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At least 19 recordsLinked to original sources
PubMed2004

Column sorting: rapid calculation of the phylogenetic likelihood function.

Likelihood applications have become a central approach for molecular evolutionary analyses since the first computationally tractable treatment two decades ago. Although Felsenstein's original pruning algorithm makes likelihood calculations feasible, it is usually possible to take advantage of repetitive structure present in the data to arrive at even greater computational reductions. In particular, alignment columns with certain similarities have components of the likelihood calculation that are identical and need not be recomputed if columns are evaluated in an optimal order. We develop an algorithm for exploiting this speed improvement via an application of graph theory. The reductions provided by the method depend on both the tree and the data, but typical savings range between 15%and 50%. Real-data examples with time reductions of 80%have been identified. The overhead costs associated with implementing the algorithm are minimal, and they are recovered in all but the smallest data sets. The modifications will provide faster likelihood algorithms, which will allow likelihood methods to be applied to larger sets of taxa and to include more thorough searches of the tree topology space.

Algorithms↗
PubMed2000

The likelihood function in fiber diffraction

The likelihood function is an appropriate target function for refinement of molecular structures using fiber diffraction data. However, its practical application to fiber diffraction faces two significant obstacles: (i) the intensities of layer lines in a fiber diffraction pattern usually arise from the superposition of several terms, each equivalent to a crystallographic structure factor, thereby making the calculation significantly more complex than for the crystallographic case; (ii) to describe a molecular structure at the atomic level based on fiber diffraction data, the radial and phase parts of the atomic coordinates must be treated separately owing to the uniaxial symmetry of the structure. These issues are addressed here in order to derive equations of likelihood functions for fiber diffraction. The special case of a single term on a layer line is treated first followed by extension of the method to the multiterm case. A practical difficulty in implementation of likelihood for the multiterm case is that each term has a different variance. An analytical technique is described that allows the conversion of the unequal-variance case to an equal-variance case. This makes it possible to express the likelihood by an explicit formula, allowing a direct implementation of the likelihood calculation. A cylindrically symmetric model is proposed for error distribution of the atomic coordinates in a helical structure. Variances and offset coefficients of the contributing terms in the likelihood functions are expressed in terms of the variance of the atomic coordinates in the cylindrical reference system.

Journal Article↗
PubMed2007

Maximum-likelihood methods in wavefront sensing: stochastic models and likelihood functions.

Maximum-likelihood (ML) estimation in wavefront sensing requires careful attention to all noise sources and all factors that influence the sensor data. We present detailed probability density functions for the output of the image detector in a wavefront sensor, conditional not only on wavefront parameters but also on various nuisance parameters. Practical ways of dealing with nuisance parameters are described, and final expressions for likelihoods and Fisher information matrices are derived. The theory is illustrated by discussing Shack-Hartmann sensors, and computational requirements are discussed. Simulation results show that ML estimation can significantly increase the dynamic range of a Shack-Hartmann sensor with four detectors and that it can reduce the residual wavefront error when compared with traditional methods.

Electrons↗
PubMed2002

Using exact Poisson likelihood functions in Bayesian interpretation of counting measurements.

A technique for computing the exact marginalized (integrated) Poisson likelihood function for counting measurement processes involving a background subtraction is described. An empirical Bayesian method for determining the prior probability distribution of background count rates from population data is recommended and would seem to have important practical advantages. The exact marginalized Poisson likelihood function may be used instead of the commonly used Gaussian approximation. Differences occur in some cases of small numbers of measured counts, which are discussed. Optional use of exact likelihood functions in our Bayesian internal dosimetry codes has been implemented using an interpolation-table approach, which means that there is no computation time penalty except for the initial setup of the interpolation tables.

Bayes Theorem↗
PubMed1996

Computation of the full likelihood function for estimating variance at a quantitative trait locus.

The proportion of alleles identical by descent (IBD) determines the genetic covariance between relatives, and thus is crucial in estimating genetic variances of quantitative trait loci (QTL). However, IBD proportions at QTL are unobservable and must be inferred from marker information. The conventional method of QTL variance analysis maximizes the likelihood function by replacing the missing IBDs by their conditional expectations (the expectation method), while in fact the full likelihood function should take into account the conditional distribution of IBDs (the distribution method). The distribution method for families of more than two sibs has not been obvious because there are n(n - 1)/2 IBD variables in a family of size n, forming an n x n symmetrical matrix. In this paper, I use four binary variables, where each indicates the event that an allele from one of the four grandparents has passed to the individual. The IBD proportion between any two sibs is then expressed as a function of the indicators. Subsequently, the joint distribution of the IBD matrix is derived from the distribution of the indicator variables. Given the joint distribution of the unknown IBDs, a method to compute the full likelihood function is developed for families of arbitrary sizes.

Alleles↗
PubMed2004

Joining genetic linkage maps using a joint likelihood function.

We present an efficient method to join genetic maps derived from different crosses, which is especially appropriate for dominant markers. In contrast to the "JoinMap" algorithm, which estimates information about recombination in a given cross from the LOD values and then combines estimates among crosses assuming a binomial sampling distribution, we construct a joint likelihood function that combines information across all crosses, to obtain a joint estimate of recombination. Simulations indicated that the difference between these two approaches is small when codominant markers are used, but that the joint likelihood approach shows substantially improved estimates when dominant or a mixture of dominant and codominant markers are used. This is because the joint likelihood implicitly finds the optimal weights among different classes of data, while the former method does not accurately predict the information from crosses involving dominant markers. Application of our method is illustrated by the construction of a linkage map for Linanthus using both backcrosses and the F2 of a cross between Linanthus jepsonii and L. bicolor, assayed with amplified fragment length polymorphisms (AFLP).

Chromosome Mapping↗
PubMed2026

Likelihood-based optimization enables accurate copy number estimation for paralogous genes using exome data.

MOTIVATION: Exome sequencing is widely used for genetic studies; however, accurate detection of copy number variants (CNV) in paralogous genes is challenging due to short-read mapping ambiguity and extensive copy-number variation. The human genome contains several hundred paralogous genes, many of which are known to harbor disease-associated CNVs. Existing exome CNV callers are primarily designed for rare CNV detection in uniquely mappable regions and are not well-suited for paralogous genes. METHODS: We describe a computational method (EdgeCopy) for copy number profiling of paralogous genes using whole-exome sequence data. EdgeCopy aggregates reads mapped to all copies of paralogous genes and relates observed read depth to copy number for multiple exome samples using an approximate composite likelihood function. The likelihood function is optimized using numerical optimization to obtain gene-level fractional copy number estimates that are discretized and refined using a Hidden Markov Model to obtain exon-level copy number estimates. RESULTS: Benchmarking of Edgecopy using experimental copy number data showed high concordance (mean = 0.973) for six disease-associated paralogous genes. We evaluated performance using whole-exome data from approximately 2400 samples across five continental populations from the 1000 Genomes Project. EdgeCopy shows robust concordance with whole-genome sequencing based estimates (0.974-0.982) across populations and 130 paralogous genes spanning a wide range of copy-number variation. In comparison, copy number analysis using a state-of-the-art exome CNV caller failed to estimate copy number for paralogous genes with very high mapping ambiguity and showed much lower concordance (0.565) for CNV events compared to EdgeCopy (0.908). AVAILABILITY: EdgeCopy is freely available at https://github.com/vibansal-lab/edgecopy.

Humans↗
PubMed2006

HAPLOFREQ--estimating haplotype frequencies efficiently.

A commonly used tool in disease association studies is the search for discrepancies between the haplotype distribution in the case and control populations. In order to find this discrepancy, the haplotypes frequency in each of the populations is estimated from the genotypes. We present a new method HAPLOFREQ to estimate haplotype frequencies over a short genomic region given the genotypes or haplotypes with missing data or sequencing errors. Our approach incorporates a maximum likelihood model based on a simple random generative model which assumes that the genotypes are independently sampled from the population. We first show that if the phased haplotypes are given, possibly with missing data, we can estimate the frequency of the haplotypes in the population by finding the global optimum of the likelihood function in polynomial time. If the haplotypes are not phased, finding the maximum value of the likelihood function is NP-hard. In this case, we define an alternative likelihood function which can be thought of as a relaxed likelihood function. We show that the maximum relaxed likelihood can be found in polynomial time and that the optimal solution of the relaxed likelihood approaches asymptotically to the haplotype frequencies in the population. In contrast to previous approaches, our algorithms are guaranteed to converge in polynomial time to a global maximum of the different likelihood functions. We compared the performance of our algorithm to the widely used program PHASE, and we found that our estimates are at least 10% more accurate than PHASE and about ten times faster than PHASE. Our techniques involve new algorithms in convex optimization. These algorithms may be of independent interest. Particularly, they may be helpful in other maximum likelihood problems arising from survey sampling.

Algorithms↗
PubMed2000

Maximum-likelihood density modification.

A likelihood-based approach to density modification is developed that can be applied to a wide variety of cases where some information about the electron density at various points in the unit cell is available. The key to the approach consists of developing likelihood functions that represent the probability that a particular value of electron density is consistent with prior expectations for the electron density at that point in the unit cell. These likelihood functions are then combined with likelihood functions based on experimental observations and with others containing any prior knowledge about structure factors to form a combined likelihood function for each structure factor. A simple and general approach to maximizing the combined likelihood function is developed. It is found that this likelihood-based approach yields greater phase improvement in model and real test cases than either conventional solvent flattening and histogram matching or a recent reciprocal-space solvent-flattening procedure [Terwilliger (1999), Acta Cryst. D55, 1863-1871].

Crystallography, X-Ray↗
PubMedDate not supplied

On non-parametric maximum likelihood estimation of the bivariate survivor function.

The likelihood function for the bivariate survivor function F, under independent censorship, is maximized to obtain a non-parametric maximum likelihood estimator &Fcirc;. &Fcirc; may or may not be unique depending on the configuration of singly- and doubly-censored pairs. The likelihood function can be maximized by placing all mass on the grid formed by the uncensored failure times, or half lines beyond the failure time grid, or in the upper right quadrant beyond the grid. By accumulating the mass along lines (or regions) where the likelihood is flat, one obtains a partially maximized likelihood as a function of parameters that can be uniquely estimated. The score equations corresponding to these point mass parameters are derived, using a Lagrange multiplier technique to ensure unit total mass, and a modified Newton procedure is used to calculate the parameter estimates in some limited simulation studies. Some considerations for the further development of non-parametric bivariate survivor function estimators are briefly described.

Computer Simulation↗
PubMedDate not supplied

The confidence profile method: a Bayesian method for assessing health technologies.

The Confidence Profile Method is a Bayesian method for adjusting and combining pieces of evidence to estimate parameters, such as the effect of health technologies on health outcomes. The information in each piece of evidence is captured in a likelihood function that gives the likelihood of the observed results of the evidence as a function of possible values of the parameter. A posterior distribution is calculated from Bayes formula as the product of the likelihood function and a prior distribution. Multiple pieces of evidence are incorporated by successive applications of Bayes' formula. Pieces of evidence are adjusted for biases to internal or external validity by modeling the biases and deriving "adjusted" likelihood functions that incorporate the models. Likelihood functions have been derived for one-, two- and multi-arm prospective studies; 2 x 2, 2 x n and matched case-control studies, and cross-sectional studies. Biases that can be incorporated in likelihood functions include crossover in controlled trials, error in measurement outcomes, patient selection biases, differences in technologies, and differences in length of follow-up. Effect measures include differences of rates, ratios of rates, and odds ratios. The elements of the method are illustrated with an analysis of the effect of a thrombolytic agent on the difference in probability of 1-year survival after a heart attack.

Bayes Theorem↗
PubMed2005

Bayesian random effects meta-analysis of trials with binary outcomes: methods for the absolute risk difference and relative risk scales.

In a recent Statistics in Medicine paper, Warn, Thompson and Spiegelhalter (WTS) made a comparison between the Bayesian approach to the meta-analysis of binary outcomes and a popular Classical approach that uses summary (two-stage) techniques. They included approximate summary (two-stage) Bayesian techniques in their comparisons in an attempt undoubtedly to make the comparison less unfair. But, as this letter will argue, there are techniques from the Classical approach that are closer-those based directly on the likelihood-and they failed to make comparisons with these. Here the differences between Bayesian and Classical approaches in meta-analysis applications reside solely in how the likelihood functions are converted into either credibility intervals or confidence intervals. Both summarize, contrast and combine data using likelihood functions. Conflating what Bayes actually offers to meta-analysts-a means of converting likelihood functions to credibility intervals-with the use of likelihood functions themselves to summarize, contrast and combine studies is at best misleading.

Bayes Theorem↗
PubMed2001

Pushing the boundaries of molecular replacement with maximum likelihood.

The molecular-replacement method works well with good models and simple unit cells, but often fails with more difficult problems. Experience with likelihood in other areas of crystallography suggests that it would improve performance significantly. For molecular replacement, the form of the required likelihood function depends on whether there is ambiguity in the relative phases of the contributions from symmetry-related molecules (e.g. rotation versus translation searches). Likelihood functions used in structure refinement are appropriate only for translation (or six-dimensional) searches, where the correct translation will place all of the atoms in the model approximately correctly. A new likelihood function that allows for unknown relative phases is suitable for rotation searches. It is shown that correlations between sequence identity and coordinate error can be used to calibrate parameters for model quality in the likelihood functions. Multiple models of a molecule can be combined in a statistically valid way by setting up the joint probability distribution of the true and model structure factors as a multivariate complex normal distribution, from which the conditional distribution of the true structure factor given the models can be derived. Tests in a new molecular-replacement program, Beast, show that the likelihood-based targets are more sensitive and more accurate than previous targets. The new multiple-model likelihood function has a dramatic impact on success.

Calibration↗
PubMed2004

A point process framework for relating neural spiking activity to spiking history, neural ensemble, and extrinsic covariate effects.

Multiple factors simultaneously affect the spiking activity of individual neurons. Determining the effects and relative importance of these factors is a challenging problem in neurophysiology. We propose a statistical framework based on the point process likelihood function to relate a neuron's spiking probability to three typical covariates: the neuron's own spiking history, concurrent ensemble activity, and extrinsic covariates such as stimuli or behavior. The framework uses parametric models of the conditional intensity function to define a neuron's spiking probability in terms of the covariates. The discrete time likelihood function for point processes is used to carry out model fitting and model analysis. We show that, by modeling the logarithm of the conditional intensity function as a linear combination of functions of the covariates, the discrete time point process likelihood function is readily analyzed in the generalized linear model (GLM) framework. We illustrate our approach for both GLM and non-GLM likelihood functions using simulated data and multivariate single-unit activity data simultaneously recorded from the motor cortex of a monkey performing a visuomotor pursuit-tracking task. The point process framework provides a flexible, computationally efficient approach for maximum likelihood estimation, goodness-of-fit assessment, residual analysis, model selection, and neural decoding. The framework thus allows for the formulation and analysis of point process models of neural spiking activity that readily capture the simultaneous effects of multiple covariates and enables the assessment of their relative importance.

Action Potentials↗
PubMed2004

Simple algorithm for a maximum-likelihood SAD function.

Recently, the multivariate complex normal distribution has been used to develop a maximum-likelihood probability function for single-wavelength anomalous diffraction phasing and refinement of heavy-atom parameters [Pannu & Read (2004), Acta Cryst. D60, 22-27]. The function accounts explicitly for the correlations between the observed and calculated Friedel mates and their errors. However, the method of derivation of the equation described by Pannu & Read (2004) leads to a complicated likelihood expression that suffers from a number of algorithmic limitations. Here, an alternative derivation of the P(SAD) function is described that leads to simplified algorithmic requirements and that allows an intuitive understanding of the expression.

Algorithms↗
PubMed2003

Likelihood approaches to sensory coding in auditory cortex.

Likelihood methods began their evolution in the early 1920s with R A Fisher, and have developed into a rich framework for inferential statistics. This framework offers tools for the analysis of the differential geometry of the full likelihood function based on observed data. We examine likelihood functions derived from inverse Gaussian (IG) probability density models of cortical ensemble responses of single units. Specifically, we investigate the problem of sound localization from the observation of an ensemble of neural responses recorded from the primary (Al) field of the auditory cortex. The problem is framed as a probabilistic inverse problem with multiple sources of ambiguity. Observed and expected Fisher information are defined for the IG cortical ensemble likelihood functions. Receptive field functions of multiple acoustic parameters are constructed and linked to the IG density. The impact of estimating multiple acoustic parameters related to the direction of a sound is discussed, and the implications of eliminating nuisance parameters are considered. We examine the degree of acuity afforded by a small ensemble of cortical neurons for locating sounds in space, and show the predicted patterns of estimation errors, which tend to follow psychophysical performance.

Acoustic Stimulation↗
PubMed2000

Nonparametric regression sinogram smoothing using a roughness-penalized Poisson likelihood objective function.

We develop and investigate an approach to tomographic image reconstruction in which nonparametric regression using a roughness-penalized Poisson likelihood objective function is used to smooth each projection independently prior to reconstruction by unapodized filtered backprojection (FBP). As an added generalization, the roughness penalty is expressed in terms of a monotonic transform, known as the link function, of the projections. The approach is compared to shift-invariant projection filtering through the use of a Hanning window as well as to a related nonparametric regression approach that makes use of an objective function based on weighted least squares (WLS) rather than the Poisson likelihood. The approach is found to lead to improvements in resolution-noise tradeoffs over the Hanning filter as well as over the WLS approach. We also investigate the resolution and noise effects of three different link functions: the identity, square root, and logarithm links. The choice of link function is found to influence the resolution uniformity and isotropy properties of the reconstructed images. In particular, in the case of an idealized imaging system with intrinsically uniform and isotropic resolution, the choice of a square root link function yields the desirable outcome of essentially uniform and isotropic resolution in reconstructed images, with noise performance still superior to that of the Hanning filter as well as that of the WLS approach.

Algorithms↗