I'm trying to create some kind of iterative Bayesian algorithm, which continuously updates as more data is gathered. However, the distribution of my data is such that there does not exist a conjugate prior, so I'm thinking of using something like Gibbs sampling to generate my posterior distribution.
However, if I want to do another iteration of this Bayesian algorithm with new data, is it possible without a conjugate prior? Because I only have the samples of the posterior distribution, not the actual distribution.
Thanks!