I am trying to calculate a posterior predictive distribution for the magnitude (Euclidean norm) of a 3D displacement vector. Displacement in each dimension is independent and normally distributed (but each may have different means and variances).
$$ X_i \sim N(\mu_i,\sigma_i^2); (i = 1:3) $$
$$ D = \sqrt{\sum\limits_{i=1}^3 X_i^2} $$
There is plenty of literature on how to deal with normally distributed data with both frequentist and bayesian approaches.
However the magnitude isn't normally distributed, and I believe it should follow a noncentral chi distribution. This is defined by number of degrees of freedom k = 3, and the lambda parameter, which combines the ratio of the mean and standard deviation for each dimension (en.wikipedia.org/wiki/Noncentral_chi_distribution).
$$ \lambda = \sqrt{\sum\limits_{i=1}^k \left(\frac{\mu_i}{\sigma_i}\right)^2} $$
I don't know any population parameters - but I have informative priors on the parameters governing displacement in each dimension, and would like to use them:
$$ \mu_i \sim N(m_i,a_i^2) $$ $$ \sigma_i^2 \sim N(s_i^2,b_i^2) $$ Where $m_i$ and $s_i^2$ are my best guesses for the mean and variances of data in each displacement dimension, and $a_i^2$ and $b_i^2$ are hyperparameters quantifying my uncertainty in these beliefs.
How can I convert prior information on the gaussian displacement parameters into a prior on the lambda parameter of the non-central chi distribution?
$$ \lambda \sim f(m_i,a_i^2,s_i^2,b_i^2)? $$
Next I will combine this prior with observed data (either in each dimension, or the magnitude directly) to obtain a posterior distribution for lambda. I will then integrate over all lambda to get my posterior predictive distribution for the magnitude. I think this is a sensible approach, but I need a decent prior first.