C. Cannings, T. E.A., and H. Skolnick. Probability functions on complex pedigrees. Advances in Applied Probability, 10:2661, 1978.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....i s father and mother, We make the standard assumption that if individual i is not a founder, then both of her parents are in the pedigree. Genotype 3 Genotype Phenotype Phenotype 2 Phenotype 3 A[1,p] B[1,p] A[3,p] F[1] C[1,p] B[3,p] G[1] C[3,p] H[1] A[1,m] B[1,m] C[1,m] SA[3,p] F[3] G[3] H[3] A[3,m] B[3,m] C[3,m] A[2,p] B[2,p] F[2] C[2,p] G[2] SA[3,m] H[2] A[2,m] B[2,m] C[2,m] SB[3,p] SC[3,p] SB[3,m] SC[3,m] Figure 2: A fragment of a probabilistic network representation of the transmission model, and the penetrance model in a 3 loci analysis. respectively, ....

....and mother, We make the standard assumption that if individual i is not a founder, then both of her parents are in the pedigree. Genotype 3 Genotype Phenotype Phenotype 2 Phenotype 3 A[1,p] B[1,p] A[3,p] F[1] C[1,p] B[3,p] G[1] C[3,p] H[1] A[1,m] B[1,m] C[1,m] SA[3,p] F[3] G[3] H[3] A[3,m] B[3,m] C[3,m] A[2,p] B[2,p] F[2] C[2,p] G[2] SA[3,m] H[2] A[2,m] B[2,m] C[2,m] SB[3,p] SC[3,p] SB[3,m] SC[3,m] Figure 2: A fragment of a probabilistic network representation of the transmission model, and the penetrance model in a 3 loci analysis. respectively, then ....

[Article contains additional citation context not shown here]

C. Cannings, E. A. Thompson, and M. H. Skolnick. Probability functions on complex pedigrees. Advances in Applied Probability, 10:26--61, 1978.

....property of com14 piling a theory into a backtrack free (i.e. greedy) theory, and their complexity is dependent on the induced width graph parameter. The algorithms are variations on known algorithms, and, for the most part, are not new in the sense that the basic ideas have existed for some time [8, 34, 31, 50, 28, 39, 32, 3, 45, 46, 48, 47]. Definition 2 (graph concepts) A directed graph is a pair, G = fV; Eg, where V = fX 1 ; Xng is a set of elements and E = f(X i ; X j )jX i ; X j 2 V; i 6= jg is the set of edges. If (X i ; X j ) 2 E, we say that X i points to X j . For each variable X i , the set of parent nodes of X i , ....

....processed, the answer is available in the first bucket. Algorithm elim bel is described in Figure 17. We conclude: Theorem 3 Algorithm elim bel computes the posterior belief P (x 1 je) for any given ordering of the variables which is initiated by X 1 . 2 The peeling algorithm for genetic trees [8], Zhang and Poole s algorithm [51] as well as the SPI algorithm by D Ambrosio et al. 39] are all variations of elim bel. Decimation algorithms in statistical physics are also related and were applied to Boltzmann trees [43] 21 B C D F A G B C D F A G D F B C A (a) b) c) ....

[Article contains additional citation context not shown here]

C. Cannings, E.A. Thompson, and H.H. Skolnick. Probability functions on complex pedigrees. Advances in Applied Probability, 10:26--61, 1978.

....by lower case letters (a) Sets of variables are denoted by bold face upper case letters (A) and their instantiations are denoted by bold face lower case letters (a) 1 time slice. Standard algorithms for inference in Bayesian networks [5, 11, 8, 9] which are variations on peeling algorithms [3], have a time and space complexity of O(N exp(w) We can always reason about DBNs using a standard structurebased algorithm by constructing a T expansion such that T covers all events of interest. This will always work, but would require time and space complexity of O(nT exp(w) This turns out ....

.... on a network of N nodes and using an elimination order of width w is O(N exp(w) We close this section by noting that variable elimination algorithms, as known in the Arti cial Intelligence literature, are structurally similar to peeling algorithms as known in the pedigree analysis literature [3]. 3 Prediction We will now utilize the above elimination algorithm to reason about DBNs. We start with the problem of prediction and then generalize to other tasks in Section 4. We rst present an algorithm that uses a slice by slice elimination order, and show how the choice of such orders ....

C. Cannings, E. A. Thompson, and M. H. Skolnick. Probability functions on complex pedigrees. Adv. Appl. Prob., 10:26-61, 1978.

.... models and projection pursuit regression led to new models and algorithms that are hybrids of statistical and neural network research (e.g. JJ94] Graph based models for efficient representation of multivariate distributions had been known for some time in areas such as genetics (e.g. [CTS78]) In the late 1980 s more general and widelyapplicable frameworks were developed independently within statistics (e.g. the acyclic directed graphical model framework of [LS88] and the largely equivalent belief network framework pioneered by Pearl [Pea88] within computer science. The last 10 ....

Cannings, C., Thompson, E. A., and Skolnick, M. H. (1978) Probability functions on complex pedigrees, Advances in Applied Probability, 10, 26--61.

.... : X X1 p(XN jpa(XN ) p(X 1 jpa(X 1 ) Because X 1 can only appear in densities p(X j jpa(X j ) for 1 It should be mentioned that many ideas developed in the SPI, variable elimination and bucket elimination algorithms were first developed for the peeling algorithm in genetics [1]. X j 2 fX 1 ; ch(X 1 )g, we can move the summation for X 1 : X XN : X X2 0 B B B B B Y X i 2 XRnfX1 ; ch(X1 )g p(X i jpa(X i ) 1 C C C C C A 0 B B B B B X X1 Y X j 2 fX1 ; ch(X1 )g p(X j jpa(X j ) 1 C C C C C A : At this point, we have ....

C. Cannings, E. A. Thompson, and M. H. Skolnick. Probability functions in complex pedigrees. Advances in Applied Probability, 10:26--61, 1978.

....name but a few. The exact method for computing such probabilities on pedigrees in which at least one of every parent pair is a founder proposed by Elston Stewart (1971) was extended by Lange Elston (1975) and finally generalised to include arbitrarily complex pedigrees and genetic models by Cannings, Thompson Skolnick (1978). This method has become known in the statistical genetics literature as peeling. Interestingly, it was re invented ten years later in the expert systems literature (Lauritzen Spiegelhalter 1988) as a means of calculating posterior probabilities on general Bayesian networks. While, in theory, ....

Cannings, C., Thompson, E. A. & Skolnick, M. H. (1978), `Probability functions on complex pedigrees', Advances in Applied Probability 10, 26--61.

....framework [20] provides a convenient and succinct language for expressing elimination algorithms in many areas. In addition to dynamic programming [3] constraint satisfaction [22] and Fourier elimination [37] there are variations on these ideas and algorithms for probabilistic inference [7, 57, 55, 59]. Our approach is inspired by adaptive consistency, a full bucket elimination algorithm whose approximation, directional i consistency and its relational variant directional relationalconsistency (i,m) DRC (i;m) enforce bounded levels of consistency [24] For example, directional relational ....

C. Cannings, E.A. Thompson, and H.H. Skolnick. Probability functions on complex pedigrees. Advances in Applied Probability, 10:26--61, 1978.

....was to formulate and discuss an interdisciplinary collaborative project involving Loughborough, Foulum and Aalborg. During a previous visit to Aalborg University, some common ground had already been established between the mathematical genetics approach to calculating probabilities on pedigrees [1] and the expert systems approach to handling general Bayesian networks [8, 11] In the case of a large complex pedigree, exact calculations are no longer possible due to the enormous storage requirements involved and genetic analyses, using full pedigree information, provide a serious ....

C. Cannings, E.A.Thompson and M. Skolnick. Probability functions on complex pedigrees. Advances in Applied Probability (1978) 10:26--61

....framework [13] provides a convenient and succinct language for expressing elimination algorithms in many areas. In addition to dynamic programming [3] constraint satisfaction [17] and Fourier elimination [28] there are variations on these ideas and algorithms for probabilistic inference [5, 50, 48, 51]. Mini bucket al..gorithms parallel directional local consistency enforcing algorithms for constraint processing. Specifically, adaptive consistency is a full bucket elimination algorithm whose approximation, directional i consistency or its relational variant directional relationalconsistency (i,m) ....

C. Cannings, E.A. Thompson, and H.H. Skolnick. Probability functions on complex pedigrees. Advances in Applied Probability, 10:26--61, 1978.

....framework [13] provides a convenient and succinct language for expressing elimination algorithms in many areas. In addition to dynamic programming [3] constraint satisfaction [17] and Fourier elimination [28] there are variations on these ideas and algorithms for probabilistic inference [5, 50, 48, 51]. Mini bucket al..gorithms parallel directional local consistency enforcing algorithms for constraint processing. Specifically, adaptive consistency is a full bucket elimination algo24 rithm whose approximation, directional i consistency or its relational variant directionalrelational ....

C. Cannings, E.A. Thompson, and H.H. Skolnick. Probability functions on complex pedigrees. Advances in Applied Probability, 10:26--61, 1978.

....framework [13] provides a convenient and succinct language for expressing elimination algorithms in many areas. In addition to dynamic programming [3] constraint satisfaction [17] and Fourier elimination [28] there are variations on these ideas and algorithms for probabilistic inference [5, 50, 48, 51]. Mini bucket al..gorithms parallel directional local consistency enforcing algorithms for constraint processing. Specifically, adaptive consistency is a full bucket elimination algorithm whose approximation, directional i consistency or its relational variant directionalrelational consistency(i,m) ....

C. Cannings, E.A. Thompson, and H.H. Skolnick. Probability functions on complex pedigrees. Advances in Applied Probability, 10:26--61, 1978.

.... ) w n w O( n exp( w n Same as worst case Elimination Conditioning Average time Space worst case better than exp( n ) O( Worst case time knowledge compilation one solution Output Figure 12: Comparing elimination and conditioning basic ideas have existed for some time [8, 35, 33, 49, 30, 39, 34, 3, 45, 46, 48, 47]. What we are presenting here is a syntactic and uniform exposition emphasizing these algorithms form as a straightforward elimination algorithm. The presentation allows ideas and techniques to flow across the boundaries between areas of research. In particular, having noted that elimination ....

....buckets are processed, the answer is available in the first bucket. Algorithm elim bel is described in Figure 17. Theorem 3 Algorithm elim bel compute the posterior belief P (x 1 je) for any given ordering of the variables which is initiated by X 1 . Both the peeling algorithm for genetic trees [8], Zhang and Poole s algorithm [50] and the SPI algorithm by D ambrosio et.al [39] are variations of elim bel. Decimation algorithms in statistical physics are also related and were applied to Boltzmann trees [43] 21 Algorithm elim bel Input: A belief network BN = fP 1 ; Png; an ordering ....

[Article contains additional citation context not shown here]

C. Cannings, E.A. Thompson, and H.H. Skolnick. Probability functions on complex pedigrees. Advances in Applied Probability, 10:26--61, 1978.

....framework [14] provides a convenient and succinct language for expressing elimination algorithms in many areas. In addition to dynamic programming [3] constraint satisfaction [18] and Fourier elimination [31] there are variations on these ideas and algorithms for probabilistic inference [5, 53, 51, 55]. Mini bucket al..gorithms parallel directional local consistency enforcing algorithms for constraint processing. Our approach is inspired by adaptive consistency, a full bucketelimination algorithm whose approximation, directional i consistency or its relational variant ....

C. Cannings, E.A. Thompson, and H.H. Skolnick. Probability functions on complex pedigrees. Advances in Applied Probability, 10:26--61, 1978.

....as a sum over all possible genotypes. We solve this problem by applying Markov chain Monte Carlo (MCMC) in a Bayesian analysis (Smith and Roberts, 1993; Besag, Green, Higdon, and Mengersen 1995) We also have considered alternative approaches such as the peeling method from pedigree analysis (Cannings, Thompson, and Skolnick, 1978) and the EM algorithm (Dempster, Laird, and Rubin, 1977) Both approaches are more difficult to implement than MCMC in our problem; however, under a restrictive setting the peeling approach provides a similar result to ours. Our proposed Metropolis Hastings (MH) algorithm includes a new updating ....

....the EM algorithm to maximize the likelihood or the peeling algorithm simply to evaluate the likelihood. The former is not applicable here for various reasons, most notably the discreteness of the parameter M and the non exponentiality of the likelihood (Lee et al. 1996a) The peeling algorithm (Cannings et al. 1978) evaluates likelihood by summing over missing genotype data in an efficient way. Noting the simplicity of the pedigree structure here, we can apply this approach to our data. From the complete likelihood (9) we can evaluate the actual likelihood function for family i, i = 1; N , at each point ....

Cannings, C., Thompson, E. A., and Skolnick, M. H. (1978). Probability Functions on Complex Pedigrees.

....graphical model, Bayesian network, junction tree, pedigree analysis, Monte Carlo. 1 Introduction Over the last decade or so, fast and exact methods have been developed for computation in graphical models (Bayesian networks) of complex stochastic systems (Cannings, Thompson Skolnick 1976, Cannings, Thompson Skolnick 1978, Lauritzen Spiegelhalter 1988, Shenoy Shafer 1990, Jensen, Lauritzen Olesen 1990, Dawid 1992, Lauritzen 1992, Spiegelhalter, Dawid, Lauritzen Cowell 1993) The success of the exact methods has become a reality despite the fact that computation in Bayesian networks is generally NP hard ....

Cannings, C., Thompson, E. A. & Skolnick, H. H. (1978). Probability functions on complex pedigrees, Advances in Applied Probability 10: 26--61.

.... of the likelihood function compel us to pursue computational analysis using Markov chain Monte Carlo (MCMC) for Bayesian inference (Smith and Roberts, 1993; Besag, Green, Higdon, and Mengersen 1995) We have considered alternative approaches such as the peeling method from pedigree analysis (Cannings, Thompson, and Skolnick, 1978) and the EM algorithm (Dempster, Laird, and Rubin, 1977) but both are more difficult to implement than MCMC in our problem; under a restrictive setting the peeling approach provides a similar result to ours. Our proposed Metropolis Hastings (MH) algorithm uses a novel vector proposal distribution ....

....Often genetic models are analyzed using the EM algorithm to maximize the likelihood or the peeling algorithm simply to evaluate the likelihood. The former is not applicable here for various reasons, most notably the discreteness of the parameter M (Lee et al. 1996a) The peeling algorithm (e.g. Cannings, Thompson, and Skolnick, 1978) evaluates likelihood by summing over missing genotype data in an efficient way. Noting the simplicity of the pedigree structure here, we can apply this approach to our data. From the complete likelihood (9) we can evaluate the actual likelihood function for family i, i = 1; N , at each point ....

Cannings, C., Thompson, E. A., and Skolnick, M. H. (1978), "Probability Functions on Complex Pedigrees," Adv. Appl. Prob., 10, 26-61.

....genes, largely because the number of IBD states grows rapidly with the number of genes involved. For example, the number of IBD states for 6 (ordered) genes is 203, and for 8 genes there are 4140 IBD states (Thompson (1974) For a development of this problem see, in addition to Thompson (1974) Cannings, Thompson and Skolnick (1978) and Whittemore and Halpern (1994) The independence structure in any pedigree may be exploited to simplify this type of calculation. We consider the relatively simple case of two noninbred individuals, dealt with in Thompson (1986) We present it here in terms of elementary probability rules. It ....

Cannings, C., Thompson, E. A., and Skolnick, M. H. (1978). Probability functions on complex pedigrees. Advances in Applied Probability 10, 26-61.

....and implemented to solve this general class of problems. In this paper, we introduce a solution method we call the Clustering Algorithm, a variant of the HUGIN algorithm (Jensen and others 1990a; Jensen and others 1990b) and closely related to a variety of algorithms (Cannings and others 1976; Cannings and others 1978; Lauritzen and others 1990; Shafer and Shenoy 1990; Shenoy 1986) The Clustering Algorithm is not an improvement over the methods This paper appeared as R. D. Shachter, S. K. Andersen and P. Szolovits. Global Conditioning for Probabilistic Inference in Belief Networks. Uncertainty in ....

....performs probabilistic inference by passing messages around cluster trees and propagating the global effects of local information. It is variation on the algorithm in HUGIN (Jensen and others 1990a; Jensen and others 1990b) but also closely related to other algorithms (Cannings and others 1976; Cannings and others 1978; Lauritzen and others 1990; Shafer and Shenoy 1990; Shenoy 1986) We show that the Polytree Algorithm (Kim and Pearl 1983; Pearl 1986b; Peot and Shachter 1991) can be viewed as a special case of the Clustering Algorithm. There are only two kinds of data which are truly local in a cluster tree: ....

Cannings, C, E. A. Thompson, and M. H. Skolnick. "Probability functions on complex pedigrees." Adv. Appl. Probabil. 10 (1978): 26- 61.

....of these algorithms are widely known. In addition to dynamic programming [ Bertele and Brioschi, 1972 ] constraint satisfaction [ Dechter and Pearl, 1987 ] and Fourier elimination [ Lassez and Mahler, 1992 ] there are variations on these ideas and algorithms for probabilistic inference in [ Canning et al. 1978; Tatman and Shachter, 1990; Shenoy, 1992; Zhang and Poole, 1996 ] Mini bucket approximation algorithms parallel consistency enforcing algorithms for constraint processing, in particular those enforcing directional consistency. Specifically, relational adaptive consistency is a full ....

C. Canning, E.A. Thompson, and M.H. Probability functions on complex pedigrees. Advances in Applied Probability, 10:26--61, 1978.

.... for belief updating in Bayesian networks 1 can be viewed as variations on a single, general algorithm involving clustering of variables in a structure called a junction tree (Jensen, Lauritzen Olesen 1990) In this category also falls peeling (Elston Stewart 1971; Lange Elston 1975; Cannings, Thompson Skolnick 1978) which is an exact method developed for belief updating in a particular type of Bayesian networks, viz. pedigrees. The junction tree is obtained from the Bayesian network in the following way, see the example in Figure 2. 3. 4. a c b d e f g h abe be bef bf bcf ef deg eg efg fg fgh a c b d e f g ....

Cannings, C., Thompson, E. A. & Skolnick, M. H. (1978). Probability functions on complex pedigrees, Advances in Applied Probability 10: 26--61.

....selective animal breeding, inference on the genetic nature of a disease, analysis of surviving genes in an endangered species and linkage analysis. The exact method for computing probabilities on pedigrees, finally generalised to include arbitrarily complex pedigrees and genetic models by Cannings, Thompson and Skolnick (1978) has become known in the statistical genetics literature as peeling. It is essentially the same algorithm as proposed by Lauritzen and Spiegelhalter (1988) as a means of calculating posterior probabilities on general Bayesian networks. While in theory every pedigree can be peeled, in practice, due ....

Cannings et al:78 Cannings, C., Thompson, E.A. and Skolnick, M. (1978) Probability functions on complex pedigrees. Advances in Applied Probability, 10, 26--61.

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C. Cannings, T. E.A., and H. Skolnick. Probability functions on complex pedigrees. Advances in Applied Probability, 10:2661, 1978.

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C. Canning, E.A. Thompson, and M.H. Probability functions on complex pedigrees. Advances in Applied Probability, 10:26--61, 1978.

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C. Cannings, E.A. Thompson, and H.H. Skolnick. Probability functions on complex pedigrees. Advances in Applied Probability, 10:26--61, 1978. 55

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Cannings, C., Thompson, E. A., Skolnick, M. H. [1978].. "Probability functions on complex pedigrees." Advances in Applied Probability, 10, 26--61.

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