DIST-EST:  Estimate the expectation of some function of state.

Dist-est reads data from one or more log files and uses it to estimate
the expectation of a function of the state variables.  If annealed
importance sampling was used, the estimate accounts for the differing
importance weights.


    dist-est formula [ temp-index ] { log-file [ range ] }

Estimates the expectation of the function of state specified by the
formula given as the first argument.  This formula may refer to state
variables, and to other variables defined in the specification given
to dist-spec.  The expectation is normally with respect to the
distribution at inverse temperature one, but if Annealed Importance
Sampling or simulated tempering was done, an index into the tempering
schedule may be given in order to produce an estimate with respect to
one of the other distributions in the schedule.

The data comes from one or more log files, at iterations within the
specified ranges.  The ranges have the form "[low][:[high]]][%mod]".
The low bound defaults to one.  If no high bound is given, the range
extends to the highest index in the log file.  If the "%mod" option is
present, only iterations within the range whose index is a multiple of
"mod" are used (e.g. "5:12%3" is equivalent to 6 9 12).  If no range
is given, the default is "1:".

The output gives the estimated mean of the function, along with the
number points used and the standard error for the estimate.  If
annealed importance sampling was used, the mean of the importance
weights is also reported, along with the adjusted sample size
(dividing by one plus the variance of the normalized weights), and the
effective sample size for the particular function whose expectation is
being estimated.

Note: The standard errors reported assume the points were generated
independently, which will not generally be true unless annealed
importance sampling was done (with independent starting points).

            Copyright (c) 1998 by Radford M. Neal