MAXIMUM-ENTROPY AND THE FOUNDATIONS OF DIRECT METHODS
MAXIMUM-ENTROPY AND THE FOUNDATIONS OF DIRECT METHODS
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DOI：
10.1107/s0108767384000866
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发表时间：
1984-01-01
影响因子：
1.8
通讯作者：
BRICOGNE, G
中科院分区：
文献类型：
--
作者：
BRICOGNE, G
本文对经典的统计定相方法进行了修正，拓宽了它们的理论基础，巩固了它们的实际应用，从而大大提高了它们的威力。在一个简短的介绍性调查（§ 1），直接方法的基本概念和数学技术进行了分析。更仔细的审查（§ 2）表明，严重的不足仍然损害这些方法的有效性。渐近性质的一系列用于近似联合分布的结构因子的要求，非常谨慎地行使，以保证其准确性，这一要求只能满足，如果他们被用于多解算法中，原子的先验分布不断更新，以便在每一个阶段，所有的相位信息假设到这一点。进一步的限制来自于传统的用单个不变量的边际分布的乘积来近似联合分布的做法。同时克服这两个困难的计划，然后提出。这个计划的关键要素是一个设备，基于Jaynes的最大熵原理，利用先验知识的一些结构因素的建设的联合分布的其他条件的知识。杰恩斯的最大熵形式主义在§ 3中被提出并系统地应用于原子的非均匀先验分布的构造。在§ 4中，通过Karle-Hauptman矩阵的“最大熵逆”的新技术解决了有效地近似大量结构因子的条件分布的问题，并且所获得的结果被证明是迄今为止获得的最复杂的概率公式的推广。这个过程在§ 5中被证明与通过Daniels鞍点近似的渐近展开的标准方法的增强一致。它与行列式方法的关系在§ 6中进行了研究。在§ 7中给出了实现这些想法的数值算法，以及对来自小蛋白Crambin的数据的应用，在§ 8中描述和讨论了从头开始使用它的统一策略。它的结论是，这里提出的相位确定策略将加快实现概率直接方法的全部潜力，并有可能使大分子结构在其范围内。
A revision of the classical statistical methods of phase determination is presented which widens their theoretical foundations and consolidates their practical implementation, thus bringing about a major increase of their power. In a brief introductory survey (§ 1), the basic concepts and mathematical techniques of direct methods are analysed. Closer scrutiny (§ 2) reveals that severe inadequacies still impair the effectiveness of these methods. The asymptotic character of the series used to approximate joint distributions of structure factors demands that great caution be exercised to guarantee their accuracy, and this requirement can only be fulfilled if they are used within a multisolution algorithm in which the prior distribution of atoms is constantly updated so as to incorporate at every stage all the phase information assumed to that point. Further limitations follow from the traditional practice of approximating joint distributions by products of marginal distributions of single invariants. A scheme for simultaneously overcoming both difficulties is then proposed. The pivotal element of this scheme is a device, based on Jaynes's maximum-entropy principle, for exploiting the prior knowledge of some structure factors in the construction of the joint distributions of others conditional to that knowledge. Jaynes's maximum-entropy formalism is presented and systematically applied to the construction of the requisite non-uniform prior distributions of atoms in § 3. The problem of effectively approximating conditional distributions of very large numbers of structure factors is solved in § 4 by a novel technique of 'maximum-entropy inversion' of Karle-Hauptman matrices, and the result obtained is shown to generalize the most sophisticated probabilistic formulae hitherto obtained. This procedure is proved in § 5 to coincide with an enhancement of the standard method of asymptotic expansions by means of Daniels's saddlepoint approximation. Its relationship to determinantal methods is investigated in § 6. A numerical algorithm for implementing these ideas is presented in § 7, together with an application to data from the small protein Crambin, and a unified strategy for its use ab initio is described and discussed in § 8. It is concluded that the phase-determination strategy proposed here will expedite the realization of the full potential of probabilistic direct methods, and is likely to bring macromolecular structures within their reach.