[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81689-en":3,"doc-seo-81689-105":29,"detail-sidebar-cat-0-en-105":90},{"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":21,"is_downloadable":21,"audit_status":21,"page_count":11,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},81689,1649267921044,"Ava Thompson","https://us-avatar.wpscdn.com/avatar/1800007509477c92dfb?_k=1782875107921204101",8,"Research & Report","Estimation–Prediction Tradeoff in Causal Probabilistic Temporal Graphs","Temporal link prediction (TLP) is commonly assessed by predictive performance on unseen edges, yet this can mix up predictive accuracy with identification of the true causal mechanism. In probabilistic stochastic models, Fisher information determines the Cramér–Rao (CR) bound for parameter estimation error. The work shows that, under comonotonicity between Fisher information and entropy, binary logistic models exhibit an estimation–prediction tradeoff: higher Fisher information, yielding tighter CR bounds, also leads to higher irreducible predictive entropy. A probabilistic causal generator for temporal graphs with transient edges and known causal ground truth is used to validate the effect empirically.","Estimation–Prediction Tradeoff in Causal Probabilistic Temporal Graphs  \nAniq Ur Rahman 1  \narXiv :2606 .28225v2 [ cs .LG] 8 Jul 2026  \nAbstract  \nTemporal link prediction (TLP) is typically evaluated by predictive performance on unseen edges, but this criterion can conflate predictive accuracy with recovery of the underlying causal mechanism. In stochastic models, Fisher information governs the Cramr–Rao (CR) bound on parameter estimation error: higher Fisher information permits more accurate parameter recovery. We show that, under comonotonicity conditions between Fisher information and entropy, binary logistic models exhibit an estimation–prediction tradeoff: regimes with higher Fisher information, and hence smaller CR bounds, also have higher irreducible predictive entropy. To study this tradeoff in TLP, we introduce a probabilistic causal generator for temporal graphs with transient edges and known ground-truth causal structure, and validate the phenomenon empirically.  \n1. Introduction  \nIn supervised learning, a model is trained on a set of input–output pairs such that the predicted outputs closely match the ground-truth, thereby minimising a prescribed error function, typically referred to as the loss. The trained model is then evaluated on previously unseen data, referred to as the test set. The resulting prediction error on the test set is reported as the predictive performance of the model, on the basis of which it is benchmarked against competing approaches. An important aspect, often overlooked by machine learning practitioners, is that the training objective is not designed to recover the true mapping, but rather to approximate a model which best fits the given training data. In other words, the learned model is inherently sensitive to the sampled training data, and a substantial body of research has focused on mitigating this sensitivity through techniques such as batch  training, stochastic gradient descent, batch  \n1Department of Engineering Science, University of Oxford, Oxford, OX1 3PJ, UK. Correspondence to: Aniq Ur Rahman \u003C[aniq.rahman@eng.ox.ac.uk](aniq.rahman@eng.ox.ac.uk)>.  \nPreliminary work.  \nnormalisation, and regularisation, to name a few.  \nIn this work, we revisit the problem of model sensitivity to the training data (Murphy, 2012) and instead of accepting low training loss as a proxy for successful learning, we ask the following question:  \n(Q1) What if the learned model could be compared directly against the true model?  \nTo this end, we assume that the true model belongs to a known parametric family, and that minimising the training loss recovers the true parameters asymptotically as the number of training samples increases (Vapnik, 2013) . Parameter recovery performance can then be quantified directly through the difference between the estimated and true parameters. Since, in practice we are limited by a finite number of samples, a question naturally arises:  \n(Q2) Does the observed data contain sufficient information to accurately recover the parameters?  \nA refined version of this question is:  \n(Q3) What is the relation between parameter estimation error and the amount of useful information in the observed data?  \nWe have tools from information theory at our disposal to answer such questions, under the additional assumption that the parametric model is probabilistic.  \nOnce the parameters have been estimated as accurately as permitted by the available data, we evaluate the predictive performance of the resulting model on unseen samples, leading to the final question:  \n(Q4) How is the predictive performance of the estimated model related to the parameter estimation error?  \nMore specifically, we study how the lowest achievable parameter estimation error is related to the predictive performance attained by the corresponding estimated model.  \nWe now motivate the problem in the context of temporal link prediction (TLP) (Longa et al., 2023), where the objective is to predict whether an edge e","cbCaikowG53nKEko","https://ap.wps.com/l/cbCaikowG53nKEko","pdf",333623,2,1,"English","en",105,"# Introduction\n## Supervised learning and evaluation\n## Parameter recovery vs. predictive performance\n## Information-theoretic view and probabilistic models\n## Motivation from temporal link prediction (TLP)\n### State-space growth and combinatorial explosion","[{\"question\":\"Why can standard temporal link prediction evaluation conflate causal recovery with predictive accuracy?\",\"answer\":\"Predictive performance on unseen edges evaluates how well the model predicts, but it may not directly reflect whether the learned model recovers the underlying causal mechanism. This separation is emphasized as often overlooked in machine learning practice.\"},{\"question\":\"How does Fisher information relate to parameter estimation error in the proposed framework?\",\"answer\":\"In stochastic models, Fisher information governs the Cramér–Rao (CR) bound on parameter estimation error. Higher Fisher information allows more accurate parameter recovery.\"},{\"question\":\"What estimation–prediction tradeoff is observed for binary logistic models, and under what condition?\",\"answer\":\"Under comonotonicity between Fisher information and entropy, regimes with higher Fisher information (and thus smaller CR bounds) also show higher irreducible predictive entropy, creating an estimation–prediction tradeoff.\"}]",1784175428,20,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":27},"estimationprediction-tradeoff-in-causal-probabilistic-temporal-graphs","",{"@graph":35,"@context":84},[36,52,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,46,49],{"item":40,"name":41,"@type":42,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":20},"https://docshare.wps.com/document/","Document",{"item":47,"name":12,"@type":42,"position":48},"https://docshare.wps.com/document/research-report/",3,{"item":50,"name":13,"@type":42,"position":51},"https://docshare.wps.com/document/estimationprediction-tradeoff-in-causal-probabilistic-temporal-graphs/81689/",4,{"url":50,"name":13,"@type":53,"author":54,"headline":13,"publisher":56,"fileFormat":59,"inLanguage":23,"description":14,"dateModified":60,"datePublished":61,"encodingFormat":59,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":55},"Person",{"url":40,"name":57,"@type":58},"DocShare","Organization","application/pdf","2026-07-24","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":20},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"Why can standard temporal link prediction evaluation conflate causal recovery with predictive accuracy?","Question",{"text":74,"@type":75},"Predictive performance on unseen edges evaluates how well the model predicts, but it may not directly reflect whether the learned model recovers the underlying causal mechanism. This separation is emphasized as often overlooked in machine learning practice.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does Fisher information relate to parameter estimation error in the proposed framework?",{"text":79,"@type":75},"In stochastic models, Fisher information governs the Cramér–Rao (CR) bound on parameter estimation error. Higher Fisher information allows more accurate parameter recovery.",{"name":81,"@type":72,"acceptedAnswer":82},"What estimation–prediction tradeoff is observed for binary logistic models, and under what condition?",{"text":83,"@type":75},"Under comonotonicity between Fisher information and entropy, regimes with higher Fisher information (and thus smaller CR bounds) also show higher irreducible predictive entropy, creating an estimation–prediction tradeoff.","https://schema.org",{"og:url":50,"og:type":86,"og:title":13,"og:site_name":57,"og:description":14},"article",{"robots":88,"canonical":50},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,114,119,122,126,129,133],{"id":21,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":45,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":51,"doc_module":4,"doc_module_name":45,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":28,"slug":125},9,"Religion & Spirituality","religion-spirituality",{"id":28,"doc_module":4,"doc_module_name":45,"category_name":127,"show_sort_weight":28,"slug":128},"World Cup","world-cup",{"id":130,"doc_module":4,"doc_module_name":45,"category_name":131,"show_sort_weight":130,"slug":132},10,"Lifestyle","lifestyle",{"id":134,"doc_module":4,"doc_module_name":45,"category_name":135,"show_sort_weight":105,"slug":136},19,"General","general"]