1.1 --- a/src/HOL/Probability/Information.thy Wed Oct 10 12:12:35 2012 +0200
1.2 +++ b/src/HOL/Probability/Information.thy Wed Oct 10 12:12:36 2012 +0200
1.3 @@ -1367,6 +1367,20 @@
1.4 finally show ?thesis ..
1.5 qed
1.6
1.7 +lemma (in information_space) mutual_information_eq_entropy_conditional_entropy':
1.8 + fixes Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real" and Pxy :: "('b \<times> 'c) \<Rightarrow> real"
1.9 + assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
1.10 + assumes Px: "distributed M S X Px" and Py: "distributed M T Y Py"
1.11 + assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
1.12 + assumes Ix: "integrable(S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Px (fst x)))"
1.13 + assumes Iy: "integrable(S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Py (snd x)))"
1.14 + assumes Ixy: "integrable(S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Pxy x))"
1.15 + shows "mutual_information b S T X Y = entropy b S X - conditional_entropy b S T X Y"
1.16 + using
1.17 + mutual_information_eq_entropy_conditional_entropy_distr[OF S T Px Py Pxy Ix Iy Ixy]
1.18 + conditional_entropy_eq_entropy[OF S T Py Pxy Ixy Iy]
1.19 + by simp
1.20 +
1.21 lemma (in information_space) mutual_information_eq_entropy_conditional_entropy:
1.22 assumes sf_X: "simple_function M X" and sf_Y: "simple_function M Y"
1.23 shows "\<I>(X ; Y) = \<H>(X) - \<H>(X | Y)"