Pseudo-Bayes smoothing of cell estimates (matrix version) ---------------------------------------------------------
^psbayesm^ data_matrix [prior_matrix] [ ^, m^atrix^(^matname^) p^rob ^f^ormat^(^%fmt^)^
Description -----------
^psbayes^ takes data_matrix, which should be a set of frequencies, and shrinks or smooths it towards a set of frequencies implied by prior probabilities. This will have the effect of replacing sampling zeros by positive estimates whenever the priors are positive.
For a matrix of data frequencies n_ij, summing to n, and a matrix of prior probabilities q_ij, the smoothed estimates are n * p_ij, where
n n_ij k p_ij = ----- ---- + ----- q_ij, n + k n n + k
and shrinkage is tuned by the constant
2 2 n - sum ( n_ij ) k = -----------------------. 2 sum (n_ij - n * q_ij )
These estimates minimise the total mean square error between estimated and estimand probabilities. For more details, see the References.
If prior_matrix is specified, it must sum to 1. If prior_matrix is not specified, it is taken to be a matrix of equal probabilities.
Remarks -------
The data matrix may be entered directly into Stata or it may be produced by a previous command, such as ^tabulate^.
Options -------
^matrix(^matname^)^ specifies that results are to be placed in matrix matname.
^prob^ indicates that probabilities rather than estimated frequencies are to be shown (and if desired kept).
^format(^%fmt^)^ controls the format with which matrix output is printed. Default ^format(%9.1f)^.
Example -------
. ^psbayesm f^
References ----------
Agresti, A. 1990. Categorical data analysis. New York: John Wiley.
Bishop, Y.M.M., Fienberg, S.E. and Holland, P.W. 1975. Discrete multivariate analysis. Cambridge, MA: MIT Press.
Fienberg, S.E. and Holland, P.W. 1970. Methods for eliminating zero counts in contingency tables. In Patil, G.P. (ed.) Random counts in scientific work. Volume 1: Random counts in models and structures. Pennsylvania State University Press, University Park, 233-260.
Fienberg, S.E. and Holland, P.W. 1972. On the choice of flattening constants for estimating multinomial probabilities. Journal of Multivariate Analysis 2, 127-134.
Fienberg, S.E. and Holland, P.W. 1973. Simultaneous estimation of multinomial cell probabilities. Journal, American Statistical Association 68, 683-691.
Good, I.J. 1965. The estimation of probabilities: an essay on modern Bayesian methods. MIT Press, Cambridge, MA.
Sutherland, M., Holland, P.W. and Fienberg, S.E. 1975. Combining Bayes and frequency approaches to estimate a multinomial parameter. In Fienberg, S.E. and Zellner, A. (eds) Studies in Bayesian econometrics and statistics in honor of Leonard J. Savage. North-Holland, Amsterdam, 585-617.
Author ------
Nicholas J. Cox, University of Durham, U.K. n.j.cox@@durham.ac.uk
Also see --------
On-line: help for @psbayes@