substitute the value y! and then treat θ as the variable Austria Vol. 1 for the probability statement, thus giving a 95 percent Ward J O 1985 Feudalism: Comparati!e Studies. Sydney Asso- ! ! ciation for Social Studies in Society & Culture, Sydney fiducial probability statement P(θ " y k1.64σ!; y ) Weber M 1921\1972 Wirtschaft und Gesellschaft. Grundriß der l 95 percent. The structural approach would consider # !erstehenden Soziologie, 5th edn. Mohr (Paul Siebeck), Tu$ b- the normal (0; σ!) distribution for the error e l ykθ ingen and might record for example the probability ! Wehler H-U 1987 Deutsche Gesellschaftsgeschichte, Vol.1: Vom P(e ! 1.64σ!) l 95 percent; then with data value y Feudalismus des Alten Reichs bis zur Defensi!en Modern- the probability statement would be applied to the isierung der ReformaW ra 1700–1815. Beck, Mu$ nchen, Germany error e l y!kθ giving the structural probability = Wunder H 1974 Der Feudalismus-Begriff. Uberlegungen zu P(y!kθ ! 1.64σ ; y!) l 95 percent or equivalently Mo$ glichkeiten der historischen Begriffsbildung. In: Wunder H ! ! ! (ed.) Feudalismus. Nymphenburger Verlag, Mu$ nchen, Ger- P(θ " y k1.64σ!; y ) l 95 percent. In a somewhat many related manner the Bayesian methodology might use a Wunder H 1989 Artikel ‘Feudalismus’. In: Sautier R H (ed.) uniform prior cdθ and obtain a posterior distribution ! ! Lexikon des Mittelalters. Artemis, Mu$ nchen, Germany, Vol. for θ that would have P(θ " y k1.64σ!; y ) l 95 4, pp. 411–15 percent. For this simple example the three methods give the same result at the 95 percent level and also at K. Schreiner other levels, thus saying essentially that with data value y! the describing the ! # unknown θ is normal (y ; σ!). With more complicated models the results from the Fiducial and Structural Statistical three methodologies can differ and philosophical Inference arguments concerning substance and relative merits arise. However, for one straightforward generaliza- is a statistical approach to interval tion the methods remain in agreement: the normal estimation first advocated by R. A. Fisher as an distribution can be replaced by some alternative alternative to the then dominant method of inverse distribution form; this is discussed in some detail in probability, i.e., using Bayes’ Theorem. Considerable Sect. 2. effort has gone into formalizing Fisher’s notions using such concepts as statistical invariance and pivotal quantities. This entry describes elements of the fiducial approach and relates them to other currently more 2. Fiducial Probability widelyusedstatisticalapproachestoinference.Section1 introduces some basic inferential ideas via a simple Fisher (1922, 1925; see also Fisher, Ronald A example. (1890–1962)) had already introduced most of the

5616 Fiducial and Structural fundamental concepts of , such as with a sample y ,…, y from the normal (µ; σ#) " n #! sufficiency, likelihood, efficiency, exhaustiveness distribution, the reduction would be to t(y) l (y` , sy). (minimal sufficiency), when he chose (Fisher 1930) to The methodology then suggests the use of a pivotal address directly the aspiration mentioned above. He quantity p l p(t; θ) with a fixed distribution and a took Laplace and Gauss to task for ‘fall(ing) into error one-one relationship between any two of p, t, θ; recall on a question of prime theoretical importance’ by more generally that a is a function of adopting the Bayesian approach that ‘Bayes (had) the variable and parameter that provides a measure of tentatively wished to postulate in a special case’ and departure of variable value from parameter value, and which was published posthumously (Bayes, ibid). Be has a fixed distribution which allows an assessment of then proposed in a restricted context the fiducial an observed departure. Thus for the example a natural method, as discussed. pivotal is Neyman and Pearson (1933) then gave a math- ematical formulation of fiducial probability that be- 1 y` µ (n 1) s# 5 came known as confidence intervals. Fisher (1956) 2 k # k y 6 3 z l , χ l 7 however treated Neyman and Pearson’s formulation σ\Nn n−" σ# as a ‘misconception having some troubling conse- 4 8 quences …’; logical and philosophical arguments be- tween the two sides were intense for many years. In which has independent components, normal (0, 1) and particular the slight to Laplace and Gauss may well chi-square with nk1 degrees of freedom. Fisher (1956, have affected the views of the more mathematical p. 172), however, rather deviously rejected this as a participants. legitimate part of the fiducial methodology, but he was Fisher (1930) entitled his paper ‘’ somewhat less explicit about what would be legitimate. and examined a statistic t(y) whose distribution This pivotal reduction procedure is now however a depended on a single parameter θ. Let P l F (t; θ) be rather familiar component of standard inference the distribution function of t, and let P itself be what theory and in particular of confidence theory. we might now call a p-value for assessing θ; of course The final step is to invert the pivotal quantity, that in the usual continuous case P has the uniform is, to insert the observed values for the variables and distribution on (0, 1). ‘If now we give P any particular then transform the distribution of the pivotal quantity value such as 0.95, we have … the perfectly objective to the parameter. For the example this gives fact that in 5 percent of samples’ t ‘will exceed the 95 percent value corresponding to the actual value of µ y z (n 1)−"/#χ s Nn θ ….’ Then to ‘any value of’ t ‘there will moreover be l ` ko \ k n−"q y\ usually a particular value of θ to which it bears this σ# l (nk1) s#\χ# relationship; we may call this the ‘fiducial 5 percent y n−" value of θ’ corresponding to’ the given t. This led to # Neyman and Pearson’s (ibid) confidence methodology where the y` , sy have their observed values. We can then but Fisher treated this as a misconception and he write followed different directions and interpretations for the fiducial methodology; for a view on related µ l y` kts \Nn approaches see Estimation: Point and Inter!al. For this y present simple case with scalar t and scalar θ, there seems little difference between the fiducial and the where t is Student with nk1 degrees of freedom. This confidence approaches and interpretations. fiducial calculation closely parallels that for the confi- This discussion effectively ascribes a distribution to dence approach, except that the limits here are θ based on an observed t; this is called the fiducial calculated from the Student distribution for µ rather distribution for θ. Just as the density of t for given θ is than from the Student distribution of the pivotal t. In obtained as (c\ct)F (t; θ) so also the fiducial density is quite wide generality fiducial regions can correspond obtained as (kc\cθ)F (t;θ); the negative sign is in- to confidence regions; it is just a matter of whether the consequential and is merely the result of F (t; θ) being limits are calculated before or after the data are examined typically for the case that is increasing with observed, a non-issue from the Bayesian viewpoint. θ. The fiducial distribution for cases like the present There are however cases where routinely obtained can also be called the confidence distribution, which confidence and fiducial regions can differ. For ex- recently has a variety of uses in statistical inference. ample, consider a sample (y"", y"#),…, (yn", yn#) from This alternative name emphasizes its more conven- a bivariate with µ , µ , # # " # tional role and attempts to avoid the inappropriate σ", σ#, and correlation ρ, and suppose stigma that surrounds the fiducial concept. primary interest centers on the regression coefficient β For the fiducial approach in more general contexts, l ρσ#\σ" of the second variable on the first variable. Fisher (1935) recommended a maximum reduction to Standard of the second variable on a statistic t(y) by the use of sufficiency. For example, the first variable can produce an interval that differs

5617 Fiducial and Structural Statistical Inference from that obtained from the joint fiducial or structural we get it without any reference to sufficiency. This with distribution for all the by integrating out some other results suggests (Fraser and Reid 2000) the unwanted parameters; see for example, Fraser that the primary concept sufficiency can be replaced (1979, pp. 189, 204, 280, and 293). Also there are other quite widely by appropriate conditioning, and with a familiar cases where fiducial, structural and Bayesian major gain in generality of viewpoint. methods can differ; see for example, the Behrens- Fisher (1935) did not use maximum likelihood Fisher and the Fieller-Creasy problems, and for some notation for this location scale model. The notation discussion see Wallace (1980). here adapts to more general contexts (Fraser 1968, Now suppose we generalize and examine a sample 1979) but needs to be modified if the error distribution y",…, yn from some distribution with location µ and involves a which would then make scale σ. For example the basic form f (z) of the the maximum likelihood estimates parameter-depen- distribution might be, say, the longer-tailed logistic, or dent. the Student with 6 degrees of freedom often cited as a For more general problems the fiducial can have realistic error pattern. The distribution for the under- non-uniqueness difficulties, and these tend to arise lying errors is then f (z")(f (zn) and for the sample is from the choice of what pivotal to use. For example with the bivariate normal discussed, different pivotals −n −" σ  f oσ (yikµ)q can produce different fiducial distributions (Fisher 1956, p. 172, where he elusively denigrates the pivotal With a typical general error shape we do not have a approach). simple minimal sufficient statistic as with the normal There are also issues connected with finding a error pattern above. Fisher (1934) then recommended marginal fiducial for a component parameter. For a conditioning on an and suggested general discussion of marginalization paradoxes con- the statistic nected with methodologies concerning distributions for parameters, including the Bayesian methodology, see David et al. (1973). y kµV y kµV d l " ,…, d l n For a mathematical analysis that avoids some of the " σV n σV difficulties with the fiducial methodology, see the transformation group framework in Fraser (1961a, called the configuration statistic, whose coordinates 1961b); this mathematical analysis led to the de- are standardized residuals, standardized here with velopment of the structural approach described below. respect to maximum likelihood values. The con- The marginalization issues still remain, however, and ditional distribution of (µ# , σ# ) is fairly easily expressed this is related to the common assessment technique of in terms of likelihood. Let examining a procedure with many repetitions from the same distribution with fixed parameter value. Altern- ! −n −" ! atives to this are suggested in Fraser and Reid (2000). L (µ, σ) l cσ  foσ (yi kµ)q be the observed from data ! ! ! ! 3. Structural Probability (y",…, yn), and (µ# , σ# ) be the observed maximum likelihood values. Then the conditional distribution In many applied problems it is possible to ascribe an ! ! underlying error distribution to the model. For ex- given the observed configuration statistic (d",…, d n) is ample with measurements that are normal with un- known error scaling we might write dµV dσV g(µV , σV Qd!; µ, σ) l σV # yi l µjσzi i l 1,…, n

dµV dσV where the z",…, zn, are independent standard normal cL!(µV !j(µkµV )σV !\σV , σσV !\σV ) variables. If we then examine (Fraser 1966) these σV # expressions with observed data we can note rather easily that many characteristics of the underlying which can be used to give confidence intervals; also a realized errors can be numerically evaluated. Specifi- straightforward fiducial argument following that for cally, let the normal case gives y"ky` ynky` d" l ,…, dn l dµdσ sy sy L!(µ, σ) σ be standardized residuals; then the observed y",…, yn and the underlying realized (z",…, zn) have the same for the fiduicial distribution. If f is replaced by the standardized residuals. In fact, nk2 characteristics as standard normal we get the result discussed earlier and presented by the residuals are available concerning the

5618 Fiducial and Structural Statistical Inference underlying errors. Most theory concerning appli- D(Z) l D! then describes the reduced model [Y ] l cations of probability would then say the analysis θ[Z] with data [Y!]. The structural distribution of θ is should be conditional on these observed charac- then given as θ l [Y!][Z]−" using the conditional teristics, that is, on what you know concerning the distribution [Z] just described. realized error (Fraser 1976, Chap. 4, pp. 161–2 and In most contexts the support measure dZ can be Chap. 11, pp. 456–66). This for quite general problems replaced by an invariant measure dM(Z) automatically gives the conditioning used by Fisher and avoids the need to invoke a conditioning principle dZ and seek an ancillary such as the configuration dM(Z) l statistic. J(Z) This present type of analysis carries through equally (Fraser 1966) if the normal distribution is replaced by where J(Z) is an appropriate Jacobian (Fraser 1979). some other error pattern. The conditional error Expressions are simpler if we use density f`(Z) with distribution can be viewed from an invariance argu- respect to the measure dM(Z):f`(Z) l f (Z)J(Z). ment as being valid after the full data are observed. The distributions of [Y ] and [Z] conditional on This distribution then gives the structural distribution D(Y ) l D(Z) l D! are for µ and σ. The mechanics follow closely what is presented for the fiducial case and the resulting ` −" ! ` ! structural distribution agrees with the fiducial distri- cf (θ [Y ]D )dµ[Y ], cf ([Z]D )dµ[Z] bution, L!(µ, σ)dµdσ\σ. The difference here is that a rather strict invariance structure is assumed as part of where dµ[Z] is the corresponding invariant measure the argument. This is summarized in general notation. on the group Ω. The inverted distribution, called the The structural approach is possible in cases where structural distribution for θ, is the underlying error can be directly modeled and the observed response is obtained as some transformation cf`(θ−"[Y !]D!)dν(θ) l cL!(θ)dν(θ) of this. Let Z designate a general error variable with n known density f (Z) on say R . Let Y designate the using the right invariant measure ν on the group Ω. In response as obtained by a transformation or re- the Bayesian context the typically preferred nonin- expression θ in some space Ω. Then we have formative prior is the right invariant measure dν(θ). Thus the structural distribution coincides with this Y l θZ, f (Z)dZ, preferred Bayesian posterior. This group type model covers a wide of regression multivariate and spherical distribution For the invariance we assume that a transformation θ n n models but not in the simplified notation used here; carries R into R and that the collection Ω of such for further details see Fraser (1968, 1979). transformations is closed under composition and Recent likelihood asymptotics has been able to inversion. Also for details here we assume that the obtain many aspects of these results in a general transformations are smooth and at most one trans- asymptotic context; see Fraser et al. (1999) and Fraser formation carries a point Z into a point Y. For some et al. (1999). general background on transformations and invari- ance, see Causal Inference and Statistical Fallacies. If we consider all the transformation Ω and apply ! ! them to a data point Y we obtain a set ΩY of images, 4. Comment of possible antecedent error values. If Z! is the unknown realization of error that produced Y! then Fiducial inference led to confidence intervals, with we have ΩZ! l ΩY! or equivalently we have that Z! both having evolved from the Bayesian procedures lies in the set ΩY!. This means that a probability initiated in the eighteenth century. In terms of freedom assessment should be made conditional on the ob- of usage, we have as the more served set ΩY!. Let D(Z) be a reference point in the set liberal. The foundational issues center on validity and ΩZ and let [Z] in Ω be the transformation in Ω that interpretation of the probabilities for the parameters. carries D(Z) to Z. Then we have that the model Y l Here the conditions are strictest for the structural and θZ with data Y! can be rewritten as minimal for the Bayesian. In the nice cases they agree; and in the less standard cases they pressure the proponents to clarify their assumptions. [Y ] θ[Z], D(Y) D!, l l For some recent discussion of confidence and fiducial interconnections plus references see Barnard giving the conditioning D(Y ) l D(Y!) l D! that (1995), with complementing views in Fraser (1996). appears as an observed ancillary in the fiducial Also see Fraser and Reid (2000) for a discussion of the framework. The conditional distribution of [Z] given interplay with objective Bayesian methods.

5619 Fiducial and Structural Statistical Inference

Bibliography scene of the human behavior she or he describes. It involves ‘being there,’ or ‘I-witnessing.’ So also, of Barnard G A 1995 Pivotal models and the fiducial argument. International Statistical Re!iew 63: 309–24 course, do other forms of documentation: journalism, Bayes T 1763 An essay towards solving a problem in the doctrine film, travel writing, explorer or missionary’s report, of chance. Philosophical Transactions of the Royal Society 53: even background research by a novelist. What makes 370–418 [Reprinted (1958) in 45: 293–315] ethnography distinctive as a research practice is that it Dawid A P, Stone M, Zidek J V 1973 Journal of the Royal forms one point of a triangular process of knowledge Statistical Society 35: 189–233 construction which also includes comparative social Fisher R A 1922 On the mathematical foundations of theoretical theory and documentary contextualization. . Philosophical Transactions of the Royal Society of The fieldnotes that result from ethnographic dis- London A222: 309–68 covery procedures are not merely ‘written up,’ but Fisher R A 1925 Theory of statistical estimation. Proceedings of the Cambridge Philosophical Society 22: 700–25 are filtered and interpreted against theoretical prop- Fisher R A 1930 Inverse probability. Proceedings of the Cam- ositions and the comparative record of other field bridge Philosophical Society 26: 528–35 observational studies; they are also grounded and Fisher R A 1934 Two new properties of maximum likelihood. enhanced by historical, ecological, demographic, Proceedings of the Royal Society of London A144: 285–307 economic, and other documentary sources that pro- Fisher R A 1935 The fiducial argument in statistical inference. vide background and context. As the products of this Annals Eugenics 6(4): 391–8 of research are read—monographs and articles Fisher R A 1956 Statistical Methods and Scientific Inference. also referred to as ethnography—they stimulate new Oliver and Boyd, Edinburgh, UK comparative theoretical thinking, which in turn sug- Fraser D A S 1961a The fiducial method and invariance. Biometrika 48: 261–80 gests further problems and interpretations to be Fraser D A S 1961b On fiducial inference. Ann. Math. Statist. resolved through more field observational research. 32: 661–76 Ethnographies also regularly lead to new demands Fraser D A S 1966 Structural probabilities and a generalization. and rising standards for documentary contextualiz- Biometrica 53: 1–9 ation—more history, more ecological or demographic Fraser D A S 1968 The Structure of Inference. Wiley, New backgrounding, more attention to state policy, econ- York omic trends or the world system. This triangle of Fraser D A S 1976 Probability and Statistics, Theory and ethnography, comparison, and contextualization is, in Applications. ITS, Toronto essence, the process by which field observational Fraser D A S 1979 Inference and Linear Models. McGraw Hill, New York research in anthropology and sociology is utilized to Fraser D A S 1996 Some remarks on pivotal models and the explain and interpret human cultures and social life. financial argument in relation to structural models. Inter- national Statistical Re!iew 64: 23–5 Fraser D A S, Reid N 2000 Strong of frequentist and 1. History and Scope in Anthropology and Bayesian inference. Journal of Statistical Planning and in- ference, to appear Sociology Fraser D A S, Reid N, Wu J 1999 A simple general formula for In 1851 Morgan published his League of the Ho-de- tail probabilities for frequentist and Bayesian inference. no-sau-nee, or Iroquois; combining fieldwork and Biometrica 86: 249–64 Fraser D A S, Wong A, Wu J 1999 Regression analysis, comparative and theoretical interests in political nonlinear or nonnormal simple and accurate p values for organization, this book was the first anthropological likelihood analysis. Journal of the American Statistical As- ethnography. It depicted the structure and operation sociation 94: 1286–95 of Iroquois society, detailing matrilineal kinship, pol- Neyman J, Pearson E S 1933 The testing of statistical hypotheses itical and ceremonial life, material culture and religion. in relation to probabilities a priori. Proceedings of the After formation of the US government Bureau of Cambridge Philosophical Society 29: 492–10 American Ethnology in 1879, a stream of ethnographic Wallace D L 1980 The Behrers–Fisher and Creasy–Fieller accounts of native American societies began, including problems. In: Fienberg S E, Henlele D V (eds.) R. A. Fisher: landmark field observational studies of the Zuni by An Appreciation. Springer Verlag, New York, pp. 139–45 Cushing, the Baffin Island Inuit by Boas, and, after D. A. S. Fraser 1900, ethnographies of Plains, Eastern Woodland, and California groups by Boas’ students trained at Copyright # 2001 Elsevier Science Ltd. Columbia University. During the 1930s the scope All rights reserved. broadened to include rural and urban studies in Latin America, the Caribbean, Africa, and Europe, and also Field Observational Research in in contemporary US settings. Anthropology and Sociology In Great Britain’s colonies, local government official or missionary fieldworkers followed the topical guide- Field observational research in anthropology and book Notes and Queries on Anthropology, drafted by sociology—or the process of doing ethnography— Oxford University theorist Tylor and others, and first brings the social science investigator directly to the published in 1874. This armchair scholar\man-on-the-

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International Encyclopedia of the Social & Behavioral Sciences ISBN: 0-08-043076-7