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\bibcite{he2016deep}{{14}{2016}{{He et~al.}}{{He, Zhang, Ren, and Sun}}}
\bibcite{hochreiter1997long}{{15}{1997}{{Hochreiter and Schmidhuber}}{{}}}
\bibcite{kalra2018online}{{16}{2018}{{Kalra et~al.}}{{Kalra, Rashwan, Hsu, Poupart, Doshi, and Trimponias}}}
\bibcite{koenker1978regression}{{17}{1978}{{Koenker and Bassett~Jr}}{{}}}
\bibcite{li2019Enhancing}{{18}{2019}{{Li et~al.}}{{Li, Jin, Xuan, Zhou, Chen, Wang, and Yan}}}
\bibcite{Liang05}{{19}{2005}{{Liang}}{{}}}
\bibcite{makridakis1993accuracy}{{20}{1993}{{Makridakis}}{{}}}
\bibcite{makridakis2020m4}{{21}{2020}{{Makridakis et~al.}}{{Makridakis, Spiliotis, and Assimakopoulos}}}
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\bibcite{melibari2016dynamic}{{23}{2016}{{Melibari et~al.}}{{Melibari, Poupart, Doshi, and Trimponias}}}
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\bibcite{oreshkin2019n}{{25}{2019}{{Oreshkin et~al.}}{{Oreshkin, Carpov, Chapados, and Bengio}}}
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\bibcite{recknagel2001applications}{{32}{2001}{{Recknagel}}{{}}}
\bibcite{rumelhart1985learning}{{33}{1985}{{Rumelhart et~al.}}{{Rumelhart, Hinton, and Williams}}}
\bibcite{SALINAS20201181}{{34}{2020}{{Salinas et~al.}}{{Salinas, Flunkert, Gasthaus, and Januschowski}}}
\bibcite{seeger2004gaussian}{{35}{2004}{{Seeger}}{{}}}
\bibcite{siami2018comparison}{{36}{2018}{{Siami-Namini et~al.}}{{Siami-Namini, Tavakoli, and Namin}}}
\bibcite{stankeviciute2021conformal}{{37}{2021}{{Stankeviciute et~al.}}{{Stankeviciute, M~Alaa, and van~der Schaar}}}
\bibcite{Stankevi2021}{{38}{2021}{{Stankevičiūtė et~al.}}{{Stankevičiūtė, Alaa, and van~der Schaar}}}
\bibcite{sun2014monthly}{{39}{2014}{{Sun et~al.}}{{Sun, Wang, and Xu}}}
\bibcite{tank2015bayesian}{{40}{2015}{{Tank et~al.}}{{Tank, Foti, and Fox}}}
\bibcite{trabelsi2018deep}{{41}{2018}{{Trabelsi et~al.}}{{Trabelsi, Bilaniuk, Zhang, Serdyuk, Subramanian, Santos, Mehri, Rostamzadeh, Bengio, and Pal}}}
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\bibcite{vaswani2017attention}{{43}{2017}{{Vaswani et~al.}}{{Vaswani, Shazeer, Parmar, Uszkoreit, Jones, Gomez, Kaiser, and Polosukhin}}}
\bibcite{whittle1953analysis}{{44}{1953}{{Whittle}}{{}}}
\bibcite{wolter2018complex}{{45}{2018}{{Wolter and Yao}}{{}}}
\bibcite{wolter2020sequence}{{46}{2020}{{Wolter et~al.}}{{Wolter, Gall, and Yao}}}
\bibcite{yang2020complex}{{47}{2020}{{Yang et~al.}}{{Yang, Ma, Li, Tsai, and Salakhutdinov}}}
\bibcite{yu2021icml_wspn}{{48}{2021{a}}{{Yu et~al.}}{{Yu, Ventola, and Kersting}}}
\bibcite{yu2021uai_momogps}{{49}{2021{b}}{{Yu et~al.}}{{Yu, Zhu, Trapp, Skryagin, and Kersting}}}
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