{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:58:41Z","timestamp":1760234321933,"version":"build-2065373602"},"reference-count":76,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2021,4,29]],"date-time":"2021-04-29T00:00:00Z","timestamp":1619654400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"German Research Foundation (DFG)","award":["352015383","390895286"],"award-info":[{"award-number":["352015383","390895286"]}]},{"name":"German Ministry of Research and Education (BMBF)","award":["05M2020"],"award-info":[{"award-number":["05M2020"]}]},{"name":"Wolfgang Gentner scholarship","award":["05E12CHA"],"award-info":[{"award-number":["05E12CHA"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Latent Variable Models (LVMs) are well established tools to accomplish a range of different data processing tasks. Applications exploit the ability of LVMs to identify latent data structure in order to improve data (e.g., through denoising) or to estimate the relation between latent causes and measurements in medical data. In the latter case, LVMs in the form of noisy-OR Bayes nets represent the standard approach to relate binary latents (which represent diseases) to binary observables (which represent symptoms). Bayes nets with binary representation for symptoms may be perceived as a coarse approximation, however. In practice, real disease symptoms can range from absent over mild and intermediate to very severe. Therefore, using diseases\/symptoms relations as motivation, we here ask how standard noisy-OR Bayes nets can be generalized to incorporate continuous observables, e.g., variables that model symptom severity in an interval from healthy to pathological. This transition from binary to interval data poses a number of challenges including a transition from a Bernoulli to a Beta distribution to model symptom statistics. While noisy-OR-like approaches are constrained to model how causes determine the observables\u2019 mean values, the use of Beta distributions additionally provides (and also requires) that the causes determine the observables\u2019 variances. To meet the challenges emerging when generalizing from Bernoulli to Beta distributed observables, we investigate a novel LVM that uses a maximum non-linearity to model how the latents determine means and variances of the observables. Given the model and the goal of likelihood maximization, we then leverage recent theoretical results to derive an Expectation Maximization (EM) algorithm for the suggested LVM. We further show how variational EM can be used to efficiently scale the approach to large networks. Experimental results finally illustrate the efficacy of the proposed model using both synthetic and real data sets. Importantly, we show that the model produces reliable results in estimating causes using proofs of concepts and first tests based on real medical data and on images.<\/jats:p>","DOI":"10.3390\/e23050552","type":"journal-article","created":{"date-parts":[[2021,4,29]],"date-time":"2021-04-29T10:30:41Z","timestamp":1619692241000},"page":"552","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Inference and Learning in a Latent Variable Model for Beta Distributed Interval Data"],"prefix":"10.3390","volume":"23","author":[{"given":"Hamid","family":"Mousavi","sequence":"first","affiliation":[{"name":"Machine Learning Lab, Department of Medical Physics and Acoustics and Cluster of Excellence Hearing4all, University of Oldenburg, 26129 Oldenburg, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7204-5844","authenticated-orcid":false,"given":"Mareike","family":"Buhl","sequence":"additional","affiliation":[{"name":"Medical Physics Group, Department of Medical Physics and Acoustics and Cluster of Excellence Hearing4all, University of Oldenburg, 26129 Oldenburg, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Enrico","family":"Guiraud","sequence":"additional","affiliation":[{"name":"Machine Learning Lab, Department of Medical Physics and Acoustics and Cluster of Excellence Hearing4all, University of Oldenburg, 26129 Oldenburg, Germany"},{"name":"European Organization for Nuclear Research, (CERN), 1211 Meyrin, Switzerland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jakob","family":"Drefs","sequence":"additional","affiliation":[{"name":"Machine Learning Lab, Department of Medical Physics and Acoustics and Cluster of Excellence Hearing4all, University of Oldenburg, 26129 Oldenburg, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J\u00f6rg","family":"L\u00fccke","sequence":"additional","affiliation":[{"name":"Machine Learning Lab, Department of Medical Physics and Acoustics and Cluster of Excellence Hearing4all, University of Oldenburg, 26129 Oldenburg, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,4,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Wang, F., Zhang, P., Qian, B., Wang, X., and Davidson, I. 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