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Register a new userLimits of accuracy for parameter estimation and localisation in Single-Molecule Microscopy via sequential Monte Carlo methods
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Assessing the quality of parameter estimates for models describing the motion of single molecules in cellular environments is an important problem in fluorescence microscopy. We consider the fundamental data model, where molecules emit photons at random times and the photons arrive at random locations on the detector according to complex point spread functions (PSFs). The random, non-Gaussian PSF of the detection process and random trajectory of the molecule make inference challenging. Moreover, the presence of other nearby molecules causes further uncertainty in the origin of the measurements, which impacts the statistical precision of estimates. We quantify the limits of accuracy of model parameter estimates and separation distance between closely spaced molecules (known as the resolution problem) by computing the Cramer-Rao lower bound (CRLB), or equivalently the inverse of the Fisher information matrix (FIM), for the variance of estimates. This fundamental CRLB is crucial, as it provides a lower bound for more practical scenarios. While analytic expressions for the FIM can be derived for static molecules, the analytical tools to evaluate it for molecules whose trajectories follow SDEs are still mostly missing. We address this by presenting a general SMC based methodology for both parameter inference and computing the desired accuracy limits for non-static molecules and a non-Gaussian fundamental detection model. For the first time, we are able to estimate the FIM for stochastically moving molecules observed through the Airy and Born & Wolf PSF. This is achieved by estimating the score and observed information matrix via SMC. We sum up the outcome of our numerical work by summarising the qualitative behaviours for the accuracy limits as functions of e.g. collected photon count, molecule diffusion, etc. We also verify that we can recover known results from the static molecule case.
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Statistical signal processing applications usually require the estimation of some parameters of interest given a set of observed data. These estimates are typically obtained either by solving a multi-variate optimization problem, as in the maximum likelihood (ML) or maximum a posteriori (MAP) estimators, or by performing a multi-dimensional integration, as in the minimum mean squared error (MMSE) estimators. Unfortunately, analytical expressions for these estimators cannot be found in most real-world applications, and the Monte Carlo (MC) methodology is one feasible approach. MC methods proceed by drawing random samples, either from the desired distribution or from a simpler one, and using them to compute consistent estimators. The most important families of MC algorithms are Markov chain MC (MCMC) and importance sampling (IS). On the one hand, MCMC methods draw samples from a proposal density, building then an ergodic Markov chain whose stationary distribution is the desired distribution by accepting or rejecting those candidate samples as the new state of the chain. On the other hand, IS techniques draw samples from a simple proposal density, and then assign them suitable weights that measure their quality in some appropriate way. In this paper, we perform a thorough review of MC methods for the estimation of static parameters in signal processing applications. A historical note on the development of MC schemes is also provided, followed by the basic MC method and a brief description of the rejection sampling (RS) algorithm, as well as three sections describing many of the most relevant MCMC and IS algorithms, and their combined use.
We propose approaches for testing implementations of Markov Chain Monte Carlo methods as well as of general Monte Carlo methods. Based on statistical hypothesis tests, these approaches can be used in a unit testing framework to, for example, check if individual steps in a Gibbs sampler or a reversible jump MCMC have the desired invariant distribution. Two exact tests for assessing whether a given Markov chain has a specified invariant distribution are discussed. These and other tests of Monte Carlo methods can be embedded into a sequential method that allows low expected effort if the simulation shows the desired behavior and high power if it does not. Moreover, the false rejection probability can be kept arbitrarily low. For general Monte Carlo methods, this allows testing, for example, if a sampler has a specified distribution or if a sampler produces samples with the desired mean. The methods have been implemented in the R-package MCUnit.
Here, we discuss a collection of cutting-edge techniques and applications in use today by some of the leading experts in the field of correlative approaches in single-molecule biophysics. A key difference in emphasis, compared with traditional single-molecule biophysics approaches detailed previously, is on the emphasis of the development and use of complex methods which explicitly combine multiple approaches to increase biological insights at the single-molecule level. These so-called correlative single-molecule biophysics methods rely on multiple, orthogonal tools and analysis, as opposed to any one single driving technique. Importantly, they span both in vivo and in vitro biological systems as well as the interfaces between theory and experiment in often highly integrated ways, very different to earlier traditional non-integrative approaches. The first applications of correlative single-molecule methods involved adaption of a range of different experimental technologies to the same biological sample whose measurements were synchronised. However, now we find a greater flora of integrated methods emerging that include approaches applied to different samples at different times and yet still permit useful molecular-scale correlations to be performed. The resultant findings often enable far greater precision of length and time scales of measurements, and a more understanding of the interplay between different processes in the same cell. Many new correlative single-molecule biophysics techniques also include more complex, physiologically relevant approaches as well as increasing number that combine advanced computational methods and mathematical analysis with experimental tools. Here we review the motivation behind the development of correlative single-molecule microscopy methods, its history and recent progress in the field.
Bayesian inference methods rely on numerical algorithms for both model selection and parameter inference. In general, these algorithms require a high computational effort to yield reliable estimates. One of the major challenges in phylogenetics is the estimation of the marginal likelihood. This quantity is commonly used for comparing different evolutionary models, but its calculation, even for simple models, incurs high computational cost. Another interesting challenge relates to the estimation of the posterior distribution. Often, long Markov chains are required to get sufficient samples to carry out parameter inference, especially for tree distributions. In general, these problems are addressed separately by using different procedures. Nested sampling (NS) is a Bayesian computation algorithm which provides the means to estimate marginal likelihoods together with their uncertainties, and to sample from the posterior distribution at no extra cost. The methods currently used in phylogenetics for marginal likelihood estimation lack in practicality due to their dependence on many tuning parameters and the inability of most implementations to provide a direct way to calculate the uncertainties associated with the estimates. To address these issues, we introduce NS to phylogenetics. Its performance is assessed under different scenarios and compared to established methods. We conclude that NS is a competitive and attractive algorithm for phylogenetic inference. An implementation is available as a package for BEAST 2 under the LGPL licence, accessible at https://github.com/BEAST2-Dev/nested-sampling.
Quantum Monte Carlo approaches such as the diffusion Monte Carlo (DMC) method are among the most accurate many-body methods for extended systems. Their scaling makes them well suited for defect calculations in solids. We review the various approximations needed for DMC calculations of solids and the results of previous DMC calculations for point defects in solids. Finally, we present estimates of how approximations affect the accuracy of calculations for self-interstitial formation energies in silicon and predict DMC values of 4.4(1), 5.1(1) and 4.7(1) eV for the X, T and H interstitial defects, respectively, in a 16(+1)-atom supercell.
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