  Bayesian P-splines and basis determination through Bayesian model selection
are both commonly employed strategies for nonparametric regression using spline
basis expansions within the Bayesian framework. Despite their widespread use,
each method has particular limitations that may introduce potential estimation
bias depending on the nature of the target function. To overcome the
limitations associated with each method while capitalizing on their respective
strengths, we propose a new prior distribution that integrates the essentials
of both approaches. The proposed prior distribution assesses the complexity of
the spline model based on a penalty term formed by a convex combination of the
penalties from both methods. The proposed method exhibits adaptability to the
unknown level of smoothness, while achieving the minimax-optimal posterior
contraction rate up to a logarithmic factor. We provide an efficient Markov
chain Monte Carlo algorithm for implementing the proposed approach. Our
extensive simulation study reveals that the proposed method outperforms other
competitors in terms of performance metrics or model complexity.
