On information gain, Kullback-Leibler divergence, entropy production and the involution kernel

It is well known that in Information Theory and Machine Learning the Kullback-Leibler divergence, which extends the concept of Shannon entropy, plays a fundamental role. Given an {\it a priori} probability kernel $\hat{\nu}$ and a probability $\pi$ on the measurable space $X\times Y$ we consider an appropriate definition of entropy of $\pi$ relative to $\hat{\nu}$, which is based on previous works. Using this concept of entropy we obtain a natural definition of information gain for general measurable spaces which coincides with the mutual information given from the K-L divergence in the case $\hat{\nu}$ is identified with a probability $\nu$ on $X$. This will be used to extend the meaning of specific information gain and dynamical entropy production to the model of thermodynamic formalism for symbolic dynamics over a compact alphabet (TFCA model). In this case, we show that the involution kernel is a natural tool for better understanding some important properties of entropy production.