Maximum entropy theory

Abstract

A maximum entropy method (MEM) map of electron density is obtained by maximizing S=-~p log p subject to whatever data are available. MEM is derived as the only reconstruction technique that is consistent with simple and general requirements. The method is very widely applicable, but, in this paper, attention is focused on the problem of producing electron density maps in crystallography. The entropy formula can also be derived by analogy with a thermodynamic system of quanta, but it is shown that this model can be misleading, and can break down in practice. MEM applied to a different problem related to quantum fluctuations in the thermodynamic model is shown to lead to formulae equivalent to the maximum determinant method. It is argued that direct MEM will produce superior maps. AbstractTwo independent determinations of the same structure may be compared by means of statistical techniques such as normal probability plots and X 2 hypothesis tests. Computer simulations show that errors may arise in the application of these techniques 0108-7673/85/020122-07501.50 if rounded estimates of structural parameters and their e.s.d.s are used in the calculations. Round-off errors are particularly serious in goodness-of-fit hypothesis tests, since they increase the probability of making type I errors, i.e. falsely rejecting the null hypothesis.

Accurate structure analysis by the maximum-entropy method

“…There have been many theoretical studies (Narayan & Nityanada, 1982;Wilkins, Varghese & Lehmann, 1983;Livesey & Skilling, 1985;Navaza, 1985Navaza, , 1986Bricogne, 1988)to apply the method to problems in crystallography and, in particular, to Fourier inversion. There are many cases in crystallography which involve Fourier inversion.…”

Analyzing the Distribution of Decay Constants in Pulse Fluorimetry Using the Maximum Entropy Method

“…This proof is given in Section 3 of Livesey and Skilling (1985). Shore and Johnson (1980) originally developed this argument but only consider probability distributions.…”

A maximum entropy derivation of the integrated direct methods–isomorphous replacement or integrated direct methods–anomalous scattering probability distributions

“…One aspect of these developments has been to interpret the entropy of a density map as the logarithm of the probability of the distribution of the atoms within the unit cell, and thus show the equivalence to direct methods when intensity constraints are applied (Bricogne, 1984). Arguments have, however, been made against such a literal interpretation Livesey & Skilling, 1985). Taking the appropriate marginals of this p.d.f.…”

“…Collins, 1982;Wilkins, O 1988 International Union of Crystallography Varghese & Lehmann, 1983;Bricogne, 1984;Livesey & Skilling, 1985;Navaza, 1985). One aspect of these developments has been to interpret the entropy of a density map as the logarithm of the probability of the distribution of the atoms within the unit cell, and thus show the equivalence to direct methods when intensity constraints are applied (Bricogne, 1984).…”