I have noisy data points, where the peak signal-to-noise ratio (PSNR) may sometimes be less than unity (hence, more noise than signal may be present). I am fitting a model with fitting parameters to this noisy data, using MCMC (Markov Chain Monte Carlo) methods. I want to know if using a noise filter on the noisy data points (such as a Wiener filter in real space or a bandpass filter in Fourier space), before doing the MCMC fitting, would cause the 90% HPDI contour (highest posterior density interval) of the joint posterior probability distribution of the fitting parameters to be tighter or wider (precision), and closer or farther away from the true parameter values (accuracy)?
As Ray Kidd mentioned, filtering data is futile. First, the noise is part of the data. In some cases noise can be informative. Filtering can not increase the information content of the data. The information content is an unalterable state of nature. Second, if the filter happens to be inappropriate, the parameter estimates can be meaningless. Third, sometimes the data information content is too low to compute meaningful parameter estimates. Using a filter tells you nothing about parameter estimate uncertainties.
One approach is to include a model for the noise in the parameter estimate model. If you know a lot about the noise, include all that information in the model for the data instead of using a data filter. Never include ad-hoc, indefensible assumptions about the noise (or the parameters).
I noticed Bayesian terms in your keywords. Under no circumstance should you filter the data with Bayesian methods. Bayesian methods include one or more models for the noise in the model for the data. This means there will be well-designed, prior probability, distributions for the noise. Objective, prior probability distributions consistent with maximum entropy principles are the best one can do.
The width of the posterior probability distributions for the parameters will tell you if the data information content is simply too low to compute meaningful parameter estimates.This can be explored by adding noise to simulated data (or empirical data). At some point the signal-to-noise ratio will no longer support meaningful parameter estimates.
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