Bayesian estimation of tridiagonal covariance I want to estimate covariance of a multivariate normal distribution from data using a Bayesian method. I want to force the result to be tridiagonal. I am looking for an appropriate prior or method. Is there any prior or method for this?
 A: Some notation: let $\mathbf{Y}$ be a random vector of length $m$ with a $MVN(\mu, \Sigma)$ distribution. For notational simplicity, I'll assume $m$ is even, but nothing hangs on this.
First off, the implications of the tridiagonal covariance: from the properties of the marginals of multivariate normal distributions, a tridiagonal covariance implies that $[Y_1 \,\,\, Y_3 \,\,\, ... Y_{m-1}]^T$ is a vector of independent normal random variables, and likewise for $[Y_2 \,\,\, Y_4 \,\,\, ... Y_{m}]^T$. Assuming this makes sense, and assuming that the prior on $\mu$ is flat, then the link I posted in comments gives the posterior distribution; with appropriate modifications for the fact that your problem has no covariates, we get, let's see...
$$\log\big(p(\Sigma|\mathbf{Y}_1,...,\mathbf{Y}_n)\big) = \log\big(p(\Sigma)\big) -\frac{n-1}{2}\log\big(|\Sigma|\big) -\frac{1}{2}\sum_{i=1}^n(\mathbf{Y}_i-\bar{\mathbf{Y}})^T\Sigma^{-1}(\mathbf{Y}_i-\bar{\mathbf{Y}})\bigg].$$
And ignoring the log-prior term, that, if I haven't gone crazy, is the the kernel of the inverse-Wishart density. Of course it isn't the inverse-Wishart density because, by definition, it's only supported on the set of symmetric tridiagonal matrices. 
As for the prior: your choice of a tridiagonal covariance matrix is very convenient in the sense that the requirement of positive definiteness is trivial to satify: any set of  (marginal) variances $\{\sigma_1^2\, \sigma_2^2, ..., \sigma_m^2\},\, \mathrm{all}\,\, \sigma_j^2 > 0,$ and correlations $\{\rho_1, \rho_2, ..., \rho_{m-2} \},\,\mathrm{all}\,\,\rho_j \in [-1,1]$ will do the job. So it's safe to give the variances independent half-Cauchy priors and the correlations uniform priors. Try that first and see what you get.
If you need more help with the math and/or the implementation of posterior sampling, just tell me in a comment and I'll go into more detail.
