Determining Optimal number of Seasonal Fourier terms using Bayesian Method

I am trying to model annual seasonality using a Fourier Series. In the frequentist approach, the optimal number of Fourier terms to keep (truncated Fourier) is determined by comparing AIC values. Is there a way to do this easily with the Bayesian approach?

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Yes, you can use either WAIC or LOO information criteria to compare between models fitted with different number of terms, like what is done in this GLM example. These criteria are equivalent to AIC if all the involved distributions are gaussian, but are more widely applicable to other probability distributions than AIC.