In fact, applying the Metropolis-within-Gibbs code to the relatively simple hierarchical problem we have devised to infer properties about the magnetic field in central stars of planetary nebulae was a success. Success means that the marginal posteriors are similar to the analytical expressions we have obtained.
The model is quite simple. We use the observed Stokes V and Stokes I profiles to infer the magnetic field. We model the circular polarization of planetary nebulae $i$ as
$$V_i=\alpha B_i \mu_i \frac{dI}{d\lambda}$$
The statistical model that we propose is the following, once all the observations we have have been taken into account:
$$V_i \sim N(\alpha B_i \mu_i dI/d\lambda, \sigma^2)$$
$$B_i \sim Maxwell(b_0)$$
$$\mu_i \sim U(-1,1)$$
$$b_0 \sim Gamma(\nu,\nu)$$
This is a hierarchical model in which we have assumed that the magnetic field strength is described by a Maxwell distribution with a scale $b_0$, over which we put a Gamma prior (which, for very small $\nu$ is equivalent to a Jeffreys prior).
The model is quite simple. We use the observed Stokes V and Stokes I profiles to infer the magnetic field. We model the circular polarization of planetary nebulae $i$ as
$$V_i=\alpha B_i \mu_i \frac{dI}{d\lambda}$$
The statistical model that we propose is the following, once all the observations we have have been taken into account:
$$V_i \sim N(\alpha B_i \mu_i dI/d\lambda, \sigma^2)$$
$$B_i \sim Maxwell(b_0)$$
$$\mu_i \sim U(-1,1)$$
$$b_0 \sim Gamma(\nu,\nu)$$
This is a hierarchical model in which we have assumed that the magnetic field strength is described by a Maxwell distribution with a scale $b_0$, over which we put a Gamma prior (which, for very small $\nu$ is equivalent to a Jeffreys prior).
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