# Monte Carlo Integration with Computationally expensive Integrant

#### pemfir

##### New Member
Consider $$\mathcal{R}_x = \int_{x'} C(x|x')P(x')dx'$$, where
$$\mathcal{R}_x$$ = risk of following input x
$$C(x|x')$$ = cost of following input x when in fact input x' occured
$$P(x')$$ = known pdf of inputs evaluated for input x'

Objective : obtain a decent estimate of the integral
Challenge : $$C(x|x')$$ is computationally expensive and can only be evaluated for a handful (less than 50) of x,x's.
Context : x is the input of a complex optimization program. To evaluate $$C(x|x')$$ we have to find the optimal solution of the optimization program once for x and once for x'. optimal solutions are compared against each other to find their relative costs ,i.e., $$C(x|x')$$.