Importance sampling is a term from the field of stochastic processes describing the technique for generating samples on the basis of a probability distribution. Importance sampling is one of several options for variance reduction, ie to increase the efficiency of Monte Carlo simulations.
Monte Carlo simulations are often used to expectation values of a variable ( here designated, otherwise - especially in mathematics - often as shown)
Calculating, with a normalized statistical weight such as a weight is Boltzmann. is the value of the quantity in the state. The summation (integration) in this case runs through a space, such as the phase space of the particles in the system.
For the limit of the average value of a number of conditions applies:
For the simplest case (simple sampling) randomly selected states gives the average
This method is usually not very effective because often received only a few relevant states in the averaging. To work around this problem and thus reduce the standard deviation of the measured mean value for the same number of samples, one tries to let go states with a larger weight more frequently in the averaging as states with a lower weight.
If states with a probability generated ( importance sampling ), then the average is calculated to
If the system states generated by the probability, we obtain
To achieve this in practice, one starts from a launch configuration and generates a Markov chain of system states.