# Problem in Probability Ratio in M-H algorithm

I attempt to sample the posterior distribution of parameters($\alpha,\alpha_1,\alpha_2,\lambda_1,\lambda_2$) using the Metropolis-Hastings within Gibbs.

Steps of Random Walk Metropolis-Hastings

1. start with t=1 and the initial values {($\alpha=1,\alpha_1=1,\alpha_2=1,\lambda_1=1,\lambda_2$=1)}
2. using the initial values generate candidate points from proposal densities

alpha.can<-abs(rnorm(1,alpha[t-1],0.5)); alpha1.can<-abs(rnorm(1,alpha1[t-1],0.5)); alpha2.can<-abs(rnorm(1,alpha2[t-1],0.5)); lamda1.can<-abs(rnorm(1,lamda1[t-1],0.5)); lamda2.can<-abs(rnorm(1,lamda2[t-1],0.005))

3. Generate u.$\alpha$~$\mathscr U$[0,1], u.$\alpha_1$~$\mathscr U$[0,1], u.$\alpha_2$~$\mathscr U$[0,1], u.$\lambda_1$~$\mathscr U$[0,1], u.$\lambda_2$~$\mathscr U$[0,1].

4. Calculate the ratio of candidate point alpha.can and previous point alpha[t-1]

R1=$\frac{g(alpha.can \quad given \quad {\alpha_1[t-1],\alpha_2[t-1],\lambda_1[t-1],\lambda_2[t-1]})}{g(alpha[t-1] \quad given \quad {\alpha_1[t-1],\alpha_2[t-1],\lambda_1[t-1],\lambda_2[t-1]})}$ where g is the full conditional posterior of $\alpha$.

1. If u.$\alpha \;$ <= min (R1,1), accept the candidate point, otherwise set alpha.can=alpha[t-1]

2. Similarly from step 4, the ratio of candidate point alpha1.cand and previous point alpha1 [t-1] is

R2=$\frac{g(alpha1.can \quad given \quad {\alpha[t],\alpha_2[t-1],\lambda_1[t-1],\lambda_2[t-1]})}{g(alpha1[t-1] \quad given \quad {\alpha[t],\alpha_2[t-1],\lambda_1[t-1],\lambda_2[t-1]})}$ where g is the full conditional posterior of $\alpha_1$.

1. If u.$\alpha_1 \;$ <= min (R2,1), accept the candidate point, otherwise set alpha1.can=alpha1[t-1]$\qquad$ and so on for $\alpha_2,\lambda_1,\lambda_2$.

2. Repeat steps 2-7 (T=10000) times and obtain sample of {$\alpha,\alpha_1,\alpha_2,\lambda_1,\lambda_2$}.

• @Xi'an, Thanks very much for your attention. Your correction give me results but Auto-correlation plot for[ four parameters (alpha,alpha1,alpha2 and lamda1)] decreases slowly. Also Histogram plot for alpha1 and alpha2 Concentrated around first bar in histogram and approximately zero for others. – Manal Sep 3 '17 at 16:19
• I searched for decreases the Auto-correlation plot and I found that I need to Thinning the MCMC chain – Manal Sep 3 '17 at 16:25
• In Acceptance probability, I used log posterior and multiplied Acceptance probability by the ratio suggested by you whereas for the parameters (parameter.can e.g. alpha.can I used alpha.can<-abs(rnorm(1,alphat,sd.Normal)) . when I try alpha.can<-dnorm((-1)^(0:1)*alphat,alphat,sd.Normal) give me this warning message In alpha1[t] <- alpha1.can number of items to replace is not a multiple of replacement length – Manal Sep 3 '17 at 16:40

The proposal of a absolute Gaussian/Normal variate $|X|$ when $X\sim\mathcal{N}(\mu,\sigma^2)$ has density $$\frac{1}{2}\varphi(x|\mu,\sigma^2)+\frac{1}{2}\varphi(-x|\mu,\sigma^2)$$ Hence, if you use the absolute value of a normal simulation for proposed move, your proposal density is no longer symmetric and it must appear in the Metropolis-Hastings acceptance ratio. Therefore, the Metropolis-Hastings ratios in your R code must be multiplied by the ratios $$\dfrac{\varphi(\theta_i^t|\theta_i,\sigma^2)+\varphi(-\theta_i^t|\theta_i,\sigma^2)}{\varphi(\theta_i|\theta_i^t,\sigma^2)+\varphi(-\theta_i|\theta_i^t,\sigma^2)}$$ For instance,

 R1=Post.alpha(Data,alpha.can,alpha1t,alpha2t,lamda1t,lamda2t)/
Post.alpha(Data,alphat,alpha1t,alpha2t,lamda1t,lamda2t)*
sum(dnorm((-1)^(0:1)*alphat,alpha.can,sd.Normal))/
sum(dnorm((-1)^(0:1)*alpha.can,alphat,sd.Normal))


As for the appearance of NaN in the outcome of the Post.alpha functions, it is no surprise as they involve powers and exponentials. For instance,

> Post.alpha(Data,.0001,.0002,.0001,.0002,.0003)
 8.512619e-223


You should work with log posteriors to reduce the possibility of underflows.