MIX-QUANTITIES:  Quantities from log files relating to mixture models.

The quantities below relating the mixture models can be obtained from
log files (eg, for use in mix-plt).  The generic quantities documented
in quantities.doc are also available.  The Markov chain quantities
documented in mc-quantities.doc are defined as well, but at present
most of them are not meaningful for mixture models, since the standard
Markov chain operations are not supported.

The quantities specific to mixture models are as follows (if "n" is
present after the letter, it represents a numeric modifier):

    u    An array containing the mean offset for each target value.

    un   An array containing the offset parameters for component n,
         numbered (starting at 1) in order of decreasing frequency
         in the training set.  If n is greater than the number of
         components currently represented in the training set, the
         value of un is the same as the mean offset (quantity u).

    V    As an array, the variances of offsets for each target value.
         As a scalar, the top-level variance hyperparameter that 
         controls these variances.

    v    Same as for V, but expressed in terms of standard deviations
         rather than variances.

    N    As an array, the hyperparameters controlling the variances 
         for each of the (real) target values in the components of the
         mixture. As a scalar, the top-level variance hyperparameter
         that controls these variances.  Valid only when the targets 
         are real-valued.  

    n    Same as for N, but expressed in terms of standard deviations
         rather than variances.

    Nn   Array of noise variances for the n'th component, or array of
         noise-variance hyperparameters if n is greater than the number
         of components currently active in the training set.

    nn   As for Nn, but in terms of standard deviations.

    Cn   The fraction of training cases that are associated with the 
         first n mixture components.  The components are ordered according    
         to the number of training cases with which they are associated 
         (most frequent first).

    cn   The frequency of the n'th most common mixture component among the 
         training cases.  The components are ordered according to the number 
         of training cases with which they are associated (most frequent 
         first).  The value is zero if there are fewer than n active 
         components at present.

    a[n] The number of components needed to account for all but n% of the
         training cases.  The default is n=0 - ie, just "a" is the total
         number of currently active components.

            Copyright (c) 1997 by Radford M. Neal