Importance Sampling: A Caveat

What happens when one sets up a computer simulation using Importance Sampling theory and then, during the simulation, knowingly or unknowingly, generates the simulation uncertainty (e.g. channel noise) with statistics other than those assumed in the simulation setup? By exploring this simple question, we nd some interesting aspects of Importance Sampling that provide insight into its properties and have important implications. Both analytical and numerical results indicate that Importance Sampling estimates are heavily biased towards the assumed statistics, while the actual statistics have little or no eeect. The consequences of this behavior are two-pronged. If the mismatch between actual and assumed statistics is unknown to the programmer, as can arise under several scenarios, then the simulation result may be misleading. If, on the other hand, the mismatch is deliberate, then the result may be simpler simulations. That is, under certain conditions, the noise samples in an Importance Sampling simulation need not be generated precisely according to their theoretical density, potentially eliminating complex random number generation. Monte Carlo (MC) simulation is a widely used computer-aided method of obtaining estimates of probabilistic quantities, such as the bit-error-rate (BER) in communication systems, when the system is suuciently complex to render analytic calculations intractable. However, MC simulations often require prohibitively many simulations to provide accurate estimates. Importance Sampling (IS) 1{7] is a variant of the Monte Carlo method that can obtain estimates of a given accuracy with many orders of magnitude fewer simulation runs than MC. Consequently, IS is frequently used to simulate communication systems.