PRIOR: Form of hierarchical prior specifications. Priors for parameters such as network weights, biases, and offsets, or for the noise levels in a regression model are specified using a common syntax, described here. The prior for a group of parameters is hierarchical. At the lowest level, each parameter is picked either from a Gaussian distribution with mean zero and with some precision (inverse of the variance), or from a two point distribution concentrated at the corresponding plus or minus standard deviation points (the latter is meant primarily for debugging). At the next level, the precision is picked from a Gamma distribution with a specified shape parameter and with a mean given by a hyperparameter common to all parameters of the same sub-group (eg, weights out of the same unit). This common value for the precision of parameters in a sub-group is in turn picked from a Gamma distribution with another specified shape parameter and with a mean given by a hyperparameter common to all parameters in the group. Finally, this top-level hyperparameter is picked from a Gamma distribution with a specified mean, and with yet another specified shape parameter. Priors for groups without sub-groups (eg, network biases and offsets) are similar, but go down only two levels. These priors are specified using the following syntax: [x]Width[:[Alpha-group][:[Alpha-sub-group][:[Alpha-parameter]]]][!] The Width part of the specification is used to specify the mean of the precision at the top level, as described below. The Alpha parts gives the shape parameters of the Gamma distributions. Alpha-group is used when picking the common precision for all parameters in the group, Alpha-sub-group is used when picking the precision common to all parameters in one of the sub-groups, and Alpha-parameter is used when picking the precision for a single parameter. If an Alpha is omitted, it is taken to be infinite, giving a distribution for the precision concentrated at the mean. A Gamma distribution with given mean and Alpha has a density proportional to p^{Alpha/2-1} * exp(-p*Alpha/2*mean), where p is the precision (always positive). Width specifies the top-level mean for the precision as follows. If an "x" is not present, the mean precision is 1/Width^2. If the prior does start with "x", which is meaningful only for network weights, the mean precision is determined as follows: Alpha infinite: N / Width^2 Alpha > 2: N * (Alpha/(Alpha-2)) / Width^2 Alpha = 2: N * log(N) / Width^2 Alpha < 2: N^(2/Alpha) / Width^2 Here N is the number of like weights going into a single unit, i.e. the number of units in a source layer for this type of weight. Alpha is Alpha-parameter, if that is finite, and is otherwise Alpha-sub-group. This scheme is intended to give proper scaling behaviour as N goes to infinity (but it may not work when both Alpha-sub-group and Alpha-parameter are finite). For Alpha >= 2, convergence is to a Gaussian process, for Alpha < 2, to a stable process of index Alpha. If the prior specification ends with "!", the lowest-level distribution is concentrated at the two standard deviation points, rather than being Gaussian. This is primarily a debugging tool - it makes it easy to see whether each parameter was derived from the correct hyperparameter. A prior specification for the level of Gaussian noise in a regression model has the same syntax as that used to specify priors for model parameters. The Width part gives the mean precision (inverse of the noise variance) at the top of the hierarchical prior. Alpha-group gives the shape parameter for picking a precision with this mean that is common to all outputs. Alpha-sub-group gives the shape parameter for picking precisions for each target with the common precision as mean. Alpha-paramerer gives the shape parameter for picking a precision for a particular output in a particular case, using that targets's precision as the mean. Specifying a finite value for Alpha-parameter has the effect of changing the distribution of the noise for a particular output in a particular case from a Gaussian to a t distribution. In network models, the distributions for weights may be modified by an "adjustment" sigma value associated with the destination unit for the weight. This adjustment multiplies the sigma for that weight which would otherwise apply. The precisions for the adjustments themselves are drawn from a Gamma distribution with mean one and shape parameter given by an alpha associated with the unit's layer. Note that hyperparameters are generally input and displayed in terms of the square root of the inverse of the precision (a 'sigma' value), not in terms of the precisions themselves, even though those are the values in terms of which the priors are mathematically expressed. Copyright (c) 1995, 1996 by Radford M. Neal