ReviewNew applications of maximum likelihood and Bayesian statistics in macromolecular crystallography
Introduction
Maximum likelihood is a general concept and can be explained simply. Consider a bag containing four dice, one 4-sided, one 6-sided, one 8-sided and one 10-sided. Blindfolded, and wearing thick gloves, you select a die from the bag and throw. I tell you that the result is a 7. Which of the dice are you most likely to have picked? To work it out, you need to know the probability of rolling a 7 from each of the dice. In this case, you are most likely to have selected the 8-sided die, as this has the highest probability (1 in 8) of rolling a 7. This example clearly illustrates the distinction between two important probabilities, the probability of the model given the data and the probability of the data given the model (known as the likelihood). For the probability of the model given the data, the variable quantity is the model (number of sides of the die) and the outcome is fixed (7). For the likelihood, the model is fixed (the die has 8 sides) and the outcome varies (we could calculate the likelihood of throwing a 7 or a 1 or any other value). In our dice-throwing example, we want to know the probability of the model given the data. The process of maximum likelihood finds the correct answer by, in effect, testing all the possible models and selecting the one with the highest likelihood of generating the data.
Bayes’ theorem gives the mathematical relationship between the two probabilities described above: the probability of the model given the data is proportional to the likelihood multiplied by the probability of the model [1]. The probability of the model is called the prior probability. In our dice-throwing example, the use of the prior probability was implicit. As you had an equal chance of picking each die from the bag, the prior probability for all the models was the same and was not a factor. But if there had been many 10-sided dice in the bag, instead of only one, your prior probability of picking a 10-sided die would outweigh the higher likelihood of rolling a 7 from an 8-sided die, and the probability would be that you had picked a 10-sided die.
When you try to calculate the likelihood of a set of data, the calculation is a lot simpler if the data are independent. If this is the case, the likelihood of obtaining the whole data set is simply the product of the individual likelihoods for the set's constituent data. In our dice-throwing example, independence would mean that throwing a 7 the first time would not alter the chance of throwing a 7, or any other number, the second time. The likelihood of throwing 7 twice from a 10-sided die would then be 1 in 10 multiplied by 1 in 10, that is, 1 in 100. Independence makes the mathematics so much simpler that it is often assumed even when the assumption is not justified. Independence also leads to the common use of the log-likelihood rather than the likelihood. Instead of quoting a likelihood of 10−2 for a 10-sided die in the example above, it would be usual to quote a log-likelihood of −2 (in fact, usually a minus log-likelihood of 2). The log-likelihood can be substituted for the likelihood because the log of a function has a maximum at the same point as the original function. The log-likelihood is preferred because, if independence is assumed, the log-(total likelihood) equals the log-(product of constituent likelihoods), which is mathematically the same as the sum of the log-(constituent likelihoods), except that the latter involves larger numbers and is easier for computers to handle.
These general principles of likelihood can be applied to crystallography in any number of ways. Bricogne [2], in particular, has presented a philosophical case for likelihood as a mechanism to unify disparate sources of crystallographic information. Likelihood methods are now well established for experimental phasing [3] and structure refinement 4., 5., 6., 7.. In part, this is because increases in computer speed have allowed the calculations to be performed in a reasonable time. Their success has recently encouraged the development of maximum likelihood and Bayesian methods in the areas of molecular replacement and density modification. Here, I review the maximum likelihood molecular replacement formulation developed recently by Read [8••] and implemented in the program BEAST, and the map probability function for statistical density modification developed by Terwilliger 9., 10••., 11••., 12••. (with additions by Cowtan [13••]) and implemented in the program RESOLVE.
Section snippets
Traditional methods
Traditionally, molecular replacement has involved finding the orientation of the model using correlations of Patterson maps and then finding the position of the oriented model in the unit cell using the correlation between the structure factor amplitudes 14., 15., 16., 17., 18., 19.. A proper likelihood formulation treats the problem in terms of the probabilities of generating the structure factors given a model of the unit cell.
Translation likelihood function
When thinking about molecular replacement in real space, it is
Traditional methods
Density modification is based on the premise that, if an electron density map is modified to possess the characteristics it ought to possess, the corresponding phases will improve. Traditionally, density modification (e.g. solvent flattening, noncrystallographic symmetry averaging, histogram matching) has involved taking the electron density map, forcing it to have the desired characteristic, calculating the phases for the modified map and then combining the modified phases with the
Conclusions
Practical maximum likelihood methods have now been developed for inclusion in all traditional areas of macromolecular crystallography: data processing [21], phasing by isomorphous replacement [3], phasing by molecular replacement [8••], density modification 9., 10••., 11••., 12••. and refinement 4., 5., 6., 7.. It has put these methods on a firmer statistical foundation, with impressive improvements over traditional methods in each case 37., 38., 39., 40., 41.. But, with the full statistical
Acknowledgements
AJM thanks Randy Read and Tom Terwilliger for critical reading of the manuscript, Kevin Cowtan for useful discussions, and David Owen, Brett Collins and Laurent Storoni for road-testing the manuscript.
References and recommended reading
Papers of particular interest, published within the annual period of review, have been highlighted as:
• of special interest
•• of outstanding interest
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