[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-121610-en":3,"doc-seo-121610-105":30,"detail-sidebar-cat-0-en-105":92},{"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":20,"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},121610,7971461741311,"Ophelia","https://ap-avatar.wpscdn.com/avatar/74000253aff267980c6?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779345379180704826",6,"Technology","Bayesian Modelling in Machine Learning - A Tutorial Review - Central ideas and interconnections","Bayesian Modelling in Machine Learning connects probability theory with statistical inference to address uncertainty in learning problems. The tutorial review surveys core ideas and methods in a high-level way, clarifying how Bayesian statistics represents observed and latent variables as random quantities. It highlights tasks such as estimation, prediction, model comparison, and validation, and explains why ignoring uncertainty can cause issues like overdispersion and overfitting. Readers are guided through interconnections to explore specialized references further.","Bayesian Modelling in Machine Learning: A Tutorial Review  \nMatthias Seeger  \nProbabilistic Machine Learning and Medical Image Processing  \nSaarland University  \nRoom 116, Campus E1.4, 66123 Saarbruecken  \n[mseeger@mmci.uni-saarland.de](mseeger@mmci.uni-saarland.de)  \nMarch 21, 2009  \nAbstract  \nMany facets of Bayesian Modelling are 􀀌rmly established in Machine Learning and give rise to state-of-the-art solutions to application problems. The sheer number of techniques, ideas and models which have been proposed, and the terminology, can be bewildering. With this tutorial review, we aim to give a wide high-level overview over this important 􀀌eld, concentrating on central ideas and methods, and on their interconnections. The reader will gain a basic understanding of the topics and their relationships, armed with which she can branch to details of her interest using the references to more specialized textbooks and reviews we provide here.  \n1 Introduction  \nMachine Learning is a hybrid of Statistics and algorithmic Computer Science. In this review, we will mostly be concerned with the statistical side. Statistics is about managing and quantifying uncertainty. Uncertainty may arise due to many di􀀋erent reasons, for example:  \n􀀏 Measurement noise: Measurements of physical processes are always subject to inaccuracies. Sometimes, low quality data may be obtained more economically. Data items may be missing  \n􀀏 Model uncertainty: Models are almost never exact, we abstract away complexity in order to allow for predictions to be feasible  \n􀀏 Parameter uncertainty: Variables in a model can never be identi􀀌ed exactly and without doubt from 􀀌nite data  \nThe calculus of uncertainty is probability theory, [22] gives a good introduction. Some phenomenon of interest is mapped to a model, being a set of random variables and probabilistic relationships between them. Variables are observed or latent (unobserved) . To give an example, consider the linear model  \ny = wT 􀀞 (x) + \"; (1)  \nwhere \" is independent noise. This model describes a functional relationship x ! y 2 R. It is a cornerstone of Statistics, and we will see much of it in the following. Suppose we can measure (x; y) repeatedly and independently. Here, x and y are observed, w and \" are latent. Latent variables are query or nuisance, we want to know only about the former. For example, w may be query (is there a linear trend in the data? are some features more relevant than others?), \" is nuisance. w is also called parameter or weights. It is important to note that the classi􀀌cation of model variables into observed, query, or nuisance depends on the task which is addressed by the model.  \nSome statistical tasks for this model: What is the \\best\"value for w representing our data? We are looking for an estimate ^w for w computed from the data. But maybe weare uncertain about w (we should always be!) . How does the data change our uncertainty from before to after having seen data? For example, can we specify a region that w lies  \nin with \\high con􀀌dence\"? Bayesian Statistics goes beyond estimators in order to address the second problem.  \nMaybe we merely want to predict y for x not seen in our measurements. Now, y is latent query, while w becomes nuisance. This ambivalence in the role of variables is resolved in the Bayesian paradigm by treating all variables as random variables. We note that the failure to acknowledge uncertainty can lead to unexpected problems. A good example is the phenomenon of overdispersion. When 􀀌tting basic models to data with maximum likelihood (see Section 2.1), it is often noted that the variance of responses predicted from the 􀀌tted model is signi􀀌cantly smaller than the variance apparent in the observed data. This \\phenomenon\" indicates that uncertainties have been ignored in the process of 􀀌tting. \\Best\"parameter values have been plugged in rather than admitting to uncertainties in this choice. If these sources of uncertainty are properly accounted for, the e","cbCaikWnmc2Q0bsL","https://ap.wps.com/l/cbCaikWnmc2Q0bsL","pdf",726846,1,38,"English","en",105,"# Introduction\n## Uncertainty sources in statistics\n## Probability theory and latent variables\n## Estimation, prediction, and task-dependent variable roles\n## Overdispersion and the need to model uncertainty\n## Model comparison, hierarchy, and validation\n## Overfitting and Bayesian complexity trade-offs","[{\"question\":\"What problem does Bayesian statistics aim to solve beyond point estimation?\",\"answer\":\"It addresses how data updates uncertainty, not only how to compute a best estimate. By treating variables probabilistically, it models uncertainty explicitly in inference and learning.\"},{\"question\":\"Why can ignoring uncertainty lead to overdispersion?\",\"answer\":\"When fitting basic models with maximum likelihood, uncertainties in parameters and model components may be disregarded. The resulting predictions can show variance smaller than what is observed, and proper Bayesian uncertainty accounting typically reduces the discrepancy.\"},{\"question\":\"How does the document relate model comparison to Bayesian hierarchical ideas?\",\"answer\":\"For model comparison, latent variables are treated as nuisance and models are ranked using a data-dependent score. This relates to hierarchical Bayesian modeling where hyperparameters index model variants and must be estimated to determine a good model form.\"}]","Bayesian Modelling in Machine Learning - A Tutorial Review - Central ideas and interconnections | PDF",1785736467,96,{"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":87,"head_meta":89,"extra_data":91,"updated_unix":28},"bayesian-modelling-in-machine-learning-a-tutorial-review-central-ideas-and-interconnections","",{"@graph":36,"@context":86},[37,54,69],{"@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/technology/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/bayesian-modelling-in-machine-learning-a-tutorial-review-central-ideas-and-interconnections/121610/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-04","2026-08-03",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What problem does Bayesian statistics aim to solve beyond point estimation?","Question",{"text":76,"@type":77},"It addresses how data updates uncertainty, not only how to compute a best estimate. By treating variables probabilistically, it models uncertainty explicitly in inference and learning.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"Why can ignoring uncertainty lead to overdispersion?",{"text":81,"@type":77},"When fitting basic models with maximum likelihood, uncertainties in parameters and model components may be disregarded. The resulting predictions can show variance smaller than what is observed, and proper Bayesian uncertainty accounting typically reduces the discrepancy.",{"name":83,"@type":74,"acceptedAnswer":84},"How does the document relate model comparison to Bayesian hierarchical ideas?",{"text":85,"@type":77},"For model comparison, latent variables are treated as nuisance and models are ranked using a data-dependent score. 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