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Tumor segmentation in oncological PET is challenging, a major reason being the partial-volume effects that arise due to low system resolution and finite voxel size. The latter results in tissue-fraction effects, i.e. voxels contain a mixture of tissue classes. Conventional segmentation methods are typically designed to assign each voxel in the image as belonging to a certain tissue class. Thus, these methods are inherently limited in modeling tissue-fraction effects. To address the challenge of accounting for partial-volume effects, and in particular, tissue-fraction effects, we propose a Bayesian approach to tissue-fraction estimation for oncological PET segmentation. Specifically, this Bayesian approach estimates the posterior mean of fractional volume that the tumor occupies within each voxel of the image. The proposed method, implemented using a deep-learning-based technique, was first evaluated using clinically realistic 2-D simulation studies with known ground truth, in the context of segmenting the primary tumor in PET images of patients with lung cancer. The evaluation studies demonstrated that the method accurately estimated the tumor-fraction areas and significantly outperformed widely used conventional PET segmentation methods, including a U-net-based method, on the task of segmenting the tumor. In addition, the proposed method was relatively insensitive to partial-volume effects and yielded reliable tumor segmentation for different clinical-scanner configurations. The method was then evaluated using clinical images of patients with stage IIB/III non-small cell lung cancer from ACRIN 6668/RTOG 0235 multi-center clinical trial. Here, the results showed that the proposed method significantly outperformed all other considered methods and yielded accurate tumor segmentation on patient images with Dice similarity coefficient (DSC) of 0.82 (95 % CI: [0.78, 0.86]).