Statistics Bayesian

The complexity of infonnation that can be incorporated into the model gives Bayesian statistics much of its power. 

Mixmre models have come up frequently in Bayesian statistical analysis in molecular and structural biology [16,28] as described below, so a description is useful here. Mixture models can be used when simple forms such as the exponential or Dirichlet function alone do not describe the data well. This is usually the case for a multimodal data distribution (as might be evident from a histogram of the data), when clearly a single Gaussian function will not suffice. A mixture is a sum of simple forms for the likelihood ... 

There is some confusion in using Bayes rule on what are sometimes called explanatory variables. As an example, we can try to use Bayesian statistics to derive the probabilities of each secondary structure type for each amino acid type, that is p( x r), where J. is a, P, or Y (for coil) secondary strucmres and r is one of the 20 amino acids. It is tempting to writep( x r) = p(r x)p( x)lp(r) using Bayes rule. This expression is, of course, correct and can be used on PDB data to relate these probabilities. But this is not Bayesian statistics, which relate parameters that represent underlying properties with (limited) data that are manifestations of those parameters in some way. In this case, the parameters we are after are 0 i(r) = p( x r). The data from the PDB are in the form of counts for y i(r), the number of amino acids of type r in the PDB that have secondary structure J.. There are 60 such numbers (20 amino acid types X 3 secondary structure types). We then have for each amino acid type a Bayesian expression for the posterior distribution for the values of xiiry. 

For the second term, we use the same trick Leave n on the right-hand side of the vertical and use the Bayesian statistics equation to invert C and a ... 

Unfortunately, some authors describing their work as Bayesian inference or Bayesian statistics have not, in fact, used Bayesian statistics rather, they used Bayes rule to calculate various probabilities of one observed variable conditional upon another. Their work turns out to comprise derivations of informative prior distributions, usually of the form piQi, 02,..., 0 1 = which is interpreted as the posterior distribution... 

For example, Stolorz et al. [88] derived a Bayesian formalism for secondary structure prediction, although their method does not use Bayesian statistics. They attempt to find an expression for / ( j. seq) = / (seq j.)/7( j.)//7(seq), where J. is the secondary structure at the middle position of seq, a sequence window of prescribed length. As described earlier in Section II, this is a use of Bayes rule but is not Bayesian statistics, which depends on the equation p(Q y) = p(y Q)p(Q)lp(y), where y is data that connect the parameters in some way to observables. The data are not sequences alone but the combination of sequence and secondary structure that can be culled from the PDB. The parameters we are after are the probabilities of each secondary structure type as a function of the sequence in the sequence window, based on PDB data. The sequence can be thought of as an explanatory variable. That is, we are looking for... 

A similar formalism is used by Thompson and Goldstein [90] to predict residue accessibilities. What they derive would be a very useful prior distribution based on multiplying out independent probabilities to which data could be added to form a Bayesian posterior distribution. The work of Arnold et al. [87] is also not Bayesian statistics but rather the calculation of conditional distributions based on the simple counting argument that p(G r) = p(a, r)lp(r), where a is some property of interest (secondary structure, accessibility) and r is the amino acid type or some property of the amino acid type (hydro-phobicity) or of an amino acid segment (helical moment, etc). 

Analysis and prediction of side-chain conformation have long been predicated on statistical analysis of data from protein structures. Early rotamer libraries [91-93] ignored backbone conformation and instead gave the proportions of side-chain rotamers for each of the 18 amino acids with side-chain dihedral degrees of freedom. In recent years, it has become possible to take account of the effect of the backbone conformation on the distribution of side-chain rotamers [28,94-96]. McGregor et al. [94] and Schrauber et al. [97] produced rotamer libraries based on secondary structure. Dunbrack and Karplus [95] instead examined the variation in rotamer distributions as a function of the backbone dihedrals ( ) and V /, later providing conformational analysis to justify this choice [96]. Dunbrack and Cohen [28] extended the analysis of protein side-chain conformation by using Bayesian statistics to derive the full backbone-dependent rotamer libraries at all... 

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. 

JS Shoemaker, IS Painter, BS Weir. Bayesian statistics in genetics A guide for the uninitiated. Trends Genet 15 354-358, 1999. 

GW Carter Ir. Entropy, likelihood and phase determination. Structure 3 147-150, 1995. RL Dunbrack Ir, EE Cohen. Bayesian statistical analysis of protein sidecham rotamer preferences. Protein Sci 6 1661-1681, 1997. 

DV Lindley. The 1988 Wald Memorial Lecture The present position of Bayesian statistics. Stat Sci 5 44-89, 1990. 

MJ Thompson, RA Goldstein. Predicting solvent accessibility Higher accuracy using Bayesian statistics and optimized residue substitution classes. Proteins Struct Funct Genet 25 38-47, 1996. 

In the framework of Bayesian statistics, this can be done by maximising the posterior probability of the Lagrange multipliers defining the distribution [51] Bayes s... 

Gilmore, J.C. (1996) Maximum-entropy and Bayesian statistics in crystallography a review ofpractical applications, Acta Cryst., A52, 561-589. 

Bricogne, G. (1988) A Bayesian statistical theory ofthe phase problem. I. A multichannel maximum-entropy formalism for constructing generalized joint probability distributions of structure factors, Acta Cryst., A44, 517-545. 

Bricogne. G. (1997) The Bayesian statistical viewpoint on structure determination basic concepts and examples, In Macromolecular Crystallography, Vol. 276 of Methods in Enzymology, Carter Jr., C.W. and Sweet, R.M. (Eds.), Academic Press, pp. 361 123. 

Lindley, S.V (1971). Bayesian Statistics A Review. SIAM, Philadelphia. 

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