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This problem is best understood by considering an example with binary variables: consider the graph and the probability distributions shown in Figure 112 (it is pABC pF DE and pBD pCE ) Suppose that we observe A = 1 Since this enforces B = C and thus D = E, we should obtain, as an inference result, P(F = 0) = 0 and P(F = 1) = 1 However, the marginal distributions on the individual variables B and C do not change due to this observation (they are still P(B = 0) = P(B = 1) = P(C = 0) = P(C = 1) = 05) As a consequence no information is transmitted to the right half of the network, leading to the incorrect result P(F = 0) = P(F = 1) = 05 This also illustrates the potential pitfalls of loopy propagation (cf Section 42.



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The lifetime family size is especially important in iteroparous sh (Nelson & Soul , 1987); rstly, as a result of the large fecundity of many sh and associated density-independent mortality, resulting in orders-of-magnitude variation in cohort strength (Cushing, 1973), and secondly, individuals surviving longer have more offspring than those dying shortly after maturity In sh, this effect is enhanced by the fact that the fecundity increases with body size (Bagenal, 1973) and therefore with age As such, a sh surviving twice as long after maturity has more than double the number of offspring If the size of the population varies largely between generations, the overall Ne is more affected by the smaller-sized generations than the large ones The breeding survivors of a population crash contain only a sample of the original genetic variation; all future generations will have a corresponding de cit in genetic variability (Frankel & Soul , 1981).





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3) We encounter basically the same problem, though in more complex form, for the electrical circuit in the graph structure shown in Figure 111 For instance, if we set the variables for the three resistors to the same value, all voltages U1 , U2 , and U3 must be equal However, this information does not, or does not completely, reach the center node To cope with this problem we would have to merge nodes in order to obtain an acyclic structure, which, if done inappropriately, can lead to fairly large cliques (and doing it optimally is a nontrivial issue actually it is NP-hard, since finding an optimal triangulation of an undirected graph is NP-hard, cf Section 422).

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Combination of the above factors means that except in heavily managed hatchery populations, Ne is often an order of magnitude smaller than the census number of individuals (Nelson & Soul , 1987) In a review of terrestrial and freshwater animals, Frankham (1995) estimated Ne /N, that is the effective population size as a proportion of census numbers, to vary between 005 and 08, with a mean of 011 However, there is evidence that this ratio may be considerably lower in marine species, where reproductive success may be very biased among individual spawners (Hedgecock, 1994) and estimates suggest Ne /N ratios of 0003 in red drum (Sciaenops ocellatus) (Turner et al, 1999) and of 10 5 in New Zealand snapper (Pagrus auratus) (Hauser et al, 2002).

The second problem we encounter if we try to construct a graphical model for an electrical circuit results from the fact that the electrical laws and the prior information about marginal probability distributions over, for example, the resistor values do not allow for a direct initialization of the quantitative part of the network In a way, we have too little information To see this, consider how one may try to build a Bayesian network for the very simple electrical circuit shown in Figure 111 We would like to have parentless nodes only for those variables for which we can specify marginal distributions, that is, for the resistor values and maybe the supply voltage Every other variable should be a child of one or more variables, with the conditional probability distribution encoding the electrical law that governs the dependence, because we cannot easily specify a marginal distribution for it..

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