# Assign an error to the parameters of MAP estimate

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.

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:
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.