We have the following model: $$y_t=Mx_t+\epsilon_t$$ with $M$ being a matrix such that $M\sim F_{\lambda}$(assume it's a conjugate prior). The $\lambda$ does not appear in $M$, only in its distribution.

The problem is that I would like the data to determine $\lambda$, not the user. So, I'm thinking of using a Metropolis-Hastings algorithm. In this case, I'm thinking of assuming $\lambda \sim F$, and then usually I would use the likelihood and from its gradient get a mean for the proposal, i.e. the new lambda, $\lambda^*$, would be drawn from the following proposal density $q$:

$$q(\lambda^*\mid \mu^*,\Sigma)$$

where $\mu^*=\mu+\frac{\partial}{\partial \lambda}p(Y\mid X, \lambda)$ and $\Sigma$ is a scale matrix.

However, since the likelihood doesn't depend on the $\lambda$, I was thinking of using $p(M\mid \lambda)$ instead and do the gradient, i.e.

$$\mu^*=\mu+\frac{\partial}{\partial \lambda}p(M\mid \lambda)$$

I think in this case I'm making $\lambda$ behave as an hyperparameter.

Is my plan acceptable?


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