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It is well known that in Information Theory and Machine Learning theKullback-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 $X\times Y$ weconsider an appropriate definition of entropy of $\pi$ relative to $\hat{\nu}$,which is based on previous works. Using this concept of entropy we obtain anatural definition of information gain for general measurable spaces whichcoincides 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 toextend the meaning of specific information gain and dynamical entropyproduction to the model of thermodynamic formalism for symbolic dynamics over acompact alphabet (TFCA model). In this case, we show that the involution kernelis a natural tool for better understanding some important properties of entropyproduction.

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