# Finishing Comparison Parametric And Nonparametric

Implemented in BUGS/allen.md

• [x] Inverse gamma priors on variances 36888da
• [x] Uniform prior on standard deviations 75df941
• [x] Mean plot for parametric fit 75df941
• [x] Convergence diagnostics for both parametric and nonparametric MCMC, using similar visual layout 75df941

• Oh, Jeromy Anglim has a rather nice collection of jags links

## On Variance Priors for the Parametric MCMC

• Gelman recommends uniform priors on standard deviations for the noise terms: in the .bugs file we have
stdQ ~ dunif(0,100)
stdR ~ dunif(0,100)
# as "precision" tau instead of stdev sigma
iQ <- 1/(stdQ*stdQ);
iR <- 1/(stdR*stdR);

Where these enter the model as

for(t in 1:(N-1)){
mu[t] <- x[t] + exp(r0 * (1 - x[t]/K)* (x[t] - theta) )
x[t+1] ~ dnorm(mu[t],iQ)
}

for(t in 1:(N)){
y[t] ~ dnorm(x[t],iR)
}

Note that dnorm in JAGS notation is defined in terms of the mean and the precision (reciprocal of the variance), rather than standard deviation (e.g. in R’s dnorm function), see the JAGS manual, particularly Table 6.1 pg 29, below. Note that the standard deviation has the uniform prior, not the precision.

• Also try inverse gamma priors,
iQ ~ dgamma(0.0001,0.0001)
iR ~ dgamma(0.0001,0.0001)

(Precision is Gamma distributed when the variances are inverse-gamma distributed.

• Both are quite uniformative, results appear quite comparable.

## Markov Chain Monte Carlo Convergence Analysis

• Add for both parametric and non-parametric cases
• Extra long runs before finalizing analysis
• Some examples with R code

A variety of tools from coda package. Of course these methods are designed from toy problems and cannot guarentee convergence.

1. Traceplots, density plots
2. Running mean, autocorrelation plots autocorr.plot
3. Metropolis acceptance/rejection rate rejectionRate
4. Multiple chain diagonostics: Gelman-Rubin gelman.diag, gelman.plot
5. Geweke (compare means between different sections) geweke.diag
6. Raferty-Lewis raferty.diag
7. Heidelberg and Welch heidel.diag