Hello I have a question about the accuracy in Bayesian updating. I use the following procedure to compute a posterior distribution:
- Generate synthetic measurement data, i.e. I know the true mean and all about the added noise distribution (number of observations, st. dev, mean, Gaussian)
- Use Bayes theorem and a Metropolis Hastings to compute a posterior distribution.
- Convergence test (autocorrelation, acceptance rate, Monte Carlo standard error)
Now the problem appears, that the true mean is usually not a member of my posterior (however, it is close to the mean)
So I have the following questions:
Is there a formula, that shows the effect of the number of observations and the st. dev of the noise distribution and the accuracy?
Is it true, that a particle filter (recursive Bayesian) \begin{equation} p(\theta^n \vert \mathbf{y}^n) = \frac{p(y^n \vert \theta^n) p(\theta^n \vert \mathbf{y}^{n-1})}{p(y^n \vert \mathbf{y}^{n-1})} \end{equation} with the unknown parameter $\theta$ and the data $\mathbf{y}$, is able to filter out the noise error completely? How can I see this and is this also possible in a time independent problem (so number of repeatings interpreted as time)?