Dear Stan Community,
My question in short:
Is it possible to update the parameters of a stan model with every iteration?
My question:
I am a student trying to implement the MGARCH-DPM methodology of Maheu and Shamsi’s Nonparametric Dynamic Conditional Beta (see attachments). It involves a hierarchical model where the returns are modeled by an infinite normal mixture model with a mixed \mu parameter and mixed component B, the latter component mixes the the covariance matrix H_t. The mixing parameters, \mu and B, are generated by a Dirichlet process.
p\left(r_{t} \mid \mu, B, W, H_{t}\right)=\sum_{j=1}^{\infty} \omega_{j} N\left(r_{t} \mid \mu_{j}, H_{t}^{1 / 2} B_{j}\left(H_{t}^{1 / 2}\right)^{\prime}\right)
The hierarchical model:
\begin{array}{ll}
r_{t} & \mid \phi_{t}, H_{t} \sim N\left(\xi_{t}, H_{t}^{1 / 2} \Lambda_{t}\left(H_{t}^{1 / 2}\right)^{\prime}\right), t=1, \ldots, T \\
\phi_{t} & \equiv\left\{\xi_{t}, \Lambda_{t}\right\} \mid G \sim G, \\
G \mid & \alpha, G_{0} \sim D P\left(\alpha, G_{0}\right) \\
G_{0} & \equiv N\left(\mu_{0}, D\right) \times \mathcal{W}^{-1}\left(B_{0}, \nu_{0}\right) \\
H_{t} & =\Gamma_{0}+\Gamma_{1} \odot\left(r_{t-1}-\eta\right)\left(r_{t-1}-\eta\right)^{\prime}+\Gamma_{2} \odot H_{t-1} .
\end{array}
In order to implement this I want to make use of Philippe Rast’s bmgarch package to handle the Multivariate Garch process (I am opting for a DCC parametrization, deviating from the attached paper). The bmgarch package models the MGARCH process using stan (see attachments).
It would be ideal to run the entire MGARCH-DPM model in Stan, but I did find out how stan does not handle discrete sampling, which is needed for a Dirichlet Process Mixture.
Hence to optimize both the MGARCH parameters and the Dirichlet process, I thought of using Ross’ and Markwick’s dirichletprocess package to optimize for \mu and B. I wrote my own distribution ‘dirichletprocess_mvnormalgarchdpm.R’ (attached) to work with the package. This distribution takes, in addition to the regular multivariate normal prior parameters, also the covariance matrix \mathbf{H} as input as generated by bmgarch.
However, with the many iterations and samples, optimizing both processes separately is not the most efficient.
Thus I think a better approach would be to nest both processes so that for every iteration of the bmgarch optimization it can use updated parameters generated by the dirichletprocess \mu and B for the next iteration round, and vice versa. Hence, I would really like to know if updating the parameters would be possible in stan!
Any suggestions how to approach this issue are greatly appreciated.
Regards,
Floor
P.s. tagging @Bob_Carpenter as kindly suggested by @ph-rast :)
Attachments:
maheu_paper.pdf (1.7 MB)
Package bmgarch (specifically to the DCCMGARCH.stan)
Package dirichletprocess
dirichletprocess_mvnormalgarchdpm.R (7.4 KB)