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Pseudo-Bayes smoothing of cell frequencies 

        psbayes datavar [priorvar] [if exp] [in range] [ , by(rowvar [colvar
                 [layervar]]) generate(newvar) prob tabdisp_options ]


Description

    psbayes takes datavar, 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 set of data frequencies n_i, summing to n, and a set of prior
    probabilities q_i, the smoothed estimates are n * p_i, where

                 n   n_i       k
        p_i =  ----- ---  +  ----- q_i,
               n + k  n      n + k

    and shrinkage is tuned by the constant

              n²  - SUM ( n_i²)
        k = --------------------.
            SUM (n_i - n * q_i)²

    These estimates minimise the total mean square error between estimated
    and estimand probabilities. For more details, see the References.

    If priorvar is specified, it must sum to 1 for the data used. If priorvar
    is not specified, it is taken to be a set of equal probabilities.


Options 

    by(rowvar colvar layervar) indicates that datavar refers to a table with
        rows (and columns if specified (and layers if specified)) indexed by
        the variable(s) named, which will structure a display of cell
        estimates using tabdisp. If by() is not specified, cell estimates
        will be displayed according to observation numbers.

    generate(newvar) generates a new variable containing results.

    prob indicates that probabilities rather than estimated frequencies are
        to be shown (and if desired kept).

    tabdisp_options are options of tabdisp.  Default center format(%9.1f).


Examples

    . psbayes f prior, by(row col) g(sf)

    . contract foreign rep78, zero nomiss
    . psbayes _freq, by(foreign rep78) prob


Author

    Nicholas J. Cox, University of Durham, U.K.
    n.j.cox@durham.ac.uk


References 

    Agresti, A. 2002. Categorical data analysis. Hoboken, NJ: 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.


Also see

    On-line:  help for tabdisp