MC Integration

Classical Monte Carlo Integration
​Robert and Casella (2013) introduces the classical Monte Carlo integration:
Toy Example
为了计算积分  我们通常采用MC模拟:
计算区域的volume：​
近似：​
根据大数律有
并且由中心极限定理有
其中​
另外，注意到
I=∫Dg(x)dx=∫DVg(x)1Vdx=Ef[Vg(x)],I = \int_Dg(\mathbf x)d\mathbf x = \int_D Vg(\mathbf x)\frac 1V d\mathbf x = {\mathbb E}_f[Vg(\mathbf x)],I=∫D​g(x)dx=∫D​Vg(x)V1​dx=Ef​[Vg(x)],
则可以直接利用上一节的结论。
## estimate pi
##
## I = \int H(x, y)dxdy
## where H(x, y) = 1 when x^2+y^2 <= 1;
## otherwise H(x, y) = 0
​
## volume
V = 4
​
n = 100000
x = runif(n, -1, 1)
y = runif(n, -1, 1)
H = x^2+y^2
H[H<=1] = 1
H[H>1] = 0
I = V* mean(H)
cat("I = ", I, "\n")
​
## n = 100, I = 2.96
## n = 1000, I = 3.22
## n = 10000, I = 3.1536
## n = 100000, I = 3.14504
Polar Simulation
Note: Unless otherwise stated, the algorithms and the corresponding screenshots are adopted from Robert and Casella (2013).
The following Julia program can be used to do polar simulation.
using Statistics
​
function polarsim(x::Array)
    while true
        phi1 = rand()*2*pi
        phi2 = rand()*pi - pi/2
        u = rand()
        xdotxi = x[1]*cos(phi1) + x[2]*sin(phi1)*cos(phi2) + x[3]*sin(phi1)*sin(phi2)
        if u <= exp(xdotxi - sum(x.^2)/2)
            return(randn() + xdotxi)
        end
    end
end
​
# approximate E^pi
function Epi(m, x = [0.1, 1.2, -0.7])
    rhos = ones(m)
    for i = 1:m
        rhos[i] = 1/(2*polarsim(x)^2+3)
    end
    return(mean(rhos))
end
​
# example
Epi(10)
MC Approximation
MC Optimization
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