I am working on a Bayesian serial correlation model for binary and ordinal logistic models (proportional odds model). I am modeling the serial correlation structure on the random effects of the model (on the logit scale) so that it is easy to handle ordinal outcome variables. If all subjects are measured at regular, say integer-valued, times, the following thinking works:

* Let $T$ be the integer maximum observation time over all subjects
* Suppose that observation times are $t=1, \dots, T$
* Let $\gamma\_i$ be a $n(0, \sigma\_\gamma$) random effect for the $i$th subject.
* Let $\epsilon\_{i,1}, ... \epsilon\_{i,T}$ be the within-subject white noise
for the $i$th subject that is $n(0, \sigma\_w)$, where $T$ is the maximum follow-up time (we may only use the first few of these for a given subject)
* $\epsilon\_{i,1} = \gamma\_i$ so the usual random effect as generated
for a hierarchical (compound symmetric correlation pattern) repeated
measures model is the starting white noise for a given subject (this random effect may have a different standard deviation $\sigma\_\gamma$)
* Then the random effect for subject $i$ at time $t$ is $r\_{i,t} = \rho^{t}\gamma\_i + \rho^{t-1}\epsilon\_{i,1} + \rho^{t-2}\epsilon\_{i,2} + ... \epsilon\_{i,T}$
* The random effects must all be defined regardless of which
observations are actually observed, so we can re-write the model as
using a matrix of random effects $r\_{i,1} = \gamma\_i, r\_{i,2} = \rho r\_{i,1} + \epsilon\_{i,2}, r\_{i,3} = \rho r\_{i, 2} + \epsilon\_{i,3}, ...$.

I want to generalize this to an irregular-spaced continuous time AR(1) structure. For that purpose I would rather not envision an $N \times T$ white noise (or random effects) matrix but would like to develop an irregular continuous-time AR(1) structure or something that is similar to AR(1) in inducing higher correlation between two nearby measurements when compared to the correlation between two distant measurements. Thoughts for how to envision and develop this would be welcomed.