  We present a probabilistic framework for both (i) determining the initial
settings of kernel adaptive filters (KAFs) and (ii) constructing fully-adaptive
KAFs whereby in addition to weights and dictionaries, kernel parameters are
learnt sequentially. This is achieved by formulating the estimator as a
probabilistic model and defining dedicated prior distributions over the kernel
parameters, weights and dictionary, enforcing desired properties such as
sparsity. The model can then be trained using a subset of data to initialise
standard KAFs or updated sequentially each time a new observation becomes
available. Due to the nonlinear/non-Gaussian properties of the model, learning
and inference is achieved using gradient-based maximum-a-posteriori
optimisation and Markov chain Monte Carlo methods, and can be confidently used
to compute predictions. The proposed framework was validated on nonlinear time
series of both synthetic and real-world nature, where it outperformed standard
KAFs in terms of mean square error and the sparsity of the learnt dictionaries.
