Can we apply importance sampling in VS .NET

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Can we apply importance sampling
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Given the methodological background provided in previous chapters of this book, the most obvious approach is to apply importance sampling Indeed, as we are
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concerned with CTMCs, importance sampling can be appropriately formulated for biological networks [63, 64] For CTMCs the relevant probability measures are path distributions, and absolute continuity corresponds to the condition that all paths that are possible under the original measure must remain possible under the importance sampling measure This can be obviously achieved by the condition that for all positive rates in the original model the corresponding rates under importance sampling are positive Since we are dealing with CTMCs given in terms of biochemical notation as described in Section 1121, we need an appropriate framework for the application of importance sampling to that type of model speci cation In particular, we need to express the distribution or density, respectively, of reaction paths Let t1 < t2 < denote the successive time instants at which reactions occur and Rmi , mi {1, , M} the reaction type that occurs at time ti De ne i := ti+1 ti to be the time between the ith and the (i + 1)th reactions Hence, state x(ti ) is reached due to the ith reaction Rmi at time ti and remains unchanged for a sojourn time of i , after which the (i + 1)th reaction Rmi+1 occurs at time ti+1 and changes the state to x(ti+1 ) Thus, the time evolution of the system is completely described by the sequence of states and corresponding sojourn times, and in compact form (x(t0 ), 0 ), (x(t1 ), 1 ), (x(t2 ), 2 ), describes a trajectory For a trajectory up to the Rth reaction, considering the Markovian property which in turn implies exponentially distributed sojourn times, the reaction path density is given by dP ((x(t0 ), 0 ), , (x(tR ), R ))
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= p (t0 ) (x0 )
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mi 1 (x(ti 1 )) exp ( 0 (x(ti 1 )) i 1 ) ,
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(118)
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where 0 (x(ti 1 )) := 1 (x(ti 1 )) + + M (x(ti 1 )) Note that for a given time horizon over which the system is observed (and should be simulated) the number R of reactions is not known in advance and not deterministic Formally, it is a random stopping time, which is in accordance with the requirement of dP being a density of a probability measure P de ned on the path space of the Markov process With importance simulation, the underlying probability measure determined by the propensity functions is changed Since the only requirement is absolute continuity of the probability measures involved, there is great freedom of choice in changing the measure It is only necessary that all reaction paths that are possible (have positive probability) under the original measure remain possible Each probability measure on the path space that meets the aforementioned condition can be considered even non-Markovian models are allowed as long as they assign positive probabilities to all possible reaction paths Nevertheless, we should avoid a large increase in trajectory generation efforts compared to the original measure Thus, obviously the most natural (and valid)
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change of measure is to remain in the Markovian world The easiest way is to change the original propensity functions to importance sampling propensity functions m such that for all m {1, , M} we have m (x) = 0 m (x) = 0, x X , or equivalently, starting with the original propensity functions, m (x) > 0 m (x) > 0, x X One then generates trajectories according to the changed propensity functions and multiplies the results with the likelihood ratio to get unbiased estimates for the original system The trajectory generation is thus carried out as before, for instance by applying the direct method, where now the changed propensity functions are used, yielding a sequence of states with associated sojourn times and reaction path density as in (118) Thus, denoting by p (t0 ) the initial distribution for the states, the likelihood ratio becomes p (t0 ) (x0 ) L( ) = (t ) p 0 (x0 )
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mi 1 (x(ti 1 )) exp ( 0 (x(ti 1 )) i 1 )
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