We develop a robust Bayesian functional principal component analysis (RB-FPCA) method that utilizes the skew elliptical class of distributions to model functional data, which are observed over a continuous domain. This approach effectively captures the primary sources of variation among curves, even in the presence of outliers, and provides a more robust and accurate estimation of the covariance function and principal components. The proposed method can also handle sparse functional data, where only a few observations per curve are available. We employ annealed sequential Monte Carlo for posterior inference, which offers several advantages over conventional Markov chain Monte Carlo algorithms. To evaluate the performance of our proposed model, we conduct simulation studies, comparing it with well-known frequentist and conventional Bayesian methods. The results show that our method outperforms existing approaches in the presence of outliers and performs competitively in outlier-free datasets. Finally, we demonstrate the effectiveness of our method by applying it to environmental and biological data to identify outlying functional observations. The implementation of our proposed method and applications are available at https://github.com/SFU-Stat-ML/RBFPCA .
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