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1. A tutorial on the Bayesian statistical approach to inverse problems

   https://doi.org/10.1063/5.0154773  

   Waqar, Faaiq_G; Patel, Swati; Simon, Cory_M (November 2023, APL Machine Learning)

   Inverse problems are ubiquitous in science and engineering. Two categories of inverse problems concerning a physical system are (1) estimate parameters in a model of the system from observed input–output pairs and (2) given a model of the system, reconstruct the input to it that caused some observed output. Applied inverse problems are challenging because a solution may (i) not exist, (ii) not be unique, or (iii) be sensitive to measurement noise contaminating the data. Bayesian statistical inversion (BSI) is an approach to tackle ill-posed and/or ill-conditioned inverse problems. Advantageously, BSI provides a “solution” that (i) quantifies uncertainty by assigning a probability to each possible value of the unknown parameter/input and (ii) incorporates prior information and beliefs about the parameter/input. Herein, we provide a tutorial of BSI for inverse problems by way of illustrative examples dealing with heat transfer from ambient air to a cold lime fruit. First, we use BSI to infer a parameter in a dynamic model of the lime temperature from measurements of the lime temperature over time. Second, we use BSI to reconstruct the initial condition of the lime from a measurement of its temperature later in time. We demonstrate the incorporation of prior information, visualize the posterior distributions of the parameter/initial condition, and show posterior samples of lime temperature trajectories from the model. Our Tutorial aims to reach a wide range of scientists and engineers. 

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2. Estimating repeat spectra and genome length from low-coverage genome skims with RESPECT

   https://doi.org/10.1371/journal.pcbi.1009449  

   Sarmashghi, Shahab; Balaban, Metin; Rachtman, Eleonora; Touri, Behrouz; Mirarab, Siavash; Bafna, Vineet (November 2021, PLOS Computational Biology)

   Segata, Nicola (Ed.)

   The cost of sequencing the genome is dropping at a much faster rate compared to assembling and finishing the genome. The use of lightly sampled genomes (genome-skims) could be transformative for genomic ecology, and results using k -mers have shown the advantage of this approach in identification and phylogenetic placement of eukaryotic species. Here, we revisit the basic question of estimating genomic parameters such as genome length, coverage, and repeat structure, focusing specifically on estimating the k -mer repeat spectrum. We show using a mix of theoretical and empirical analysis that there are fundamental limitations to estimating the k -mer spectra due to ill-conditioned systems, and that has implications for other genomic parameters. We get around this problem using a novel constrained optimization approach (Spline Linear Programming), where the constraints are learned empirically. On reads simulated at 1X coverage from 66 genomes, our method, REPeat SPECTra Estimation (RESPECT), had 2.2% error in length estimation compared to 27% error previously achieved. In shotgun sequenced read samples with contaminants, RESPECT length estimates had median error 4%, in contrast to other methods that had median error 80%. Together, the results suggest that low-pass genomic sequencing can yield reliable estimates of the length and repeat content of the genome. The RESPECT software will be publicly available at https://urldefense.proofpoint.com/v2/url?u=https-3A__github.com_shahab-2Dsarmashghi_RESPECT.git&d=DwIGAw&c=-35OiAkTchMrZOngvJPOeA&r=ZozViWvD1E8PorCkfwYKYQMVKFoEcqLFm4Tg49XnPcA&m=f-xS8GMHKckknkc7Xpp8FJYw_ltUwz5frOw1a5pJ81EpdTOK8xhbYmrN4ZxniM96&s=717o8hLR1JmHFpRPSWG6xdUQTikyUjicjkipjFsKG4w&e= . 

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3. Submodular Observation Selection and Information Gathering for Quadratic Models

   Hashemi, Abolfazl; Ghasemi, Mahsa; Vikalo, Haris and (April 2019, Proceedings of the 36th International Conference on Machine Learning)

   We study the problem of selecting most informative subset of a large observation set to enable accurate estimation of unknown parameters. This problem arises in a variety of settings in machine learning and signal processing including feature selection, phase retrieval, and target localization. Since for quadratic measurement models the moment matrix of the optimal estimator is generally unknown, majority of prior work resorts to approximation techniques such as linearization of the observation model to optimize the alphabetical optimality criteria of an approximate moment matrix. Conversely, by exploiting a connection to the classical Van Trees’ inequality, we derive new alphabetical optimality criteria without distorting the relational structure of the observation model. We further show that under certain conditions on parameters of the problem these optimality criteria are monotone and (weak) submodular set functions. These results enable us to develop an efficient greedy observation selection algorithm uniquely tailored for quadratic models, and provide theoretical bounds on its achievable utility. 

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4. A Bayesian complex-valued latent variable model applied to functional magnetic resonance imaging

   https://doi.org/10.1093/jrsssc/qlae046  

   Sakitis, Chase_J; Brown, D_Andrew; Rowe, Daniel_B (September 2024, Journal of the Royal Statistical Society Series C: Applied Statistics)

   Abstract In linear regression, the coefficients are simple to estimate using the least squares method with a known design matrix for the observed measurements. However, real-world applications may encounter complications such as an unknown design matrix and complex-valued parameters. The design matrix can be estimated from prior information but can potentially cause an inverse problem when multiplying by the transpose as it is generally ill-conditioned. This can be combat by adding regularizers to the model but does not always mitigate the issues. Here, we propose our Bayesian approach to a complex-valued latent variable linear model with an application to functional magnetic resonance imaging (fMRI) image reconstruction. The complex-valued linear model and our Bayesian model are evaluated through extensive simulations and applied to experimental fMRI data. 

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5. On Submodularity of Quadratic Observation Selection in Constrained Networked Sensing Systems

   https://doi.org/10.23919/ACC.2019.8814899  

   Ghasemi, Mahsa; Hashemi, Abolfazl; Topcu, Ufuk; Vikalo, Haris (October 2019, 2019 American Control Conference (ACC))

   We study the problem of observation selection in a resource-constrained networked sensing system, where the objective is to select a small subset of observations from a large network to perform a state estimation task. When the measurements are gathered using nonlinear systems, majority of prior work resort to approximation techniques such as linearization of the measurement model to utilize the methods developed for linear models, e.g., (weak) submodular objectives and greedy selection schemes. In contrast, when the measurement model is quadratic, e.g., the range measurements in a radar system, by exploiting a connection to the classical Van Trees' inequality, we derive new optimality criteria without distorting the relational structure of the measurement model. We further show that under certain conditions these optimality criteria are monotone and (weak) submodular set functions. These results enable us to develop an efficient greedy observation selection algorithm uniquely tailored for constrained networked sensing systems following quadratic models and provide theoretical bounds on its achievable utility. Extensive numerical experiments demonstrate efficacy of the proposed framework. 

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