[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-125445-en":3,"doc-seo-125445-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},125445,687197207919,"Theodora","https://ap-avatar.wpscdn.com/avatar/a000253d6f5f7c60be?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779446848396160552",8,"Research & Report","Quantifying Aleatoric and Epistemic Uncertainty in Machine Learning: Are Conditional Entropy and Mutual Information Appropriate Measures? - Supplementary Material","Supplementary material detailing experimental configurations and additional analyses for evaluating uncertainty estimates in machine learning. The work specifies software and training setups in Python, including datasets spanning CIFAR10, MNIST, and synthetic image and tabular benchmarks, along with ensemble-based probabilistic classifiers. It reports how aleatoric and epistemic uncertainty measures behave under controlled changes in class overlap, injected label noise, ensemble size, and base learner complexity, comparing deep ensembles, random forests, and Laplace-style approximations. Results highlight measure-dependent stability, volatility, and robustness across tasks.","Quantifying Aleatoric and Epistemic Uncertainty in Machine Learning: Are Conditional Entropy and Mutual Information Appropriate Measures?  \n(Supplementary Material)  \nLisa Wimmer 1,3 Yusuf Sale2,3 Paul Hofman2,3 Bernd Bischl 1,3 Eyke Hüllermeier2,3  \n1Department of Statistics, LMU Munich, Germany  \n2Institute of Informatics, LMU Munich, Germany  \n3Munich Center for Machine Learning (MCML), Germany  \n1 EXPERIMENTAL DETAILS  \nIn the following, we list the most important training configurations used to generate our results. The full experimental code is hosted in a public repository 1.  \nSoftware Our codebase is written in Python. It chiefly relies on the PyTorch [Paszke et al., 2019], PyTorch Lightning [Lightning AI, 2023], Laplace Redux [Daxberger et al., 2021], and scikit-learn [Pedregosa et al., 2011] libraries.  \nDatasets The real-world computer vision tasks are CIFAR10 [Krizhevsky, 2009] and MNIST [LeCun et al., 1998] . Both contain ten balanced classes. We further synthesize rectangles (white-on-black), where the class label is determined by whether height > width or vice versa, and random non-convex polygons (white-on-black) with 3–5 vertices. These datasets comprise 60k (10k) training (test) samples. The tabular classification problem is created via scikit-learn’s make_classification function, using two features (and four classes. Here, we generate 6k (1k) training (test) samples.  \nBase learners Our probabilistic classifiers all combine some base learners into an explicit (deep ensemble, random forest) or implicit (Laplace approximation) ensemble. We train EfficientNet-B7 (approx. 64m parameters; Tan and Le [2019]) for CIFAR10 and a small convolutional network (three convolutional layers with ReLU activation; approx. 62k parameters) for MNIST and the rectangle/polygon images. In the tabular classification problem, we use a random forest with a maximum tree depth of ten as well as single-hidden-layer MLPs with a hidden layer size of ten, adopting the default parameters from scikit-learn unless stated otherwise. Ensemble size is set to M = 10 .  \n1 [https://github.com/lisa-wm/entropybaseduq](https://github.com/lisa-wm/entropybaseduq)  \nTraining Configurations We use an SGD optimizer (momentum 0.9), a learning rate schedule with cosine annealing, where the initial learning rate is set to 10 −2, and weight decay (5 × 10 −4) . Training runs for a maximum of 200 epochsat batch size 256 with early stopping if validation loss does not improve over five consecutive epochs (evaluated on a validation set containing 10% of the training data) .  \n2 ADDITIONAL RESULTS  \n2.1 INCREASING DATA NOISE  \nCompared to the ensemble of MLPs2 , the random forest (Fig. 1) reacts in both uncertainty components when class overlap is increased.  \nFigure 1: Entropy-based uncertainty for increasing class overlap (tabular data) .  \nIn order to simulate label noise, we randomly change classes for a varying share (1%–75%) of observations in the tabular classification task, leading to datasets as depicted in Fig. 2.  \n2In the tabular classification task, we bootstrap the data for the MLP ensemble to make it directly comparable to the random forest that relies on this technique.  \nAccepted for the 39th Conference on Uncertainty in Artificial Intelligence (UAI 2023) .  \nFigure 2: Tabular data with two features and four classes for increasing noise level.  \nExpected Behavior AU picks up with increasing noise level. Since learner capacity remains fixed, it is reasonable to assume that EU also rises to some extent when the decision boundaries become more complex with mounting degree of dataset contamination.  \nObserved Behavior As observed in the experiments modifying image resolution and class overlap, we find that AU duly increases for a rising noise level, though it remains moderate for the random forest even in the most extreme scenario (Fig. 3), where three out of four labels are assigned randomly. EU goes up slightly for the random forest, as presumed, b","cbCaiuw2HgMfHZU0","https://ap.wps.com/l/cbCaiuw2HgMfHZU0","pdf",614661,1,7,"English","en",105,"# Experimental details\n## Software\n## Datasets\n## Base learners\n## Training configurations\n# Additional results\n## Increasing data noise\n## Number of ensemble members\n## Base learner complexity","[{\"question\":\"What uncertainty components are analyzed in this supplementary material?\",\"answer\":\"The experiments evaluate aleatoric (AU) and epistemic uncertainty (EU) and compare how both respond across learners under controlled dataset changes.\"},{\"question\":\"Which datasets and synthetic data types are used for the experiments?\",\"answer\":\"Real datasets include CIFAR10 and MNIST. Synthetic benchmarks are generated with white-on-black rectangles and random non-convex polygons, as well as a tabular classification task constructed with scikit-learn.\"},{\"question\":\"How do the uncertainty measures change when label noise increases?\",\"answer\":\"AU increases with rising noise level, while EU behaves differently across learners: it grows slightly for the random forest but remains ultra-low for ablations using the MLP ensemble.\"}]","Quantifying Aleatoric and Epistemic Uncertainty in Machine Learning: Are Conditional Entropy and Mutual Information Appropriate Measures? - Supplementary Material | PDF",1785899039,18,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"quantifying-aleatoric-and-epistemic-uncertainty-in-machine-learning-are-conditional-entropy-and-mutual-information-appropriate-measures-supplementary-material","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/quantifying-aleatoric-and-epistemic-uncertainty-in-machine-learning-are-conditional-entropy-and-mutual-information-appropriate-measures-supplementary-material/125445/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-05",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What uncertainty components are analyzed in this supplementary material?","Question",{"text":75,"@type":76},"The experiments evaluate aleatoric (AU) and epistemic uncertainty (EU) and compare how both respond across learners under controlled dataset changes.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Which datasets and synthetic data types are used for the experiments?",{"text":80,"@type":76},"Real datasets include CIFAR10 and MNIST. Synthetic benchmarks are generated with white-on-black rectangles and random non-convex polygons, as well as a tabular classification task constructed with scikit-learn.",{"name":82,"@type":73,"acceptedAnswer":83},"How do the uncertainty measures change when label noise increases?",{"text":84,"@type":76},"AU increases with rising noise level, while EU behaves differently across learners: it grows slightly for the random forest but remains ultra-low for ablations using the MLP ensemble.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":21,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]