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is to apply the Russian roulette principle [22] and modify the weight of the chain accordingly Several versions of this are proposed in [25, 26], including the following (where we also select a positive integer and assume that the chain tries to reach level k): Probabilistic truncation applies Russian roulette each time a trajectory crosses a level k 1 j downward, for any j We select real numbers rk 1,j [1, ) for j = , , k 1 Whenever a chain crosses level k 1 j downward from level k 1, for j , it is killed with probability 1 1/rk 1,j If it survives, its weight is multiplied by rk 1,j When a chain of weight w > 1 reaches level k, it is cloned into w 1 additional copies and each copy is given weight 1 (if w is not an integer, we make w additional copies with probability = w w and w 1 additional copies with probability 1 ) The latter is done to reduce the variance introduced by the weights Periodic truncation [26] reduces the variability due to the Russian roulette in probabilistic truncation by adopting a more systematic selection of the chains that we retain Otherwise it works similarly to probabilistic truncation and also uses positive integers rk 1,j It also uses a random integer Dk 1,j generated uniformly in {1, , rk 1,j }, for each k and j When a chain crosses a level k 1 j downward, if it is the (irk 1,j + Dk 1,j )th chain that does that for some integer i, it is retained and its weight is multiplied by rk 1,j , otherwise it is killed Tag-based truncation [26] xes beforehand the level at which a chain would be killed Each chain is tagged to level k 1 j with probability qk 1,j = (rk 1,j 1)/(rk 1, rk 1,j ) for j = , , k 1, and it is killed if it reaches that level With the remaining probability, it is never killed By properly choosing integers rk 1,j , the proportion of chains tagged to level Lk 1 j can be exactly qk 1,j , while the probability of receiving a given tag is the same for all chains To get rid of the weights, which carry additional variance, we can let the chain resplit when it crosses some levels upward after having gone down and having its weight increased The idea is to keep the weights close to 1 The above truncation schemes can be adapted to t that framework One of the best-known versions of splitting is the RESTART method [31 33] Here, when a chain hits a level upward, xed splitting is used (ie, the chain is split by a xed factor), but one of the copies is tagged as the original for that level Truncation is used to reduce the work: when a non-original copy hits its creation level downward, it is killed Only the original chain continues its path (to avoid starvation) The weight of the original chain accounts (in some sense) for those that are killed, to keep the estimator unbiased This rule applies recursively, and the method is implemented in a depth- rst fashion as follows: whenever there is a split, all the copies are simulated completely, one after the

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SPLITTING TECHNIQUES

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other, then simulation continues for the original chain The gain in work reduction is counterbalanced by the loss in terms of a higher variance in the number of chains, and a stronger positive correlation between the chains due to resplits [15] In the discrete-time situation, another implementation does not make use of levels, but applies splitting and Russian roulette at each step of the simulation [3, 5, 13, 28] The number of splits and the killing probabilities are determined in terms of the importance function h De ne = (x, y) = h(y)/ h(x) to be the ratio of importance values for a transition from x to y If 1, the chain is split into C copies where E[C] = , whereas if < 1 it is killed with probability 1 (this is Russian roulette) A weight is again associated with each chain to keep the estimator unbiased: whenever a chain of weight w is split into C copies, the weight of all the copies is set to w/E[C] When Russian roulette is applied, the weight of a surviving chain is multiplied by 1/(1 ) Yet another version, again in the discrete-time situation, mixes splitting and Russian roulette with IS The weight of a chain is rede ned as the weight due to splitting and Russian roulette (as above) times the likelihood ratio accumulated so far (see the previous chapter on IS) To reduce the variance of the weights, the idea of weight windows was introduced in [2], and further studied in [4, 14, 27] The goal is to keep the weights of chain inside a given prede ned window, with the aim of reducing the variance This is done by controlling the weighted importance of each chain, de ned as the product of its weight w and the value of the importance function h(x) at its current state, so that it remains close to = P[TB T ] for the trajectories for which TB T If these windows are selected correctly (this requires a good prior approximation of ), the main source of variance will then be the random number of chains that reach B [4] To proceed, we select three real numbers 0 < amin < a < amax Whenever the weighted importance = wh(x) of a chain falls below amin , Russian roulette is applied, killing the chain with probability 1 /a If the chain survives, its weight is set (increased) to a/ h(x) If the weighted importance rises above amax , we split the chain into c = /amax copies and give (decreased) weight w/c to each copy If a = (amin + amax )/2 P[TB T ], this number has expectation N0 (approximately), the initial number of chains

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