# Naive SE vs Time Series SE: which statistics should I report after Bayesian estimation?

I am new to Bayesian estimation.

When I do some estimations with JAGS, I find there are statistics called Naive SE and Time Series SE.

What exactly do they mean? Is it necessary that I report one or both of them as part of the estimation result?

• Where exactly did you find those statistics? I use JAGS too, but never ever seen JAGS reporting something like this on its own. Maybe it is not JAGS, but some model you use. In this case you should include the model into the question. – Adam Ryczkowski Nov 3 '13 at 9:42
• I run JAGS in R via r2jags.You can check the statistics with the following R code: "jags.mcmc<-as.mcmc(jags.estimation); jags.sum<-summary(jags.mcmc); jags.sum.stat<-jags.sum\$statistics". Here is an example where both Naive SE and Time Series SE are reported by JAGS: stats.stackexchange.com/questions/71731/… – HelloL Nov 3 '13 at 10:52
• OK. I never used r2jags. – Adam Ryczkowski Nov 3 '13 at 11:48

If we dig a little bit in the R functions summary.mcmc.list from the package coda (which uses function safespec0 and this one uses spectrum0.ar in its turn) we find that the definition of the naive SE is: $$SE_{Naive} = \sqrt{\frac{Var(X)}{C\cdot S}} \,,$$ whith $$C$$ being the number of run chains, $$X = \{X^{(c)}\}$$ being the vector of posterior samples for a certain parameter (concatenation all the chains, $$c\in 1, \ldots, C$$), and $$S$$ being the length (the number of iterations) of each chain.
We also find that the time-series SE is defined as: $$SE_{ts} = \sqrt{\frac{Var_{ts}(X)}{C\cdot S}} \,,$$ where $$Var_{ts}(X)$$ is the average of the $$Var_{ts}(X^{(c)})^{(c)}$$ for each set of samples $$X^{(c)}$$ of each chain, and each $$Var_{ts}(X^{(c)})^{(c)}$$ is obtained from the application of an auto-regressive (AR) model over $$X^{(c)}$$ by the function ar, selecting the order $$K$$ of the AR using the Akaike Information Criterion (AIC). If the fitted AR model is $$X^{(c)}_t - \bar{X^{(c)}}= \sum_{k=1}^K \rho_k (X^{(c)}_k - \bar{X^{(c)}}) + \varepsilon_t \,, \varepsilon_t \sim Norm(0, \sigma_\varepsilon^2) \,,$$ then $$Var_{ts}(X^{(c)})^{(c)} = \frac{\sigma_\varepsilon^2}{(1-\sum_{k=1}^K \rho_k)^2} \,.$$