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Through a MCMC Gibbs sampler I obtain $M$ chains of the parameters vector $\mathbf{\theta}$, meaning that each component of $\mathbf{\theta}$ is the value of one parameter at a given iteration.

Since I know the likelihood and the prior I found the maximum a posteriori estimate:

$$\hat{\mathbf{\theta}}=\underset{\mathbf{\theta}^{(1)},\,...,\,\mathbf{\theta}^{(M)}}{\operatorname{argmax}}\,p(\mathbf{\theta}),$$

I'd like to have an error on each component of $\hat{\mathbf{\theta}}$.

One other estimate of the best parameters is the mode or the mean of MCMC chains for each parameter, for which I can consider the standard deviation as an error, but this is not what I'm interested in.

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Since we are treating $\theta$ as vector of random variables i think the term error is thematically wrong, indeed one of the strength of Bayesian methods is the richness of information granted about $\theta$ through it's posterior distribution. For MCMC sampling methods (of which the Gibbs sampler is one kind) the posterior distribution of paramters $\theta$ can generally be obtained in this manner:

  1. Let the chains converge to stationary (if it's possible for the problem) - generally known as the burn-in period.
  2. Keep every x sample of $\theta$. Since the chain is correlated in time we only keep every xth value, this concept is known as thinning. The value of x is generally a trade-off between time and accuracy.
  3. The kept samples of $\theta$ describes it's posterior distribution for which any moment can be calculated if so desired.
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  • $\begingroup$ Thank you for your answer, kept samples of θ they are the marginalized posterior for each parameter. With maximum a posteriori estimate, from those generated, I take the set of parameter more likely given the model and data. No moment of its marginalized posterior is reasonable as error of one of these parameters most probable $\endgroup$ – Johnpiton Sep 27 '19 at 18:44

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