Barndor-Nielsen, O. (1978). Information and Exponential Families in Statistical Theory. Wiley, New York.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....natural exponential families. Keywords natural exponential families, variance function, umbral calculus, Sheffer polynomials. 1 Introduction Exponential families of probability measures play an important role in statistics because of their nice properties concerning estimation (see e.g. [2]or [12] In [11] Morris studied so called natural exponential families on the real line. He showed that there are six classes of natural exponential families with quadratic variance function (i.e. the variance is at most a quadratic function of the mean) We approach natural exponential families ....

O. Barndorff-Nielsen, Information and exponential families in statistical theory, Wiley, New York, 1978.

....realistic simulation model is used. A feasible order independent solution applicable whenever the data set re ects internally homogeneous information pieces is presented in Section 7. 4 Solution for exponential family Let us specialise the general result of Section 3 to the exponential family [12] and naturally conjugate prior pdf [4] This combination guarantees that the functional form of the prior pdf is preserved in the posterior pdf, too. This gave to naturally conjugate priors an alternative name self reproducing priors. Moreover, data determine this functional form through a ....

O. Barndor-Nielsen, Information and exponential families in statistical theory, Wiley, New York, 1978.

....on the explanatory variables in a corresponding family of joint models, the CG distribution models, for which a general estimation algorithm has been described by Frydenberg Edwards (1989) and implemented in the MIM software. In the special case where the explanatory variables form a cut (Barndor Nielsen, 1978, p. 50) in the CG distribution models, the CG regression models can be tted by piecing together suitable estimates obtained in the CG distribution models as described in Proposition 6.33 of Lauritzen (1996) In particular this holds when all variables are discrete or all variables are ....

....therefore takes the form 0 = arg max l( n 1 = arg max q( j n ) 4 so the M step typically solves the equation E n f l( n ) j xg = l( 3) in contrast to the EM algorithm which for the M step solves E n f l( j xg = 0. In the special case where x induces a cut (Barndor Nielsen, 1978, p. 50) no iteration is needed, because then we have = with and variation independent and l( l x ( l x ( implying that 0 = 0 ; 0 ) satis es 0 = arg max l x ; 0 = arg max l x ; and thus l x ( 0 ) 0, leading to q = l. An important property of the ....

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Barndor-Nielsen, O. E. (1978). Information and Exponential Families in Statistical Theory. New York: Wiley.

....whose density function is given by p(x; 13) 1 (2 ) n 2 (det ) 1 2 expf 1 2 (x ) T 1 (x )g = expf 1 2 x T 1 x x T 1 1 2 log(det ) 1 2 T 1 n 2 log(2 )g; where x; 2 R n and is a symmetric positive de nite matrix. Following Bandor Nielsen [4], Bandor Nielsen and Bl sild [5] and Amari [2] we de ne a potential function de ned on the manifold N by the cumulant transformation of (13) The function is represented in terms of a chart = as = 1 2 T r( 1 T ) 1 2 log(det ) n 2 log(2 ) 14) Contrary to the general ....

.... S n (25) N( 7 ( 1 ; 1 2 1 ) and I H : N , R n S n (26) N( 7 ( T ) H) H: The map I is a bijection and the map I H is an into injection. Note that the sign of our map I is di erent from that of Bandor Nielsen [4] and Amari [2] We have also the following relations. 1 2 1 = 1 2 1 = H T ; 27) 1 2 (H T ) 1 ; H T ) 1 : 28) If we employ the canonical map, then we have Proposition 4.1. Let ( I 1 ( The potential function can be ....

O.E. Barndor-Nielsen, Information and exponential families in statistical theory, Wiley,Chichester (1978).

....the ML estimation which is possible when the likelihood function factorizes into functions that are distinct with respect to the unknown parameter vector, say . Thus, the maximization reduces to the separate maximization of each factor. This procedure is justified by a general concept proposed by Barndorff Nielsen (1978). The idea is to show that P can be written as a product space P T Theta P T for a statistic T = T (X) where P T denotes the set of distributions of T and P T the set of conditional distributions of X given T = T (x) This implies on the one hand that any density p 2 P factorizes into the ....

Barndorff--Nielsen, O. E. (1978) Information and Exponential Families in Statistical Theory. John Wiley and Sons, New York.

....is as follows: E(gjV) X a2A e a (gjV) V ffl H(g) j V ffl H cn (g) Here, the subscript cn indicates the centering. Therefore, the model (1) can be written in the following equivalent form of the exponential family distribution (these families were studied in detail by BARNDORFF NIELSEN [1978]) Pr(gjV) 1 ZV Delta exp (V ffl H cn (g) 2) where H(g) fH a (g) a 2 Ag denotes the vector of the (centered) GLDHs for the image g. This representation shows that the (centered) GLDHs, collected over the clique families, C form the sufficient statistic for the model. RR n3202 12 ....

....pairwise interactions and recover most characteristic ones to represent a given texture type. Then, the desired MLE of the potentials for the chosen families C a is refined by a stochastic approximation technique similar to the one introduced by YOUNES [1988] As shown by BARNDORFF NIELSEN ([1978]) the GPD in (4) is the regular exponential family distribution with minimal canonical parameter V and minimal sufficient statistic H cn (g) if and only if the following conditions are both satisfied: i) the vectors V are affinely independent and (ii) the vectors H cn (g) are affinely ....

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Barndorff-Nielsen, O.: Information and Exponential Families in Statistical Theory. Wiley (1978)

....checked what consequences have to be drawn from the equivalence of conditional and marginal independence. Another open question concerns the discussion of estimating and testing properties. The familiy of KS distributions is no exponential family which means that common results cannot be applied (Barndor Nielsen, 1978, Frydenberg and Lauritzen, 1987) Exponential families are characterized by a special structure of the density whereas the family of KS distributions is featured by the cdf. The corresponding density is given by a complicated formula which is not easy to treat analytically. Eventually, the ....

Barndor{Nielsen, O.E. (1978). Information and Exponential Families in Statistical Theory. John Wiley and Sons, New York.

....Components of # and will be called scalar canonical and scalar expectation parameters, respectively. Also of interest are mixed parameterizations of the form ( 1) # (2) and(# (1) 2) where (1) 1 , p ) 2) p 1 , k ) # (1) # 1 , # p )and# (2) # p 1 , # k ) Barndor# Nielsen (1978, p. 183) has shown that (1) and # (2) are orthogonal as are # (1) and (2) It is shown in the following theorem that (1) and # (2) are also X (1) orthogonal, where X (1) X 1 , X p ) Note that X (1) is the MLE of (1) Theorem 2. If (1) 1 , p ) is a part of the expectation ....

O. E. Barndor#-Nielsen (1978). Information and Exponential Families in Statistical Theory. Wiley, New York.

....function we can reduce to a sufficient statistic (Lehmann (1983) and assume that we have just one observation, X, rather than a random sample. Our goal is to prove the admissibility of the maximum likelihood estimator for this problem. We need to recall various facts about exponential families. Barndorff Nielsen (1978) and Brown (1986) are good references. In particular the facts about maximum likelihood estimation can be found in section 3 of chapter 9 of Barndorff Nielsen (1978) Let H be the convex hull of X . We assume that the dimension of H is d so that the family P is minimal. By (9) the family P is ....

....the admissibility of the maximum likelihood estimator for this problem. We need to recall various facts about exponential families. Barndorff Nielsen (1978) and Brown (1986) are good references. In particular the facts about maximum likelihood estimation can be found in section 3 of chapter 9 of Barndorff Nielsen (1978). Let H be the convex hull of X . We assume that the dimension of H is d so that the family P is minimal. By (9) the family P is regular and hence steep. It follows that the range of ( Delta) is the interior of H. Moreover if X = x is observed and x is in the interior of H then x is the maximum ....

Barndorff-Nielsen, O. (1978) Information and Exponential Families in Statistical Theory, Wiley: New York.

....application of these formulae to the estimation problem in graphical models where a multivariate distribution is assumed with intractable density but tractable cdf. In comparison to the more natural way via density or likelihood functions, it is described in detail how the general concept of a cut (Barndor Nielsen, 1978) can be used in a modi ed way for cdf s. The nal section deals with a concrete family of multivariate distributions where in fact the density function in contrast to the cdf is not feasible. Let X = X i ) i2V = X 1 ; X p ) T denote a vector of random variables with related index set V ....

....of the ML estimation is possible whenever the likelihood function factorizes into functions that are distinct with respect to the unknown parameter vector, say . Thus, the maximization reduces to the separate maximization of each factor. This procedure is justi ed by a general concept proposed by Barndor Nielsen (1978). The idea is to show that the investigated family of distributions P can be written as a product space P T P T for a statistic T = T (X) where P T denotes the set of distributions of T and P T the set of conditional distributions of X given T . This implies on the one hand that any ....

[Article contains additional citation context not shown here]

Barndor{Nielsen, O.E. (1978). Information and exponential families in statistical theory. John Wiley and Sons, New York.

....Section 4 presents the Bayes solution of FDI problem. Section 5 concerns the approximation of the formal solution of FDI problem using the quasi Bayes procedure ( 6] and the references therin) The result is then elaborated for practically significant and feasible exponential family of models [7]. A simple example illustrates properties of the proposed algorithm. It deals with two faulty states only but handles both discrete and continuous data, i.e. it handles so called mixed data model. Finally, conclusions are drawn. The main contributions of the research report are: ffl adaptation of ....

O. Barndorff-Nielsen, Information and exponential families in statistical theory, Wiley, New York, 1978. 33

....not depend on . The parameter is called the natural parameter. The function G( is a normalization factor so that R x2X p(xj )dx = 1 holds, and it is called the cumulant function that characterizes the family G . We rst review some basic properties of the family. For further details, see [3, 1]. Let g( denote the gradient vector r G( It is well known that G is a strictly convex function and g( equals the mean of x, i.e. g( R x2X xp(xj )dx. We let g( and call the expectation parameter. Since G is strictly convex, the map g( has an inverse: Let f : g 1 . ....

O. Barndor-Nielsen. Information and Exponential Families in Statistical Theory. Wiley, Chichester, 1978.

....= ln(1 e ) then G ( e ; e e e ( e ) e 1 e : 3.1 The Exponential Family The features of the exponential family that are used throughout this paper include a measure of divergence between two members of the family and an intrinsic duality relationship. See [BN78, Ama85] for a more comprehensive treatment of the exponential family. A multivariate parametric family FG of distributions is said to be an exponential family when its members have a density function of the form PG (xj ) exp ( x G( P 0 (x) 3.2) where and x are vectors in R d , and ....

....the Gaussian. The function G( is a normalization factor de ned by G( ln Z exp( x) P 0 (x) dx The space R d , for which the integral above is nite, is called the natural parameter space. The exponential family is called regular if is an open subset of R d . It is well known [BN78, Ama85] that is a convex set, and that G( is a strictly convex function on . The function G( is called the cumulant function, and it plays a fundamental role in characterizing members of this family of distributions. We use g( to denote the gradient vector r G( and G ij ( to ....

O. Barndor-Nielsen. Information and Exponential Families in Statistical Theory. Wiley, Chichester, 1978.

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Barndor-Nielsen, O. (1978). Information and Exponential Families in Statistical Theory. Wiley, New York.

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Barndor#-Nielsen, O., Information and Exponential Families in Statistical Theory. Wiley, New York, 1978.

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Barndor#-Nielsen, O. (1978). Information and Exponential Families in Statistical Theory. John Wiley and Sons, New York.

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Barndor#-Nielsen, O. (1978). Information and Exponential Families in Statistical Theory. John Wiley and Sons, New York.

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O. Barndor-Nielsen. Information and Exponential Families in Statistical Theory. Wiley, Chichester, 1978.

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O. Barndor#-Nielsen. Information and Exponential Families in Statistical Theory. John Wiley & Sons, 1978.

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O. Barndor-Nielsen. Information and Exponential Families in Statistical Theory. John Wiley & Sons, 1978.

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O. Barndor-Nielsen. Information and exponential families in statistical theory. Wiley, New York, 1978.

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O. Barndor#-Nielsen. Information and Exponential Families in Statistical Theory. John Wiley & Sons, 1978.

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O. Barndor#-Nielsen, Information and Exponential Families in Statistical Theory (John Wiley & Sons, 1978).

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Barndorff-Nielsen, O. E. (1978). Information and Exponential Families in Statistical Theory, John Wiley: Chichester.

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O. Barndor-Nielsen. Information and Exponential Families in Statistical Theory. Wiley, Chichester, 1978.

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