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However, due to natural disasters, wars, material deterioration, and human errors, records documenting the construction years of many artifacts have often been lost. Historians and experts can estimate construction years within specific ranges using chemical-based analysis technologies or extensive historical research. Given the vast number of collected artifacts, applying these conventional methods to every artifact is impractical. To address this challenge, we developed a deep neural network model designed for Buddha statues to estimate an artifact\u2019s construction year from its image. One major challenge in this task is the heterogeneity of the labels: the training samples include both precise construction years and possible ranges (e.g., a dynasty or a century) estimated by historians. To unify these heterogeneous labels during training, we represent them as probabilistic distributions. In our previous work\u00a0Qian et al. (2021), we assumed that the ambiguity in heterogeneous construction year labels followed a Gaussian distribution, assigning the highest likelihood to the midpoint of the designated time range. However, this assumption does not always hold. In this paper, we propose representing heterogeneous construction year labels as a uniform distribution, assigning equal probability to all points within the designated time range. Based on this label representation, we designed a semi-supervised learning loss function to leverage both labeled and unlabeled samples during training. Our experimental results demonstrate that our method achieves a mean absolute error of 34.3 years on a test set consisting of Buddha statues constructed between 400 and 1403. These results are further analyzed in two ways. First, we compared our model\u2019s performance to the image quality BRISQUE score, revealing a correlation between higher image quality and lower prediction error rates. Second, we validated our predictions with experts, assessing the level of agreement with our model, the challenges in determining construction years, and identifying features of interest in the artifacts.<\/jats:p>","DOI":"10.1007\/s11042-025-20784-4","type":"journal-article","created":{"date-parts":[[2025,4,24]],"date-time":"2025-04-24T07:27:14Z","timestamp":1745479634000},"page":"39427-39450","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Built year prediction of buddha face with heterogeneous label modeled as probabilistic distribution"],"prefix":"10.1007","volume":"84","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1795-2038","authenticated-orcid":false,"given":"Yiming","family":"Qian","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Cheikh Brahim El","family":"Vaigh","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuta","family":"Nakashima","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Benjamin","family":"Renoust","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hajime","family":"Nagahara","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yutaka","family":"Fujioka","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,4,24]]},"reference":[{"key":"20784_CR1","doi-asserted-by":"crossref","unstructured":"Qian Y, El\u00a0Vaigh CB, Nakashima Y, Renoust B, Nagahara H, Fujioka Y (2021) Built year prediction from buddha face with heterogeneous labels. 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All participants are treated equally and ethically.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical"}}]}}