Maximum approximate likelihood estimation of general continuous-time state-space models

Continuous-time state-space models (SSMs) are flexible tools for analysing irregularly sampled sequential observations that are driven by an underlying state process. Corresponding applications typically involve restrictive assumptions concerning linearity and Gaussianity to facilitate inference on the model parameters via the Kalman filter. In this contribution, we provide a general continuous-time SSM framework, allowing both the observation and the state process to be non-linear and non-Gaussian. Statistical inference is carried out by maximum approximate likelihood estimation, where multiple numerical integration within the likelihood evaluation is performed via a fine discretization of the state process. The corresponding reframing of the SSM as a continuous-time hidden Markov model, with structured state transitions, enables us to apply the associated efficient algorithms for parameter estimation and state decoding. We illustrate the modelling approach in a case study using data from a longitudinal study on delinquent behaviour of adolescents in Germany, revealing temporal persistence in the deviation of an individual's delinquency level from the population mean.

Variational inference using approximate likelihood under the coalescent with recombination

Coalescent methods are proven and powerful tools for population genetics, phylogenetics, epidemiology, and other fields. A promising avenue for the analysis of large genomic alignments, which are increasingly common, are coalescent hidden Markov model (coalHMM) methods, but these methods have lacked general usability and flexibility. We introduce a novel method for automatically learning a coalHMM and inferring the posterior distributions of evolutionary parameters using black-box variational inference, with the transition rates between local genealogies derived empirically by simulation. This derivation enables our method to work directly with three or four taxa and through a divide-and-conquer approach with more taxa. Using a simulated data set resembling a human-chimp-gorilla scenario, we show that our method has comparable or better accuracy to previous coalHMM methods. Both species divergence times and population sizes were accurately inferred. The method also infers local genealogies and we report on their accuracy. Furthermore, we discuss a potential direction for scaling the method to larger data sets through a divide-and-conquer approach. This accuracy means our method is useful now, and by deriving transition rates by simulation it is flexible enough to enable future implementations of all kinds of population models.

Profile likelihood analysis for a stochastic model of diffusion in heterogeneous media

We compute profile likelihoods for a stochastic model of diffusive transport motivated by experimental observations of heat conduction in layered skin tissues. This process is modelled as a random walk in a layered one-dimensional material, where each layer has a distinct particle hopping rate. Particles are released at some location, and the duration of time taken for each particle to reach an absorbing boundary is recorded. To explore whether these data can be used to identify the hopping rates in each layer, we compute various profile likelihoods using two methods: first, an exact likelihood is evaluated using a relatively expensive Markov chain approach; and, second, we form an approximate likelihood by assuming the distribution of exit times is given by a Gamma distribution whose first two moments match the moments from the continuum limit description of the stochastic model. Using the exact and approximate likelihoods, we construct various profile likelihoods for a range of problems. In cases where parameter values are not identifiable, we make progress by re-interpreting those data with a reduced model with a smaller number of layers.

Polee: RNA-Seq analysis using approximate likelihood

The analysis of mRNA transcript abundance with RNA-Seq is a central tool in molecular biology research, but often analyses fail to account for the uncertainty in these estimates, which can be significant, especially when trying to disentangle isoforms or duplicated genes. Preserving uncertainty necessitates a full probabilistic model of the all the sequencing reads, which quickly becomes intractable, as experiments can consist of billions of reads. To overcome these limitations, we propose a new method of approximating the likelihood function of a sparse mixture model, using a technique we call the Pólya tree transformation. We demonstrate that substituting this approximation for the real thing achieves most of the benefits with a fraction of the computational costs, leading to more accurate detection of differential transcript expression and transcript coexpression.

A hierarchical Bayesian approach for calibration of stochastic material models

This article recasts the traditional challenge of calibrating a material constitutive model into a hierarchical probabilistic framework. We consider a Bayesian framework where material parameters are assigned distributions, which are then updated given experimental data. Importantly, in true engineering setting, we are not interested in inferring the parameters for a single experiment, but rather inferring the model parameters over the population of possible experimental samples. In doing so, we seek to also capture the inherent variability of the material from coupon-to-coupon, as well as uncertainties around the repeatability of the test. In this article, we address this problem using a hierarchical Bayesian model. However, a vanilla computational approach is prohibitively expensive. Our strategy marginalizes over each individual experiment, decreasing the dimension of our inference problem to only the hyperparameter—those parameter describing the population statistics of the material model only. Importantly, this marginalization step, requires us to derive an approximate likelihood, for which, we exploit an emulator (built offline prior to sampling) and Bayesian quadrature, allowing us to capture the uncertainty in this numerical approximation. Importantly, our approach renders hierarchical Bayesian calibration of material models computational feasible. The approach is tested in two different examples. The first is a compression test of simple spring model using synthetic data; the second, a more complex example using real experiment data to fit a stochastic elastoplastic model for 3D-printed steel.

Polee: RNA-Seq analysis using approximate likelihood

The analysis of mRNA transcript abundance with RNA-Seq is a central tool in molecular biology research, but often analyses fail to account for the uncertainty in these estimates, which can be significant, especially when trying to disentangle isoforms or duplicated genes. Preserving un-certainty necessitates a full probabilistic model of the all the sequencing reads, which quickly becomes intractable, as experiments can consist of billions of reads. To overcome these limitations, we propose a new method of approximating the likelihood function of a sparse mixture model, using a technique we call the Pólya tree transformation. We demonstrate that substituting this approximation for the real thing achieves most of the benefits with a fraction of the computational costs, leading to more accurate detection of differential transcript expression.AvailabilityThe method is implemented in a Julia package available from https://github.com/dcjones/[email protected]

Variational Inference Using Approximate Likelihood Under the Coalescent With Recombination

Coalescent methods are proven and powerful tools for population genetics, phylogenetics, epidemiology, and other fields. A promising avenue for the analysis of large genomic alignments, which are increasingly common, are coalescent hidden Markov model (coalHMM) methods, but these methods have lacked general usability and flexibility. We introduce a novel method for automatically learning a coalHMM and inferring the posterior distributions of evolutionary parameters using black-box variational inference, with the transition rates between local genealogies derived empirically by simulation. This derivation enables our method to work directly with three or four taxa and through a divide-and-conquer approach with more taxa. Using a simulated data set resembling a human-chimp-gorilla scenario, we show that our method has comparable or better accuracy to previous coalHMM methods. Both species divergence times and population sizes were accurately inferred. The method also infers local genealogies and we report on their accuracy. Furthermore, we illustrate how to scale the method to larger data sets through a divide-and-conquer approach. This accuracy means our method is useful now, and by deriving transition rates by simulation it is flexible enough to enable future implementations of all kinds of population models.