In Bayesian statistics, you have a likelihood and a prior, $f(x_1,\ldots,x_n \mid \theta)$ and $\pi(\theta)$ respectively, and you use these to obtain the posterior $\pi(\theta \mid x_1, \ldots, x_n) \propto f(x_1,\ldots,x_n \mid \theta) \pi(\theta)$. Occasionally there are situations where a) the data arrive very quickly, and b) using Markov chain Monte Carlo techniques to simulate from the posterior is difficult/slow.

It seems the only option is to use an "outdated" posterior. Say you were interested in the posterior predictive distribution's mean, and you have data up to time $n$. However, the last time you finshed an MCMC sampling was at time $m < n$. You could use something like \begin{align*} \int x_{n+1} f(x_{n+1} \mid x_1, \ldots, x_n) \text{d}x_{n_1} &= \iint x_{n+1} f(x_{n+1} \mid \theta) \pi(\theta \mid x_1, \ldots, x_n)\text{d}x_{n+1}\text{d}\theta\\ &\approx \iint x_{n+1} f(x_{n+1} \mid \theta) \pi(\theta \mid x_1, \ldots, x_m)\text{d}x_{n+1}\text{d}\theta. \end{align*}

What is this approximation called? What are some references where people address the issue of quantifying the error in this approximation?

  • $\begingroup$ I do not know if there is a name for this approximation. It is related to scalable Bayesian inference. For which there exist computational solutions to keep the approximation within a Bayesian framework. $\endgroup$ – Xi'an Mar 22 '18 at 11:02
  • $\begingroup$ Thanks, @Xi'an. The search phrase "scalable Bayesian inference" has given me plenty to look through $\endgroup$ – Taylor Mar 22 '18 at 18:18

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