MCMC for Bayesian Inference (Gibbs Sampling) Varying Observed Data

At every step $k$, a Markov chain Monte Carlo algorithm for Bayesian inference with Gibbs sampling draws a parameter of the model to fit, $\beta_i^{(k)}$, from the conditional $Pr\left(\beta_i^{(k)}|\beta_1^{(k)},...,\beta_{i-1}^{(k)},\beta_{i+1}^{(k-1)},...,\beta_N^{(k-1)},X\right)$, where $X$ is the observed data.

Do you know of some paper exploring this algorithm (or similia) where the data $X$ also vary along the chain? I do not mean at every step, just sometimes along the chain; imagine e.g. some financial transactions which happen almost continuously, and whose model always has to adapt.

Thanks!

• I do not think the data can change during the MCMC algorithm since you sample from $\beta_1, ... \beta_n | X$, then conditioning on the observed set X. If you mean that once you have estimated the model conditioning to $X$ you get another observation, say $X^*$ and you want to find the new posterior distribution of $\beta_1,... \beta_n | X, X^*$ without re-running all the chain, then you probably want to look to the particle filtering: en.wikipedia.org/wiki/Particle_filter maybe if you can explain better you problem i can help you with a more detailed answer. – niandra82 Jul 10 '14 at 11:24
• Thank you for your reply, @niandra82, and sorry for my late reply. I'll edit my question. – Pippo Jul 11 '14 at 20:06
• Dear @niandra82, I decided to open a new question given the different problem I would like to explore: stats.stackexchange.com/questions/107656/… – Pippo Jul 11 '14 at 20:26