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bayes

set up for Bayesian inference

Syntax: bayes <option>
  bayes <mod par #> <prior type> [<hyperparameters>]
  bayes smooth <par> [order 1|2] [space lin|log] [lambda <value> | lcurve | discrep] [radii <r$_1$>... <r$_N$>]
  bayes smooth clear

where <option> ::= [off |on |cons]. If a parameter number is given as the first argument then this command sets up the prior for the specified model parameter but does not turn Bayesian inference on. If the first argument to the bayes command is not a parameter number then one of the options off, on, or cons is used. The first two turn Bayesian inference off or on, while cons turns Bayesian inference on and gives all parameters a constant prior. The options for prior types are as follows:

Prior type Log(prior)      
cons 0      
exp $-p/h_1-\ln{(h_1)}$      
jeffreys $-\ln{(p)}$      
gauss $-0.5\ln{(2\pi h_2)}-0.5(h_1-p)^{2}/h_2^{2}$      
lognormal $-\ln{(p \sqrt{2\pi} h_2)} -0.5(\ln{(p)}-\ln{(h_1)})^{2}/h_2^{2}$      
gamma $h_1\ln{(h_2)} + (h_1-1) \ln{p} - h_2 p - \ln{\Gamma(h_1)}$      
cauchy $-\ln{(1+(p-h_1)^2/h_2^2)} - \ln{\pi} - \ln{h_2}$      
beta $(h_1-1)\ln{x} + (h_2-1)\ln{(1-x)} - \ln{\Gamma(h_1+h_2)} - \ln{\Gamma(h_1)} - \ln{\Gamma(h_2)}$      
studentst $\ln{\Gamma((h_1+1)/2)} - 0.5\ln{(h_1\pi)} - \ln{\Gamma(h_1/2)} - ((h_1+2)/2)\ln{(1+p^2/h_1)}$      
halfnormal $-0.5\ln{(h_1^2\pi/2)} - p^2/(2h_1^2)$      
powerlaw $-h_1\ln{(p)} - \ln{Z}$ for $h_2 \le p \le h_3$, where the prior density is $p^{-h_1}/Z$ and $Z = (h_3^{1-h_1}-h_2^{1-h_1})/(1-h_1)$, or $\ln{(h_3/h_2)}$ when $h_1=1$      
isotropic $\ln{(\sin{(p)})} + \ln{(\pi/360)}$, with p in degrees over $[0,180]$      
shiftedlognormal $-\ln{((p-h_1) \sqrt{2\pi} h_3)} -0.5(\ln{(p-h_1)}-\ln{(h_2)})^{2}/h_3^{2}$      

Where p is the parameter value, h$_{\char93 }$ the hyperparameter values, and for the beta prior x = (p-min)/(max-min) where min and max are the parameter hard limits.

Multi-parameter smoothness prior.    The priors above act on one parameter at a time. The bayes smooth subcommand instead applies a single prior that couples a chosen model parameter <par>(its local index within the model) across all datagroups, penalising shell-to-shell roughness of that parameter. It is the in-engine regularizer for projct de-projection, where the per-datagroup normalisations (or temperatures) recovered by inverting the projection are noisy and anti-correlated; see the projct model description for the underlying failure modes. The prior is a correlated Gaussian

$\displaystyle \ln p = -\tfrac{1}{2}\,\lambda\,\lVert L\,q\rVert^{2}, \qquad q_k = p_k\ (\texttt{lin})\ $   or$\displaystyle \ \ \ln p_k\ (\texttt{log}),$    

where the sum runs over the datagroups, L is a spacing-aware finite-difference operator built from the shell radii, and $\lambda$ is the regularisation strength. The options are:
order 1|2
1 penalises the gradient of the profile (biasing towards a flat profile); 2 penalises the curvature (biasing towards a power-law-like profile). Default 2.
space lin|log
whether the operator acts on the parameter (lin) or its logarithm (log). Default log, appropriate for normalisations that vary over orders of magnitude.
lambda <value> | lcurve | discrep
a fixed strength <value>; or lcurve to select it automatically at the corner of the misfit-versus-roughness L-curve; or discrep to select it by the discrepancy principle (increase the smoothing until the data misfit reaches the number of degrees of freedom). The lcurve and discrep options refit the model over a grid of $\lambda$ and leave the model fitted at the chosen value.
radii <r$_1$> ... <r$_N$>
the shell radii, one per datagroup. If omitted they default to the XFLT major axis of each datagroup (the projct convention), so cluster users need not re-enter them.
The smoothness term is consumed by the same machinery as the per-parameter priors, so it regularises Levenberg–Marquardt fits, the error calculation, and all four samplers (under nest it is sampled through a spectral transform of its rank-deficient precision). The members must be free, unlinked, and free of a per-parameter prior. bayes smooth clear removes all smoothness terms (as does bayes cons). Registered terms are written to save files and listed by show bayes.
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