We propose a universal channel coding when each output distribution forms an exponential family even in a continuous output system. We propose two types of universal codes; One has the exponentially decreasing error with explicit a lower bound for the error exponent. The other attains the $\epsilon$-capacity up to the second order. Our encoder is the same as the previous paper [CMP {\bf 289}, 1087]. For our decoding process, we invent $\alpha$-R\'{e}nyi divergence version of Clarke and Barron's formula for Bayesian average distribution, which are not required in the previous paper. Combing this formula and the information spectrum method, we propose our universal decoder. Our method enables us to treat a universal code of the continuous and discrete cases in a unified way with the second order coding rate as well as with the exponential decreasing rate.
['Masahito Hayashi']
Universal channel coding for exponential family of channels