Having trouble with Bayesian Inference model - JAGS with R

Question

I've been trying to reproduce the results of the following paper using R and JAGS with no success. I can get the model to run, but the results shown are consistently different.

Link for the paper: https://www.pmi.org/learning/library/bayesian-approach-earned-value-management-2260

The purpose of the paper is to use data gathered from project management reports to estimate project completion date or budget at completion, for instance. Project performance are mostly reported with the use of Earned Value measurement that consist basically on a ratio between actual work completed and the amount of work that was planned to be completed up to a milestone date (in order words, 'Work Done/Planned Work'). So, if I have spent on the third month of the project $300.000 to produce a amount of work that I previously planned to spend $270.000, my Cost Perfomance Index (CPI) is 300.000/270.000 = 1.111. Similarly, if by the 3rd month I had completed a amount of work that correspond with the what was planned to be completed by the 2nd month, my Schedule Performance Index (SPI) is 2/3 = 0.667.

The general problem behind the paper is how to use the performance measurement to update the prior belief about final project performance.

My code is shown bellow. I had to perform a transformation on the data (adding 1 before taking the log(), because some of them would be negative and JAGS return a error (that's why the parameters on my model is different to what's shown on paper's Table 4).

The model used on the paper was lognormal as likelihood and prior for mu and sigma on Normal and Inverse Gamma, respectively. Since BUGS syntax uses tau = 1/(variance) as parameter for Normal and Lognormal, I used the Gamma distribution on tau (that made sense to me).

model_pmi <-  function() {  
   for (i in 1:9) {
cpi_log[i] ~ dlnorm(mu_cpi, tau_cpi)
spi_log[i] ~ dlnorm(mu_spi, tau_spi)
}

tau_cpi ~ dgamma(75, 1)
mu_cpi ~ dnorm(0.734765, 558.126)
cpi_pred ~ dlnorm(mu_cpi, tau_cpi)
tau_spi ~ dgamma(75, 1.5)
mu_spi ~ dnorm(0.67784, 8265.285)
spi_pred ~ dlnorm(mu_spi, tau_spi)

}

model.file <- file.path(tempdir(), "model_pmi.txt")  
write.model(model_pmi, model.file)

cpis <- c(0.486, 1.167, 0.856, 0.770, 1.552, 1.534, 1.268, 2.369, 2.921)
spis <- c(0.456, 1.350, 0.949, 0.922, 0.693, 0.109, 0.506, 0.588, 0.525)
cpi_log <- log(1+cpis)
spi_log <- log(1+spis)

data <- list("cpi_log", "spi_log")

params <- c("tau_cpi","mu_cpi","tau_spi", "mu_spi", "cpi_pred", "spi_pred")
inits <- function() { list(tau_cpi = 1, tau_spi = 1, mu_cpi = 1, mu_spi = 1, cpi_pred = 1, spi_pred = 1) }

out_test <- jags(data, inits, params, model.file, n.iter=10000)

out_test

The 95% CI (2.5%;97.5%) found on the paper is (1.05;2.35) for CPI and (0.55;1.525). The model presented the results shown below. For CPI, the results are fairly close, but when I saw the results for SPI, I figured it should be just chance.

Inference for Bugs model at 
"C:\Users\felip\AppData\Local\Temp\RtmpSWZ70g/model_pmi.txt", fit using jags,
 3 chains, each with 10000 iterations (first 5000 discarded), n.thin = 5
 n.sims = 3000 iterations saved
         mu.vect sd.vect    2.5%     25%     50%     75%   97.5%  Rhat n.eff
cpi_pred   1.691   0.399   1.043   1.406   1.639   1.918   2.610 1.001  2200
mu_cpi     0.500   0.043   0.416   0.471   0.500   0.529   0.585 1.001  3000
mu_spi     0.668   0.011   0.647   0.660   0.668   0.675   0.690 1.001  3000
spi_pred   2.122   0.893   0.892   1.499   1.942   2.567   4.340 1.001  3000
tau_cpi   20.023   2.654  15.202  18.209  19.911  21.726  25.496 1.001  3000
tau_spi    6.132   0.675   4.889   5.657   6.107   6.568   7.541 1.001  3000
deviance 230.411  19.207 194.463 217.506 230.091 243.074 269.147 1.001  3000

For each parameter, n.eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor (at convergence, Rhat=1).

DIC info (using the rule, pD = var(deviance)/2)
pD = 184.5 and DIC = 414.9
DIC is an estimate of expected predictive error (lower deviance is better).

Been working on this for days, can't find what's missing or what's wrong.

Answer 1

When using y ~ dlnorm(mu,tau) , the y value is the original-scale value, not the log-scale value. But mu and tau are on the log scale (which is confusing).

Also, putting priors directly on mu and tau can produce bad autocorrelation in the chains. Reparameterizing helps. For details, see this blog post (that I wrote): http://doingbayesiandataanalysis.blogspot.com/2016/04/bayesian-estimation-of-log-normal.html

Finally, the mean, mode, and SD on the original scale are somewhat complex transformations of mu and tau on the log scale. Again, see the blog post linked above.