alternate density function (for ), usually referred to as a biasing density, for the simulation experiment. This density allows the event to occur more frequently, so the sequence lengths gets smaller for a given estimator variance. Alternatively, for a given , use of the biasing density results in a variance smaller than that of the conventional Monte Carlo estimate. From the definition of , we can introduce as below. where is a likelihood ratio and is referred to as the weighting function. The last equality in the above equation motivates the estimator This is the importance sampling estimator of and is unbiased. That is, the estimation procedure is to generate i.i.d. samples from and for each sample which exceeds , the estimate is incremented by the weight evaluated at the sample value. The results are averaged over trials. The variance of the importance sampling estimator is easily shown to be Now, the importance sampling problem then focuses on finding a biasing density such that the variance of the importance sampling estimator is less than the variance of the general Monte Carlo estimate. For some biasing density function, which minimizes the variance, and under certain conditions reduces it to zero, it is called an optimal biasing density function. Conventional biasing methods Although there are many kinds of biasing methods, the following two methods are most widely used in the applications of importance sampling. Scaling Shifting probability mass into the event region by positive scaling of the random variable with a number greater than unity has the effect of increasing the variance (mean also) of the density function. This results in a heavier tail of the density, leading to an increase in the event probability. Scaling is probably one of the earliest biasing methods known and has been extensively used in practice. It is simple to implement and usually provides conservative simulation gains as compared to other methods. In importance sampling by scaling, the simulation density is chosen as the density function of the scaled random variable , where usually for tail probability estimation. By transformation, and the weighting function is While scaling shifts probability mass into the desired event region, it also pushes mass into the complementary region which is undesirable. If is a sum of random variables, the spreading of mass takes place in an dimensional space. The consequence of this is a decreasing importance sampling gain for increasing ,