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case The only way of getting a bounded relative error is via state-dependent IS However, when p is replaced by p = a, the relative error increases only very slowly when n : the second moment decreases exponentially at the same exponential rate as the square of the rst moment When this property holds, the estimator is said to have logarithmic ef ciency In this example, it holds for no other value of p All these results have been proved in a more general setting by Sadowsky [24]
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A general conclusion from the previous section is that to accurately approximate the zero variance IS estimator, a key ingredient is a good approximation of the function ( ) In fact, there are several ways of nding a good IS strategy Most of the good methods can be classi ed into two large families: those that try to directly approximate the zero-variance change of measure via an approximation of the function ( ), and those that restrict a priori the change of measure to a parametric class, and then try to optimize the parameters In both cases, the choice can be made either via simple heuristics, or via a known asymptotic approximation for (y), or by adaptive methods that learn (statistically) either the function ( ) or the vector or parameters that minimizes the variance In the remainder of this section, we brie y discuss these various approaches In the scienti c literature, IS has often been applied in a very heuristic way, without making any explicit attempt to approximate the zero-variance change of measure One heuristic idea is simply to change the probabilities so that the system is pushed in the direction of the rare event, by looking at what could increase its occurrence However, Example 1 shows very well how pushing too much can have the opposite effect; in fact, it can easily lead to an in nite variance Changes of measure that may appear promising a priori can eventually lead to a variance increase In situations where the rare event can be reached in more than one direction, pushing in one of those directions may easily in ate the variance by reducing the probability or density of paths that lead to the rare event via other directions The last part of Example 1 illustrates a simpli ed case of this Other illustrations can be found in [2, 3, 11], for example Generally speaking, good heuristics should be based on a reasonable understanding of the shape of ( ) and/or the way the likelihood ratio will behave under IS We give examples of these types of heuristics in the next subsection
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241 Heuristic approaches
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Here, the idea is to use a heuristic approximation of ( ) in the change of measure (24) Example 7 We return to Example 3, with py = p Our aim is to estimate (1) Instead of looking at the case where B is large, we focus on the case where p
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is small, p 0 for xed B This could be seen as a (simpli ed) dependability model where each transition from y to y + 1 represents a component failure, each transition from y to y 1 corresponds to a repair, and B is the minimal number of failed components for the whole system to be in a state of failure If p 1, each failure transition (except the rst) is rare and we have (1) 1 as well Instead of just blindly increasing the failure probabilities, we can try to mimic the zero-variance probabilities (24) by replacing ( ) in this expression by an approximation, with c(y, z) = 0, (0) = 0 and (B) = 1 Which approximation (y) could we use instead of (y) Based on the asymptotic estimate (y) = p B y + o(p B y ), taking (y) = p B y for all y {1, , B 1}, with (0) = 0 and (B) = 1, looks like a good option This gives P (y, y + 1) = p B y p B y 1 = B y+1 + (1 p)p 1 + (1 p)p
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for y = 2, , B 2 Repairs then become rare while failures are no longer rare We can extend the previous example to a multidimensional state space, which may correspond to the situation where there are different types of components, and a certain subset of the combinations on the numbers of failed components of each type corresponds to the failure state of the system Several IS heuristics have been proposed for this type of setting [16] and some of them are examined in 6 One heuristic suggested in [20] approximates (y) by considering the probability of the most likely path to failure In numerical examples, it provides a drastic variance reduction with respect to previously known IS heuristics
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Various techniques that try to approximate the function ( ), often by adaptive learning, and plug the approximation (24), have been developed in the literature [16] Old proposals of this type can be found in the computational physics literature, for example; see the references in [5] We outline examples of such techniques taken from recent publications One simple type of approach, called adaptive Monte Carlo in [8, 17], proceeds iteratively as follows At step i, it replaces the exact (unknown) value (x) in (24) by a guess (i) (x), and it uses the probabilities P (i) (y, z) = P (y, z)(c(y, z) + (i) (z)) (i) w Y P (y, w)(c(y, w) + (w)) (26)
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in ni independent simulation replications, to obtain a new estimation (i+1) (y) of (y), from which a new transition matrix P (i+1) is de ned These iterations could go on until we feel that the probabilities have converged to reasonably good estimates A second type of approach is to try to approximate the function ( ) stochastically The adaptive stochastic approximation method proposed in [1] for the
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