Approximate Posterior Probability Research Articles

Bayesian sampling of evolutionarily conserved RNA secondary structures with pseudoknots

Today many non-coding RNAs are known to play an active role in various important biological processes. Since RNA's functionality is correlated with specific structural motifs that are often conserved in phylogenetically related molecules, computational prediction of RNA structure should ideally be based on a set of homologous primary structures. But many available RNA secondary structure prediction programs that use sequence alignments do not consider pseudoknots or their estimations consist on a single structure without information on uncertainty. In this article we present a method that takes advantage of the evolutionary history of a group of aligned RNA sequences for sampling consensus secondary structures, including pseudoknots, according to their approximate posterior probability. We investigate the benefit of using evolutionary history and demonstrate the competitiveness of our method compared with similar methods based on RNase P RNA sequences and simulated data. PhyloQFold, a C + + implementation of our method, is freely available from http://evol.bio.lmu.de/_statgen/software/phyloqfold/.

Approximate Bayesian Recursive Estimation of Linear Model with Uniform Noise

Recursive estimation forms core of adaptive prediction and control. Dynamic exponential family is the only but narrow class of parametric models that allows exact Bayesian estimation. The paper provides an approximate estimation of important autoregressive model with exogenous variables (ARX) and uniform noise. This model reflects well physical nature of modelled system: majority of signals, noise and estimated parameters are bounded. Unlike former solutions, the paper proposes an algorithm that provides a full (approximate) posterior probability density function (pdf) of unknown parameters. Behaviour of the designed algorithm is illustrated by simulations.

Pooling Design and Bias Correction in DNA Library Screening

We study the group test for DNA library screening based on probabilistic approach. Group test is a method of detecting a few positive items from among a large number of items, and has wide range of applications. In DNA library screening, positive item corresponds to the clone having a specified DNA segment, and it is necessary to identify and isolate the positive clones for compiling the libraries. In the group test, a group of items, called pool, is assayed in a lump in order to save the cost of testing, and positive items are detected based on the observation from each pool. It is known that the design of grouping, that is, pooling design is important to %reduce the estimation bias and achieve accurate detection. In the probabilistic approach, positive clones are picked up based on the posterior probability. Naive methods of computing the posterior, however, involves exponentially many sums, and thus we need a device. Loopy belief propagation (loopy BP) algorithm is one of popular methods to obtain approximate posterior probability efficiently. There are some works investigating the relation between the accuracy of the loopy BP and the pooling design. Based on these works, we develop pooling design with small estimation bias of posterior probability, and we show that the balanced incomplete block design (BIBD) has nice property for our purpose. Some numerical experiments show that the bias correction under the BIBD is useful to improve the estimation accuracy.

Decoding choice and mutual information on a moment-by-moment basis from single neurons in rat prefrontal cortex

Prefrontal cortex (PFC) neurons participate in representing a variety of stimuli, events, and processes that occur during the execution of behavioral tasks including spatial relationships as well as decision- and reward-related processes. We tested rats in a T-maze, delayed-alternation, decision-making task while simultaneously recording multiple single-unit spike trains from the prelimbic region of PFC. Animals were trained to navigate down the runway of the maze and choose one of two arms opposite to the one previously visited for food rewards. We used a recursive Bayesian approach to investigate the extent to which decision states could be decoded from the spike counts from neuronal ensembles in PFC at the time of decision-making. This approach estimates a full posterior probability distribution over states as a function of the spiking activity. We estimated the Bayesian model using a boot-strap particle filter [1]. The particle filter is useful because it provides a computationally tractable approximation of the spike-train and task-related state probability distributions without assumptions of linearity or being Gaussian. Particle filtering starts with a collection of particles, which can be thought of as a cloud of guesses of the task-related state xt. Importance weights Wt are then assigned to each guess according to how likely a neural event yt is, given the particular guess xt. By resampling we obtain a new set of particles that, in a certain sense, is consistent with the likelihood of generating yt from xt. Additionally, these resampled particles can be regarded as a draw from the approximate posterior probability distribution p(xt | yt-1) of task-related states. Calculating the expectation of the posterior probability distribution yields the point estimate of the state as the process iterates through time. To investigate the contribution of individual neurons to the decoding process, mutual information was calculated on a moment-by-moment basis between the state distribution and the neuron’s probability distribution of firing. The animal’s choice to turn left or right in order to obtain a reward was decoded during each trial based on training spike-train data from all other trials (leave-one-out cross-validation). For each neuron in the decoded ensemble, the history of spiking activity and its interaction with choice contributed to the spike-train probability model. The expected value calculated over 1000 binary particles of the filter diffused toward the observed behavioral choice in 85% and 76% of the trials of two animals. For trials where a behavioral error was observed (same arm choice on adjacent trials), the mutual information declined between the state distribution and spike train distribution indicating an increase in independence between task-related states and probability of neural firing. These results indicate that the particle filter accurately decoded the future behavioral choice of the animal during the pickup and delay intervals in two animals tested.

Bayesian Inference With Adaptive Fuzzy Priors and Likelihoods

Fuzzy rule-based systems can approximate prior and likelihood probabilities in Bayesian inference and thereby approximate posterior probabilities. This fuzzy approximation technique allows users to apply a much wider and more flexible range of prior and likelihood probability density functions than found in most Bayesian inference schemes. The technique does not restrict the user to the few known closed-form conjugacy relations between the prior and likelihood. It allows the user in many cases to describe the densities with words and just two rules can absorb any bounded closed-form probability density directly into the rulebase. Learning algorithms can tune the expert rules as well as grow them from sample data. The learning laws and fuzzy approximators have a tractable form because of the convex-sum structure of additive fuzzy systems. This convex-sum structure carries over to the fuzzy posterior approximator. We prove a uniform approximation theorem for Bayesian posteriors: An additive fuzzy posterior uniformly approximates the posterior probability density if the prior or likelihood densities are continuous and bounded and if separate additive fuzzy systems approximate the prior and likelihood densities. Simulations demonstrate this fuzzy approximation of priors and posteriors for the three most common conjugate priors (as when a beta prior combines with a binomial likelihood to give a beta posterior). Adaptive fuzzy systems can also approximate non-conjugate priors and likelihoods as well as approximate hyperpriors in hierarchical Bayesian inference. The number of fuzzy rules can grow exponentially in iterative Bayesian inference if the previous posterior approximator becomes the new prior approximator.

Lunar Cycle and the Number of Births: A Spectral Analysis of 4,071,669 Births From South-Western Germany

Lunar Cycle and the Number of Births: A Spectral Analysis of 4,071,669 Births From South-Western Germany

Lunar cycle and the number of births: A spectral analysis of 4,071,669 births from South‐Western Germany

Lunar cycle and the number of births: A spectral analysis of 4,071,669 births from South‐Western Germany

A general class of pattern mixture models for nonignorable dropout with many possible dropout times.

In this article we consider the problem of fitting pattern mixture models to longitudinal data when there are many unique dropout times. We propose a marginally specified latent class pattern mixture model. The marginal mean is assumed to follow a generalized linear model, whereas the mean conditional on the latent class and random effects is specified separately. Because the dimension of the parameter vector of interest (the marginal regression coefficients) does not depend on the assumed number of latent classes, we propose to treat the number of latent classes as a random variable. We specify a prior distribution for the number of classes, and calculate (approximate) posterior model probabilities. In order to avoid the complications with implementing a fully Bayesian model, we propose a simple approximation to these posterior probabilities. The ideas are illustrated using data from a longitudinal study of depression in HIV-infected women.

Consistent model selection based on parameter estimates

We consider model selection based on estimators that are asymptotically normal. Such a method can be applied to the context of estimating equations, since a complete specification of the probability model or likelihood function is not required. We construct a cost function for the models in consideration, and show that the minimizer of the cost function is a consistent estimator of the model. Despite the absence of a likelihood function, the cost function is shown to be related to an approximate posterior probability conditional on the parameter estimates, which enables a Bayesian-type evaluation of all candidate models and not just to present one best choice. The proposed method is modular and easily adapted to different problems, since only one set of estimates of the parameters and asymptotic variance is needed as the input, which can be obtained from very different estimation techniques for very different models, both linear and nonlinear. We also show that by ranking Z-statistics, the scope of model searching can be reduced to achieve computing efficiency. We provide data analysis examples from two clinical trials and illustrate these variable selection techniques in the contexts of partial likelihood analysis and generalized estimating equations. A third example of used automobile prices illustrates an application of the methodology in selecting graphical models.

Statistical methods for gene map construction by fluorescence in situ hybridization.

Fluorescence in situ hybridization (FISH) provides an efficient and powerful technique for ordering loci both on metaphase chromosomes and in less condensed interphase chromatin. Two-color metaphase FISH can be used to order pairs of loci relative to the centromere; two- and three-color interphase FISH can be used to accurately order trios of loci spaced within 1 Mb relative to one another. Loci separated by a distance > 1-2 Mb exhibit chromatin loops that often give rise to a statistically significant but incorrect order. We derive Bayesian methods for selecting the best locus order based on microscopic evaluation for each of these types of FISH mapping data. We then describe how the results from several two- and three-locus analyses can be combined to evaluate the approximate posterior probability of a given multilocus order within the limits of the technology utilized. These methods directly address the question of interest: What is the probability that the inferred two-, three-, or multilocus order actually is correct? We illustrate our analysis methods by applying them to previously described FISH mapping data of 14 markers in the BRCA1 region on chromosome 17q12-q21. We also propose design strategies to order a group of closely spaced (< 1 Mb) loci, two and three loci at a time, using a bisection strategy for two-color FISH data and a trisection strategy for three-color FISH data. These strategies have the best worst-case performance for ordering a new locus relative to a group of ordered loci and are nearly optimal for ordering a group of loci of unknown order. These, in conjunction with physical mapping strategies, provide efficient and reliable methods for gene map construction by FISH.

Approximate Bayesian Inference with the Weighted Likelihood Bootstrap

SUMMARY We introduce the weighted likelihood bootstrap (WLB) as a way to simulate approximately from a posterior distribution. This method is often easy to implement, requiring only an algorithm for calculating the maximum likelihood estimator, such as iteratively reweighted least squares. In the generic weighting scheme, the WLB is first order correct under quite general conditions. Inaccuracies can be removed by using the WLB as a source of samples in the sampling-importance resampling (SIR) algorithm, which also allows incorporation of particular prior information. The SIR-adjusted WLB can be a competitive alternative to other integration methods in certain models. Asymptotic expansions elucidate the second-order properties of the WLB, which is a generalization of Rubin's Bayesian bootstrap. The calculation of approximate Bayes factors for model comparison is also considered. We note that, given a sample simulated from the posterior distribution, the required marginal likelihood may be simulation consistently estimated by the harmonic mean of the associated likelihood values; a modification of this estimator that avoids instability is also noted. These methods provide simple ways of calculating approximate Bayes factors and posterior model probabilities for a very wide class of models.

Incorporating prior theory in covariance structure analysis: A bayesian approach

A Bayesian approach to the testing of competing covariance structures is developed. The method provides approximate posterior probablities for each model under consideration without prior specification of individual parameter distributions. The method is based on ayesian updating using cross-validated pseudo-likelihoods. Given that the observed variables are the samefor all competing models, the approximate posterior probabilities may be obtained easily from the chi square values and other known constants, using only a hand calculator. The approach is illustrated using and example which illustrates how the prior probabilities can alter the results concerning which model specification is preferred.

Approximately calibrated small sample inference about means from bivariate normal data with missing values

Frequency properties of approximate Bayesian posterior probability intervals are considered for a small bivariate sample with data missing on one variable. For a class of priors, approximations to the posterior distribution based on matching moments of the t distribution are derived, and compared with the true distributions computed numerically. Coverage properties of highest posterior density intervals for three choices of prior are evaluated by simulation, and compared with other solutions. The simulations suggest that a second moment t approximation combined with the Jeffreys' prior for the bivariate distribution provides intervals that are quite well calibrated, in the sense of having approximate or slightly conservative coverage for a wide range of values of the underlying parameters. The use of calibration to select a suitable reference prior seems to have potential for a large number of problems.

Calculation of maximum entropy distributions and approximation of marginalposterior distributions

It is shown how members of the broad class of Cobb-Koppstein-Chen probability density functions can be generated by maximizing Shannon's entropy subject to appropriate side conditions. A particular member of the class, the quartic exponential distribution, is obtained by maximizing Shannon's entropy subject to the conditions that the probability density function integrate to one and have given first four moments. An algorithm for computing the quartic exponential distribution is provided and applied to approximate marginal posterior probability densities. These approximations are compared with those provided by the Pearson density function approximation procedure and exact densities obtained by numerical integration. It is found that the maximum entropy method performs very well.