predictive distribution and can be achieved by making draws from the posterior distribution and from these values, making draws from the likelihood function, i.e.,

This distribution resembles the data closely for rotamer (3, 3, 3) but also forms a very reasonable distribution when there are only seven data points (3, 3, 1). A good posterior predictive distribution for any protein structural feature can be used in simulations of protein folding or structure prediction.

V.CONCLUSION

The field of statistics arose in the eighteenth and nineteenth centuries because of the need to develop good public policy based on demographic and economic data. Applications in the natural sciences were immediate, but generally natural scientists have lagged behind in their knowledge of modern statistics compared to social scientists. This is unfortunate, because many algorithms and methodologies have been developed in the last 20 years or so that make feasible sophisticated analysis of very complex data sets. Bayesian statistics has been used fruitfully in molecular and structural biology in recent years but has enjoyed more applications in genetics and clinical research and in the social sciences. Bayesian methods are particularly useful in modeling complex data, where the distribution of information may be uneven or hierarchical. This is true not only of the sequence and structure databases described in this chapter but also of more recently developed experimental methods such as DNA microarrays for analyzing mRNA expression levels over many thousands of genes [100–106]. The computational challenges for this kind of data are immense [107,108]. Particularly now, when the influx of data in biology is overwhelming, Bayesian statistical analysis promises to be an important tool.

ACKNOWLEDGMENTS

I thank Prof. Marc Sobel of Temple University for many useful discussions on Bayesian statistics. This work was funded in part by an appropriation from the Commonwealth of Pennsylvania and NIH Grant CA06927.

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