Metropois-Hastings(MH) Monte Carlo Integral

참고 A Gentle Introduction to Markov Chain Monte Carlo (MCMC)

참고(중요!!) Hastings-Metropolis for Integration Problems

$$\begin{array}{r}V={\int }_a^{b}g(x)dx={\int }_a^{b}f(x)h(x)dx\\ C={\int }_a^{b}h(x)dx\\ p(x)=\frac{h(x)}{C},\text{Probability Density : }p(x)\\ V={\int }_a^{b}g(x)dx={\int }_a^{b}f(x)h(x)dx=C{\int }_a^{b}f(x)p(x)dx=C\cdot \mathbb{E}[f(x)]\\ V=C{\mathbb{E}}_{x\sim p}[f(x)]\approx C\frac{1}{N}\sum _i^{N}f(x_i)\end{array}$$

---

예제

• mh_integral_01 : $g(z,y,z)$의 MonteCarloIntegration 패키지의 값과 수치적분 비교
  

• 아래 (1),(2),(3)의 각각에 대해 MH Monte Carlo 적분

$$\begin{array}{r}g(x,y,z)=xyz\ln (x+2y+3z)\sin (x+y+z)\\ V={\int }_a^b{\int }_a^b{\int }_a^{b}g(x,y,z)dxdydz\\ \\ \text{mh\_integral\_02()}\\ \text{(1)}& h(x,y,z)=1f(x,y,z)=xyz\ln (x+2y+3z)\sin (x+y+z)\\ \\ \text{mh\_integral\_03()}\\ \text{(2)}& h(x,y,z)=\sin (x+y+z)f(x,y,z)=xyz\ln (x+2y+3z)\\ \\ \text{mh\_integral\_04(), mh\_integral\_05()}\\ \text{(3)}& h(x,y,z)=\ln (x+2y+3z)\sin (x+y+z)f(x,y,z)=xyz\\ \\ \text{※ 주의 : 0 ≤ a,b ≤ 1 범위에서 적분값이 제대로 나옴}\end{array}$$

MonteCarloIntegration::vegas와 HCubature::hcubature 의 적분값 비교

Metropolis-Hasting function 정의

$$\begin{array}{r}f(x,y,z)=xyz\ln (x+2y+3z)\sin (x+y+z)\\ h(x,y,z)=1\\ C={\int }_a^b{\int }_a^b{\int }_a^{b}1dxdydz=(b-a{)}^3\end{array}$$

$$\begin{array}{r}f(x,y,z)=xyz\ln (x+2y+3z)\\ h(x,y,z)=\sin (x+y+z)\\ C={\int }_a^b{\int }_a^b{\int }_a^b\sin (x+y+z)dxdydz\end{array}$$

$$\begin{array}{r}f(x,y,z)=xyz\\ h(x,y,z)=\ln (x+2y+3z)\sin (x+y+z)\\ C={\int }_a^b{\int }_a^b{\int }_a^b\ln (x+2y+3z)\sin (x+y+z)dxdydz\end{array}$$

C 값을 MH로 구하고 적분하기

$$\begin{array}{r}f(x,y,z)=xyz\\ h(x,y,z)=\ln (x+2y+3z)\sin (x+y+z)\\ C=\text{MetroPolis-Hasting MonteCarlo Integral}\end{array}$$