Bayesian structure

In what follows, a robust reliabiUty procedure is applied, and the terms of the reliability integral are characterized by means of a Bayesian structural identification approach (Papadimitriou et al. 2001). 

This paper is structured as follows in section 2, we recall the statement of the forward problem. We remind the numerical model which relates the contrast function with the observed data. Then, we compare the measurements performed with the experimental probe with predictive data which come from the model. This comparison is used, firstly, to validate the forward problem. In section 4, the solution of the associated inverse problem is described through a Bayesian approach. We derive, in particular, an appropriate criteria which must be optimized in order to reconstruct simulated flaws. Some results of flaw reconstructions from simulated data are presented. These results confirm the capability of the inversion method. The section 5 ends with giving some tasks we have already thought of. 

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. 

A common use of statistics in structural biology is as a tool for deriving predictive distributions of strucmral parameters based on sequence. The simplest of these are predictions of secondary structure and side-chain surface accessibility. Various algorithms that can learn from data and then make predictions have been used to predict secondary structure and surface accessibility, including ordinary statistics [79], infonnation theory [80], neural networks [81-86], and Bayesian methods [87-89]. A disadvantage of some neural network methods is that the parameters of the network sometimes have no physical meaning and are difficult to interpret. 

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... 

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. 

The Local Structure Operator By the Kolmogorov consistency theorem, we can use the Bayesian extension of Pn to define a measure on F. This measure -called the finite-block measure, /i f, where N denotes the order of the block probability function from which it is derived by Bayesian extension - is defined by assigning t.o each cylinder c Bj) = 5 G F cti = 6i, 0 2 = 62, , ( j — bj a value equal to the probability of its associated block ... 

We take a Bayesian approach to research process modeling, which encourages explicit statements about the prior degree of uncertainty, expressed as a probability distribution over possible outcomes. Simulation that builds in such uncertainty will be of a what-if nature, helping managers to explore different scenarios, to understand problem structure, and to see where the future is likely to be most sensitive to current choices, or indeed where outcomes are relatively indifferent to such choices. This determines where better information could best help improve decisions and how much to invest in internal research (research about process performance, and in particular, prediction reliability) that yields such information. 

The Bayesian network technology embedded in the ARBITER tool is also well suited for learning both probability relationships (e.g., method reliability estimates) and the essential structure of cause and effect, from data sets where predictions and outcomes can be compared. Colleagues have already applied this capability on a large scale for risk management (selection of potentially suspect claims for further inspection and examination) in the insurance industry. 

Simons KT, Kooperberg C, Huang E, Baker D. Assembly of protein tertiary structures from fragments with similar local sequences using simulated annealing and Bayesian scoring functions. J Mol Biol 1997 268 209-25. 

Within the multichannel Bayesian formalism of structure determination, it is indeed possible to make use ofMaxEnt distributions to model systems whose missing structure can be reasonably depicted as made of random independent scatterers. This requires that the structural information absent in the diffraction data be obtained from some other experimental or theoretical source. The known substructure can be described making use of a parametrised model. 

The general computational mechanism of Bayesian crystal structure determination in presence of various sources of partial phase information was first outlined by... 

In this section we briefly discuss an approximate formalism that allows incorporation of the experimental error variances in the constrained maximisation of the Bayesian score. The problem addressed here is the derivation of a likelihood function that not only gives the distribution of a structure factor amplitude as computed from the current structural model, but also takes into account the variance due to the experimental error. 

Under general hypotheses, the optimisation of the Bayesian score under the constraints of MaxEnt will require numerical integration of (29), in that no analytical solution exists for the integral. A Taylor expansion of Ao(R) around the maximum of the P(R) function could be used to compute an analytical expression for A and its first and second order derivatives, provided the spread of the A distribution is significantly larger than the one of the P(R) function, as measured by a 2. Unfortunately, for accurate charge density studies this requirement is not always fulfilled for many reflexions the structure factor variance Z2 appearing in Ao is comparable to or even smaller than the experimental error variance o2, because the deformation effects and the associated uncertainty are at the level of the noise. 

At each stage during the structure determination process, the current structural model gives an estimate of the prediction variance Z2 to be associated with the calculated amplitude. The contribution of the random part of the structure to this prediction variance decreases while the structure determination proceeds, and uncertainty is removed by the fit to the observations. Rescaling of Z2 would be needed during the optimisation of the Bayesian score. 

We have described in this paper the first implementation of this Bayesian approach to charge density studies, making joint use of structural models for the atomic cores substructure, and MaxEnt distributions of scatterers for the valence part. Used in this way, the MaxEnt method is safe and can usefully complement the traditional modelling based on finite multipolar expansions. This supports our initial proposal that accurate charge density studies should be viewed as the late stages of the structure determination process. 

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. 

Roversi, P., Irwin, J.J. and Bricogne, G. (1998) Accurate charge density studies as an extension of bayesian crystal structure determination, A54(6(2)), 971-996. 

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. 

A thorough study by Rosenberg et al. (4) examined human population structure using 377 markers in 1056 individuals from 52 populations around the world. Without prior information about the origins of individuals, these authors used a Bayesian algorithm to identify six major genetic clusters (1) sub-Saharan Africans ... 

Peterson has presented a Bayesian approach to defining the DS, which provides design space reliability as well, as it takes into account both model parameter uncertainty and the correlation structure of data. To aid in use of this approach, he proposes a means of organizing information about the process in a sortable spreadsheet, which can be used by manufacturing engineers to aid them in making informed process changes as needed, and continue to operate in the DS. 

Burden, F. R., and Winkler, D. A. (2000) A quantitative structure-activity relationships model for the acute toxicity of substituted benzenes to Tetrahymena pyriformis using Bayesian-regularized neural networks. Chem. Res. Toxicol. 13,436-440. 

Use Bayesian or non-Bayesian updating to infer the correlation structure. 

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