author  hoelzl 
Mon, 03 May 2010 14:35:10 +0200  
changeset 36624  25153c08655e 
parent 36623  d26348b667f2 
child 36649  bfd8c550faa6 
permissions  rwrr 
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theory Information 
36623  2 
imports Probability_Space Product_Measure "../Multivariate_Analysis/Convex" 
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begin 
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section "Convex theory" 
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lemma log_setsum: 
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assumes "finite s" "s \<noteq> {}" 

9 
assumes "b > 1" 

10 
assumes "(\<Sum> i \<in> s. a i) = 1" 

11 
assumes "\<And> i. i \<in> s \<Longrightarrow> a i \<ge> 0" 

12 
assumes "\<And> i. i \<in> s \<Longrightarrow> y i \<in> {0 <..}" 

13 
shows "log b (\<Sum> i \<in> s. a i * y i) \<ge> (\<Sum> i \<in> s. a i * log b (y i))" 

14 
proof  

15 
have "convex_on {0 <..} (\<lambda> x.  log b x)" 

16 
by (rule minus_log_convex[OF `b > 1`]) 

17 
hence " log b (\<Sum> i \<in> s. a i * y i) \<le> (\<Sum> i \<in> s. a i *  log b (y i))" 

18 
using convex_on_setsum[of _ _ "\<lambda> x.  log b x"] assms pos_is_convex by fastsimp 

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thus ?thesis by (auto simp add:setsum_negf le_imp_neg_le) 

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qed 

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lemma log_setsum': 
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assumes "finite s" "s \<noteq> {}" 

24 
assumes "b > 1" 

25 
assumes "(\<Sum> i \<in> s. a i) = 1" 

26 
assumes pos: "\<And> i. i \<in> s \<Longrightarrow> 0 \<le> a i" 

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"\<And> i. \<lbrakk> i \<in> s ; 0 < a i \<rbrakk> \<Longrightarrow> 0 < y i" 

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shows "log b (\<Sum> i \<in> s. a i * y i) \<ge> (\<Sum> i \<in> s. a i * log b (y i))" 

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proof  
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have "\<And>y. (\<Sum> i \<in> s  {i. a i = 0}. a i * y i) = (\<Sum> i \<in> s. a i * y i)" 
31 
using assms by (auto intro!: setsum_mono_zero_cong_left) 

32 
moreover have "log b (\<Sum> i \<in> s  {i. a i = 0}. a i * y i) \<ge> (\<Sum> i \<in> s  {i. a i = 0}. a i * log b (y i))" 

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proof (rule log_setsum) 

34 
have "setsum a (s  {i. a i = 0}) = setsum a s" 

35 
using assms(1) by (rule setsum_mono_zero_cong_left) auto 

36 
thus sum_1: "setsum a (s  {i. a i = 0}) = 1" 

37 
"finite (s  {i. a i = 0})" using assms by simp_all 

38 

39 
show "s  {i. a i = 0} \<noteq> {}" 

40 
proof 

41 
assume *: "s  {i. a i = 0} = {}" 

42 
hence "setsum a (s  {i. a i = 0}) = 0" by (simp add: * setsum_empty) 

43 
with sum_1 show False by simp 

44 
qed 

45 

46 
fix i assume "i \<in> s  {i. a i = 0}" 

47 
hence "i \<in> s" "a i \<noteq> 0" by simp_all 

48 
thus "0 \<le> a i" "y i \<in> {0<..}" using pos[of i] by auto 

49 
qed fact+ 

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ultimately show ?thesis by simp 

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qed 
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36624  53 
section "Information theory" 
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lemma (in finite_prob_space) sum_over_space_distrib: 

56 
"(\<Sum>x\<in>X`space M. distribution X {x}) = 1" 

57 
unfolding distribution_def prob_space[symmetric] using finite_space 

58 
by (subst measure_finitely_additive'') 

59 
(auto simp add: disjoint_family_on_def sets_eq_Pow intro!: arg_cong[where f=prob]) 

60 

61 
locale finite_information_space = finite_prob_space + 

62 
fixes b :: real assumes b_gt_1: "1 < b" 

63 

64 
definition 

65 
"KL_divergence b M X Y = 

66 
measure_space.integral (M\<lparr>measure := X\<rparr>) 

67 
(\<lambda>x. log b ((measure_space.RN_deriv (M \<lparr>measure := Y\<rparr> ) X) x))" 

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69 
lemma (in finite_prob_space) distribution_mono: 

70 
assumes "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y" 

71 
shows "distribution X x \<le> distribution Y y" 

72 
unfolding distribution_def 

73 
using assms by (auto simp: sets_eq_Pow intro!: measure_mono) 

74 

75 
lemma (in prob_space) distribution_remove_const: 

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shows "joint_distribution X (\<lambda>x. ()) {(x, ())} = distribution X {x}" 

77 
and "joint_distribution (\<lambda>x. ()) X {((), x)} = distribution X {x}" 

78 
and "joint_distribution X (\<lambda>x. (Y x, ())) {(x, y, ())} = joint_distribution X Y {(x, y)}" 

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and "joint_distribution X (\<lambda>x. ((), Y x)) {(x, (), y)} = joint_distribution X Y {(x, y)}" 

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and "distribution (\<lambda>x. ()) {()} = 1" 

81 
unfolding prob_space[symmetric] 

82 
by (auto intro!: arg_cong[where f=prob] simp: distribution_def) 

83 

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context finite_information_space 

86 
begin 

87 

88 
lemma distribution_mono_gt_0: 

89 
assumes gt_0: "0 < distribution X x" 

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assumes *: "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y" 

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shows "0 < distribution Y y" 

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by (rule less_le_trans[OF gt_0 distribution_mono]) (rule *) 

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lemma 

95 
assumes "0 \<le> A" and pos: "0 < A \<Longrightarrow> 0 < B" "0 < A \<Longrightarrow> 0 < C" 

96 
shows mult_log_mult: "A * log b (B * C) = A * log b B + A * log b C" (is "?mult") 

97 
and mult_log_divide: "A * log b (B / C) = A * log b B  A * log b C" (is "?div") 

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proof  
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have "?mult \<and> ?div" 
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proof (cases "A = 0") 

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case False 

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hence "0 < A" using `0 \<le> A` by auto 

103 
with pos[OF this] show "?mult \<and> ?div" using b_gt_1 

104 
by (auto simp: log_divide log_mult field_simps) 

105 
qed simp 

106 
thus ?mult and ?div by auto 

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qed 
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lemma split_pairs: 
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shows 

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"((A, B) = X) \<longleftrightarrow> (fst X = A \<and> snd X = B)" and 

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"(X = (A, B)) \<longleftrightarrow> (fst X = A \<and> snd X = B)" by auto 

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ML {* 

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(* tactic to solve equations of the form @{term "W * log b (X / (Y * Z)) = W * log b X  W * log b (Y * Z)"} 

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where @{term W} is a joint distribution of @{term X}, @{term Y}, and @{term Z}. *) 

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val mult_log_intros = [@{thm mult_log_divide}, @{thm mult_log_mult}] 

120 
val intros = [@{thm divide_pos_pos}, @{thm mult_pos_pos}, @{thm positive_distribution}] 

121 

122 
val distribution_gt_0_tac = (rtac @{thm distribution_mono_gt_0} 

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THEN' assume_tac 

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THEN' clarsimp_tac (clasimpset_of @{context} addsimps2 @{thms split_pairs})) 

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val distr_mult_log_eq_tac = REPEAT_ALL_NEW (CHANGED o TRY o 

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(resolve_tac (mult_log_intros @ intros) 

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ORELSE' distribution_gt_0_tac 

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ORELSE' clarsimp_tac (clasimpset_of @{context}))) 

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fun instanciate_term thy redex intro = 

132 
let 

133 
val intro_concl = Thm.concl_of intro 

134 

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val lhs = intro_concl > HOLogic.dest_Trueprop > HOLogic.dest_eq > fst 

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val m = SOME (Pattern.match thy (lhs, redex) (Vartab.empty, Vartab.empty)) 

138 
handle Pattern.MATCH => NONE 

139 

140 
in 

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Option.map (fn m => Envir.subst_term m intro_concl) m 

142 
end 

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fun mult_log_simproc simpset redex = 

145 
let 

146 
val ctxt = Simplifier.the_context simpset 

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val thy = ProofContext.theory_of ctxt 

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fun prove (SOME thm) = (SOME 

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(Goal.prove ctxt [] [] thm (K (distr_mult_log_eq_tac 1)) 

150 
> mk_meta_eq) 

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handle THM _ => NONE) 

152 
 prove NONE = NONE 

153 
in 

154 
get_first (instanciate_term thy (term_of redex) #> prove) mult_log_intros 

155 
end 

156 
*} 

157 

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simproc_setup mult_log ("distribution X x * log b (A * B)"  

159 
"distribution X x * log b (A / B)") = {* K mult_log_simproc *} 

160 

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end 

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lemma KL_divergence_eq_finite: 

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assumes u: "finite_measure_space (M\<lparr>measure := u\<rparr>)" 

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assumes v: "finite_measure_space (M\<lparr>measure := v\<rparr>)" 

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assumes u_0: "\<And>x. \<lbrakk> x \<in> space M ; v {x} = 0 \<rbrakk> \<Longrightarrow> u {x} = 0" 

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shows "KL_divergence b M u v = (\<Sum>x\<in>space M. u {x} * log b (u {x} / v {x}))" (is "_ = ?sum") 

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proof (simp add: KL_divergence_def, subst finite_measure_space.integral_finite_singleton, simp_all add: u) 

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have ms_u: "measure_space (M\<lparr>measure := u\<rparr>)" 

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using u unfolding finite_measure_space_def by simp 

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show "(\<Sum>x \<in> space M. log b (measure_space.RN_deriv (M\<lparr>measure := v\<rparr>) u x) * u {x}) = ?sum" 

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apply (rule setsum_cong[OF refl]) 

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apply simp 

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apply (safe intro!: arg_cong[where f="log b"] ) 

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apply (subst finite_measure_space.RN_deriv_finite_singleton) 

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using assms ms_u by auto 

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qed 
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lemma log_setsum_divide: 
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assumes "finite S" and "S \<noteq> {}" and "1 < b" 

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assumes "(\<Sum>x\<in>S. g x) = 1" 

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assumes pos: "\<And>x. x \<in> S \<Longrightarrow> g x \<ge> 0" "\<And>x. x \<in> S \<Longrightarrow> f x \<ge> 0" 

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assumes g_pos: "\<And>x. \<lbrakk> x \<in> S ; 0 < g x \<rbrakk> \<Longrightarrow> 0 < f x" 

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shows " (\<Sum>x\<in>S. g x * log b (g x / f x)) \<le> log b (\<Sum>x\<in>S. f x)" 

186 
proof  

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have log_mono: "\<And>x y. 0 < x \<Longrightarrow> x \<le> y \<Longrightarrow> log b x \<le> log b y" 

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using `1 < b` by (subst log_le_cancel_iff) auto 

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36624  190 
have " (\<Sum>x\<in>S. g x * log b (g x / f x)) = (\<Sum>x\<in>S. g x * log b (f x / g x))" 
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proof (unfold setsum_negf[symmetric], rule setsum_cong) 

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fix x assume x: "x \<in> S" 

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show " (g x * log b (g x / f x)) = g x * log b (f x / g x)" 

194 
proof (cases "g x = 0") 

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case False 

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with pos[OF x] g_pos[OF x] have "0 < f x" "0 < g x" by simp_all 

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thus ?thesis using `1 < b` by (simp add: log_divide field_simps) 

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qed simp 

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qed rule 

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also have "... \<le> log b (\<Sum>x\<in>S. g x * (f x / g x))" 

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proof (rule log_setsum') 

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fix x assume x: "x \<in> S" "0 < g x" 

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with g_pos[OF x] show "0 < f x / g x" by (safe intro!: divide_pos_pos) 

204 
qed fact+ 

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also have "... = log b (\<Sum>x\<in>S  {x. g x = 0}. f x)" using `finite S` 

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by (auto intro!: setsum_mono_zero_cong_right arg_cong[where f="log b"] 

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split: split_if_asm) 

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also have "... \<le> log b (\<Sum>x\<in>S. f x)" 

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proof (rule log_mono) 

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have "0 = (\<Sum>x\<in>S  {x. g x = 0}. 0)" by simp 

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also have "... < (\<Sum>x\<in>S  {x. g x = 0}. f x)" (is "_ < ?sum") 

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proof (rule setsum_strict_mono) 

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show "finite (S  {x. g x = 0})" using `finite S` by simp 

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show "S  {x. g x = 0} \<noteq> {}" 

215 
proof 

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assume "S  {x. g x = 0} = {}" 

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hence "(\<Sum>x\<in>S. g x) = 0" by (subst setsum_0') auto 

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with `(\<Sum>x\<in>S. g x) = 1` show False by simp 

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qed 

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fix x assume "x \<in> S  {x. g x = 0}" 

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thus "0 < f x" using g_pos[of x] pos(1)[of x] by auto 

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qed 

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finally show "0 < ?sum" . 

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show "(\<Sum>x\<in>S  {x. g x = 0}. f x) \<le> (\<Sum>x\<in>S. f x)" 

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using `finite S` pos by (auto intro!: setsum_mono2) 

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qed 
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finally show ?thesis . 
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qed 
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36624  230 
lemma KL_divergence_positive_finite: 
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assumes u: "finite_prob_space (M\<lparr>measure := u\<rparr>)" 

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assumes v: "finite_prob_space (M\<lparr>measure := v\<rparr>)" 

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assumes u_0: "\<And>x. \<lbrakk> x \<in> space M ; v {x} = 0 \<rbrakk> \<Longrightarrow> u {x} = 0" 

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and "1 < b" 

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shows "0 \<le> KL_divergence b M u v" 

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proof  

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interpret u: finite_prob_space "M\<lparr>measure := u\<rparr>" using u . 

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interpret v: finite_prob_space "M\<lparr>measure := v\<rparr>" using v . 

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36624  240 
have *: "space M \<noteq> {}" using u.not_empty by simp 
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36624  242 
have " (KL_divergence b M u v) \<le> log b (\<Sum>x\<in>space M. v {x})" 
243 
proof (subst KL_divergence_eq_finite, safe intro!: log_setsum_divide *) 

244 
show "finite_measure_space (M\<lparr>measure := u\<rparr>)" 

245 
"finite_measure_space (M\<lparr>measure := v\<rparr>)" 

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using u v unfolding finite_prob_space_eq by simp_all 

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36624  248 
show "finite (space M)" using u.finite_space by simp 
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show "1 < b" by fact 

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show "(\<Sum>x\<in>space M. u {x}) = 1" using u.sum_over_space_eq_1 by simp 

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36624  252 
fix x assume x: "x \<in> space M" 
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thus pos: "0 \<le> u {x}" "0 \<le> v {x}" 

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using u.positive u.sets_eq_Pow v.positive v.sets_eq_Pow by simp_all 

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36624  256 
{ assume "v {x} = 0" from u_0[OF x this] show "u {x} = 0" . } 
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{ assume "0 < u {x}" 

258 
hence "v {x} \<noteq> 0" using u_0[OF x] by auto 

259 
with pos show "0 < v {x}" by simp } 

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qed 
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thus "0 \<le> KL_divergence b M u v" using v.sum_over_space_eq_1 by simp 
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qed 
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definition (in prob_space) 
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"mutual_information b s1 s2 X Y \<equiv> 
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let prod_space = 
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prod_measure_space (\<lparr>space = space s1, sets = sets s1, measure = distribution X\<rparr>) 
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(\<lparr>space = space s2, sets = sets s2, measure = distribution Y\<rparr>) 
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in 
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KL_divergence b prod_space (joint_distribution X Y) (measure prod_space)" 
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36624  272 
abbreviation (in finite_information_space) 
273 
finite_mutual_information ("\<I>'(_ ; _')") where 

274 
"\<I>(X ; Y) \<equiv> mutual_information b 

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\<lparr> space = X`space M, sets = Pow (X`space M) \<rparr> 
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\<lparr> space = Y`space M, sets = Pow (Y`space M) \<rparr> X Y" 
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36624  278 
lemma (in finite_measure_space) measure_spaceI: "measure_space M" 
279 
by unfold_locales 

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36624  281 
lemma prod_measure_times_finite: 
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assumes fms: "finite_measure_space M" "finite_measure_space M'" and a: "a \<in> space M \<times> space M'" 

283 
shows "prod_measure M M' {a} = measure M {fst a} * measure M' {snd a}" 

284 
proof (cases a) 

285 
case (Pair b c) 

286 
hence a_eq: "{a} = {b} \<times> {c}" by simp 

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36624  288 
with fms[THEN finite_measure_space.measure_spaceI] 
289 
fms[THEN finite_measure_space.sets_eq_Pow] a Pair 

290 
show ?thesis unfolding a_eq 

291 
by (subst prod_measure_times) simp_all 

292 
qed 

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36624  294 
lemma setsum_cartesian_product': 
295 
"(\<Sum>x\<in>A \<times> B. f x) = (\<Sum>x\<in>A. setsum (\<lambda>y. f (x, y)) B)" 

296 
unfolding setsum_cartesian_product by simp 

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36624  298 
lemma (in finite_information_space) 
299 
assumes MX: "finite_prob_space \<lparr> space = space MX, sets = sets MX, measure = distribution X\<rparr>" 

300 
(is "finite_prob_space ?MX") 

301 
assumes MY: "finite_prob_space \<lparr> space = space MY, sets = sets MY, measure = distribution Y\<rparr>" 

302 
(is "finite_prob_space ?MY") 

303 
and X_space: "X ` space M \<subseteq> space MX" and Y_space: "Y ` space M \<subseteq> space MY" 

304 
shows mutual_information_eq_generic: 

305 
"mutual_information b MX MY X Y = (\<Sum> (x,y) \<in> space MX \<times> space MY. 

306 
joint_distribution X Y {(x,y)} * 

307 
log b (joint_distribution X Y {(x,y)} / 

308 
(distribution X {x} * distribution Y {y})))" 

309 
(is "?equality") 

310 
and mutual_information_positive_generic: 

311 
"0 \<le> mutual_information b MX MY X Y" (is "?positive") 

312 
proof  

313 
let ?P = "prod_measure_space ?MX ?MY" 

314 
let ?measure = "joint_distribution X Y" 

315 
let ?P' = "measure_update (\<lambda>_. ?measure) ?P" 

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36624  317 
interpret X: finite_prob_space "?MX" using MX . 
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moreover interpret Y: finite_prob_space "?MY" using MY . 

319 
ultimately have ms_X: "measure_space ?MX" 

320 
and ms_Y: "measure_space ?MY" by unfold_locales 

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321 

36624  322 
have fms_P: "finite_measure_space ?P" 
323 
by (rule finite_measure_space_finite_prod_measure) fact+ 

324 

325 
have fms_P': "finite_measure_space ?P'" 

326 
using finite_product_measure_space[of "space MX" "space MY"] 

327 
X.finite_space Y.finite_space sigma_prod_sets_finite[OF X.finite_space Y.finite_space] 

328 
X.sets_eq_Pow Y.sets_eq_Pow 

329 
by (simp add: prod_measure_space_def) 

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330 

36624  331 
{ fix x assume "x \<in> space ?P" 
332 
hence x_in_MX: "{fst x} \<in> sets MX" using X.sets_eq_Pow 

333 
by (auto simp: prod_measure_space_def) 

334 

335 
assume "measure ?P {x} = 0" 

336 
with prod_measure_times[OF ms_X ms_Y, of "{fst x}" "{snd x}"] x_in_MX 

337 
have "distribution X {fst x} = 0 \<or> distribution Y {snd x} = 0" 

338 
by (simp add: prod_measure_space_def) 

339 

340 
hence "joint_distribution X Y {x} = 0" 

341 
by (cases x) (auto simp: distribution_order) } 

342 
note measure_0 = this 

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343 

36624  344 
show ?equality 
345 
unfolding Let_def mutual_information_def using fms_P fms_P' measure_0 MX MY 

346 
by (subst KL_divergence_eq_finite) 

347 
(simp_all add: prod_measure_space_def prod_measure_times_finite 

348 
finite_prob_space_eq setsum_cartesian_product') 

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349 

36624  350 
show ?positive 
351 
unfolding Let_def mutual_information_def using measure_0 b_gt_1 

352 
proof (safe intro!: KL_divergence_positive_finite, simp_all) 

353 
from ms_X ms_Y X.top Y.top X.prob_space Y.prob_space 

354 
have "measure ?P (space ?P) = 1" 

355 
by (simp add: prod_measure_space_def, subst prod_measure_times, simp_all) 

356 
with fms_P show "finite_prob_space ?P" 

357 
by (simp add: finite_prob_space_eq) 

358 

359 
from ms_X ms_Y X.top Y.top X.prob_space Y.prob_space Y.not_empty X_space Y_space 

360 
have "measure ?P' (space ?P') = 1" unfolding prob_space[symmetric] 

361 
by (auto simp add: prod_measure_space_def distribution_def vimage_Times comp_def 

362 
intro!: arg_cong[where f=prob]) 

363 
with fms_P' show "finite_prob_space ?P'" 

364 
by (simp add: finite_prob_space_eq) 

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365 
qed 
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366 
qed 
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367 

36624  368 
lemma (in finite_information_space) mutual_information_eq: 
369 
"\<I>(X;Y) = (\<Sum> (x,y) \<in> X ` space M \<times> Y ` space M. 

370 
distribution (\<lambda>x. (X x, Y x)) {(x,y)} * log b (distribution (\<lambda>x. (X x, Y x)) {(x,y)} / 

371 
(distribution X {x} * distribution Y {y})))" 

372 
by (subst mutual_information_eq_generic) (simp_all add: finite_prob_space_of_images) 

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373 

36624  374 
lemma (in finite_information_space) mutual_information_positive: "0 \<le> \<I>(X;Y)" 
375 
by (subst mutual_information_positive_generic) (simp_all add: finite_prob_space_of_images) 

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376 

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377 
definition (in prob_space) 
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378 
"entropy b s X = mutual_information b s s X X" 
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379 

36624  380 
abbreviation (in finite_information_space) 
381 
finite_entropy ("\<H>'(_')") where 

382 
"\<H>(X) \<equiv> entropy b \<lparr> space = X`space M, sets = Pow (X`space M) \<rparr> X" 

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383 

36624  384 
lemma (in finite_information_space) joint_distribution_remove[simp]: 
385 
"joint_distribution X X {(x, x)} = distribution X {x}" 

386 
unfolding distribution_def by (auto intro!: arg_cong[where f=prob]) 

387 

388 
lemma (in finite_information_space) entropy_eq: 

389 
"\<H>(X) = (\<Sum> x \<in> X ` space M. distribution X {x} * log b (distribution X {x}))" 

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390 
proof  
36624  391 
{ fix f 
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392 
{ fix x y 
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393 
have "(\<lambda>x. (X x, X x)) ` {(x, y)} = (if x = y then X ` {x} else {})" by auto 
36624  394 
hence "distribution (\<lambda>x. (X x, X x)) {(x,y)} * f x y = (if x = y then distribution X {x} * f x y else 0)" 
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395 
unfolding distribution_def by auto } 
36624  396 
hence "(\<Sum>(x, y) \<in> X ` space M \<times> X ` space M. joint_distribution X X {(x, y)} * f x y) = 
397 
(\<Sum>x \<in> X ` space M. distribution X {x} * f x x)" 

398 
unfolding setsum_cartesian_product' by (simp add: setsum_cases finite_space) } 

399 
note remove_cartesian_product = this 

400 

401 
show ?thesis 

402 
unfolding entropy_def mutual_information_eq setsum_negf[symmetric] remove_cartesian_product 

403 
by (auto intro!: setsum_cong) 

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404 
qed 
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405 

36624  406 
lemma (in finite_information_space) entropy_positive: "0 \<le> \<H>(X)" 
407 
unfolding entropy_def using mutual_information_positive . 

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408 

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409 
definition (in prob_space) 
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410 
"conditional_mutual_information b s1 s2 s3 X Y Z \<equiv> 
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411 
let prod_space = 
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412 
prod_measure_space \<lparr>space = space s2, sets = sets s2, measure = distribution Y\<rparr> 
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413 
\<lparr>space = space s3, sets = sets s3, measure = distribution Z\<rparr> 
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414 
in 
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415 
mutual_information b s1 prod_space X (\<lambda>x. (Y x, Z x))  
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416 
mutual_information b s1 s3 X Z" 
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417 

36624  418 
abbreviation (in finite_information_space) 
419 
finite_conditional_mutual_information ("\<I>'( _ ; _  _ ')") where 

420 
"\<I>(X ; Y  Z) \<equiv> conditional_mutual_information b 

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421 
\<lparr> space = X`space M, sets = Pow (X`space M) \<rparr> 
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422 
\<lparr> space = Y`space M, sets = Pow (Y`space M) \<rparr> 
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423 
\<lparr> space = Z`space M, sets = Pow (Z`space M) \<rparr> 
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424 
X Y Z" 
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425 

36624  426 
lemma (in finite_information_space) setsum_distribution_gen: 
427 
assumes "Z ` {c} \<inter> space M = (\<Union>x \<in> X`space M. Y ` {f x}) \<inter> space M" 

428 
and "inj_on f (X`space M)" 

429 
shows "(\<Sum>x \<in> X`space M. distribution Y {f x}) = distribution Z {c}" 

430 
unfolding distribution_def assms 

431 
using finite_space assms 

432 
by (subst measure_finitely_additive'') 

433 
(auto simp add: disjoint_family_on_def sets_eq_Pow inj_on_def 

434 
intro!: arg_cong[where f=prob]) 

435 

436 
lemma (in finite_information_space) setsum_distribution: 

437 
"(\<Sum>x \<in> X`space M. joint_distribution X Y {(x, y)}) = distribution Y {y}" 

438 
"(\<Sum>y \<in> Y`space M. joint_distribution X Y {(x, y)}) = distribution X {x}" 

439 
"(\<Sum>x \<in> X`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution Y Z {(y, z)}" 

440 
"(\<Sum>y \<in> Y`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Z {(x, z)}" 

441 
"(\<Sum>z \<in> Z`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Y {(x, y)}" 

442 
by (auto intro!: inj_onI setsum_distribution_gen) 

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443 

36624  444 
lemma (in finite_information_space) conditional_mutual_information_eq_sum: 
445 
"\<I>(X ; Y  Z) = 

446 
(\<Sum>(x, y, z)\<in>X ` space M \<times> (\<lambda>x. (Y x, Z x)) ` space M. 

447 
distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} * 

448 
log b (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)}/ 

449 
distribution (\<lambda>x. (Y x, Z x)) {(y, z)}))  

450 
(\<Sum>(x, z)\<in>X ` space M \<times> Z ` space M. 

451 
distribution (\<lambda>x. (X x, Z x)) {(x,z)} * log b (distribution (\<lambda>x. (X x, Z x)) {(x,z)} / distribution Z {z}))" 

452 
(is "_ = ?rhs") 

453 
proof  

454 
have setsum_product: 

455 
"\<And>f x. (\<Sum>v\<in>(\<lambda>x. (Y x, Z x)) ` space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x,v)} * f v) 

456 
= (\<Sum>v\<in>Y ` space M \<times> Z ` space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x,v)} * f v)" 

457 
proof (safe intro!: setsum_mono_zero_cong_left imageI) 

458 
fix x y z f 

459 
assume *: "(Y y, Z z) \<notin> (\<lambda>x. (Y x, Z x)) ` space M" and "y \<in> space M" "z \<in> space M" 

460 
hence "(\<lambda>x. (X x, Y x, Z x)) ` {(x, Y y, Z z)} \<inter> space M = {}" 

461 
proof safe 

462 
fix x' assume x': "x' \<in> space M" and eq: "Y x' = Y y" "Z x' = Z z" 

463 
have "(Y y, Z z) \<in> (\<lambda>x. (Y x, Z x)) ` space M" using eq[symmetric] x' by auto 

464 
thus "x' \<in> {}" using * by auto 

465 
qed 

466 
thus "joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, Y y, Z z)} * f (Y y) (Z z) = 0" 

467 
unfolding distribution_def by simp 

468 
qed (simp add: finite_space) 

469 

470 
thus ?thesis 

471 
unfolding conditional_mutual_information_def Let_def mutual_information_eq 

472 
apply (subst mutual_information_eq_generic) 

473 
by (auto simp add: prod_measure_space_def sigma_prod_sets_finite finite_space 

474 
finite_prob_space_of_images finite_product_prob_space_of_images 

475 
setsum_cartesian_product' setsum_product setsum_subtractf setsum_addf 

476 
setsum_left_distrib[symmetric] setsum_distribution 

477 
cong: setsum_cong) 

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478 
qed 
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479 

36624  480 
lemma (in finite_information_space) conditional_mutual_information_eq: 
481 
"\<I>(X ; Y  Z) = (\<Sum>(x, y, z) \<in> X ` space M \<times> Y ` space M \<times> Z ` space M. 

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482 
distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} * 
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483 
log b (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)}/ 
36624  484 
(joint_distribution X Z {(x, z)} * joint_distribution Y Z {(y,z)} / distribution Z {z})))" 
485 
unfolding conditional_mutual_information_def Let_def mutual_information_eq 

486 
apply (subst mutual_information_eq_generic) 

487 
by (auto simp add: prod_measure_space_def sigma_prod_sets_finite finite_space 

488 
finite_prob_space_of_images finite_product_prob_space_of_images 

489 
setsum_cartesian_product' setsum_product setsum_subtractf setsum_addf 

490 
setsum_left_distrib[symmetric] setsum_distribution setsum_commute[where A="Y`space M"] 

491 
cong: setsum_cong) 

492 

493 
lemma (in finite_information_space) conditional_mutual_information_eq_mutual_information: 

494 
"\<I>(X ; Y) = \<I>(X ; Y  (\<lambda>x. ()))" 

495 
proof  

496 
have [simp]: "(\<lambda>x. ()) ` space M = {()}" using not_empty by auto 

497 

498 
show ?thesis 

499 
unfolding conditional_mutual_information_eq mutual_information_eq 

500 
by (simp add: setsum_cartesian_product' distribution_remove_const) 

501 
qed 

502 

503 
lemma (in finite_information_space) conditional_mutual_information_positive: 

504 
"0 \<le> \<I>(X ; Y  Z)" 

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505 
proof  
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506 
let ?dXYZ = "distribution (\<lambda>x. (X x, Y x, Z x))" 
36624  507 
let ?dXZ = "joint_distribution X Z" 
508 
let ?dYZ = "joint_distribution Y Z" 

36080
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509 
let ?dX = "distribution X" 
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changeset

510 
let ?dZ = "distribution Z" 
36624  511 
let ?M = "X ` space M \<times> Y ` space M \<times> Z ` space M" 
512 

513 
have split_beta: "\<And>f. split f = (\<lambda>x. f (fst x) (snd x))" by (simp add: expand_fun_eq) 

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514 

36624  515 
have " (\<Sum>(x, y, z) \<in> ?M. ?dXYZ {(x, y, z)} * 
516 
log b (?dXYZ {(x, y, z)} / (?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z}))) 

517 
\<le> log b (\<Sum>(x, y, z) \<in> ?M. ?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z})" 

518 
unfolding split_beta 

519 
proof (rule log_setsum_divide) 

520 
show "?M \<noteq> {}" using not_empty by simp 

521 
show "1 < b" using b_gt_1 . 

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522 

36624  523 
fix x assume "x \<in> ?M" 
524 
show "0 \<le> ?dXYZ {(fst x, fst (snd x), snd (snd x))}" using positive_distribution . 

525 
show "0 \<le> ?dXZ {(fst x, snd (snd x))} * ?dYZ {(fst (snd x), snd (snd x))} / ?dZ {snd (snd x)}" 

526 
by (auto intro!: mult_nonneg_nonneg positive_distribution simp: zero_le_divide_iff) 

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527 

36624  528 
assume *: "0 < ?dXYZ {(fst x, fst (snd x), snd (snd x))}" 
529 
thus "0 < ?dXZ {(fst x, snd (snd x))} * ?dYZ {(fst (snd x), snd (snd x))} / ?dZ {snd (snd x)}" 

530 
by (auto intro!: divide_pos_pos mult_pos_pos 

531 
intro: distribution_order(6) distribution_mono_gt_0) 

532 
qed (simp_all add: setsum_cartesian_product' sum_over_space_distrib setsum_distribution finite_space) 

533 
also have "(\<Sum>(x, y, z) \<in> ?M. ?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z}) = (\<Sum>z\<in>Z`space M. ?dZ {z})" 

534 
apply (simp add: setsum_cartesian_product') 

535 
apply (subst setsum_commute) 

536 
apply (subst (2) setsum_commute) 

537 
by (auto simp: setsum_divide_distrib[symmetric] setsum_product[symmetric] setsum_distribution 

538 
intro!: setsum_cong) 

539 
finally show ?thesis 

540 
unfolding conditional_mutual_information_eq sum_over_space_distrib by simp 

36080
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changeset

541 
qed 
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542 

36624  543 

36080
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544 
definition (in prob_space) 
0d9affa4e73c
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545 
"conditional_entropy b S T X Y = conditional_mutual_information b S S T X X Y" 
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546 

36624  547 
abbreviation (in finite_information_space) 
548 
finite_conditional_entropy ("\<H>'(_  _')") where 

549 
"\<H>(X  Y) \<equiv> conditional_entropy b 

36080
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550 
\<lparr> space = X`space M, sets = Pow (X`space M) \<rparr> 
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551 
\<lparr> space = Y`space M, sets = Pow (Y`space M) \<rparr> X Y" 
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552 

36624  553 
lemma (in finite_information_space) conditional_entropy_positive: 
554 
"0 \<le> \<H>(X  Y)" unfolding conditional_entropy_def using conditional_mutual_information_positive . 

36080
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555 

36624  556 
lemma (in finite_information_space) conditional_entropy_eq: 
557 
"\<H>(X  Z) = 

36080
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558 
 (\<Sum>(x, z)\<in>X ` space M \<times> Z ` space M. 
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559 
joint_distribution X Z {(x, z)} * 
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560 
log b (joint_distribution X Z {(x, z)} / distribution Z {z}))" 
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561 
proof  
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562 
have *: "\<And>x y z. (\<lambda>x. (X x, X x, Z x)) ` {(x, y, z)} = (if x = y then (\<lambda>x. (X x, Z x)) ` {(x, z)} else {})" by auto 
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563 
show ?thesis 
36624  564 
unfolding conditional_mutual_information_eq_sum 
565 
conditional_entropy_def distribution_def * 

36080
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566 
by (auto intro!: setsum_0') 
0d9affa4e73c
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567 
qed 
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568 

36624  569 
lemma (in finite_information_space) mutual_information_eq_entropy_conditional_entropy: 
570 
"\<I>(X ; Z) = \<H>(X)  \<H>(X  Z)" 

571 
unfolding mutual_information_eq entropy_eq conditional_entropy_eq 

36080
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Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

572 
using finite_space 
36624  573 
by (auto simp add: setsum_addf setsum_subtractf setsum_cartesian_product' 
574 
setsum_left_distrib[symmetric] setsum_addf setsum_distribution) 

36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

575 

36624  576 
lemma (in finite_information_space) conditional_entropy_less_eq_entropy: 
577 
"\<H>(X  Z) \<le> \<H>(X)" 

578 
proof  

579 
have "\<I>(X ; Z) = \<H>(X)  \<H>(X  Z)" using mutual_information_eq_entropy_conditional_entropy . 

580 
with mutual_information_positive[of X Z] entropy_positive[of X] 

581 
show ?thesis by auto 

36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

582 
qed 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

583 

0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

584 
(* Entropy of a RV with a certain event is zero *) 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

585 

36624  586 
lemma (in finite_information_space) finite_entropy_certainty_eq_0: 
587 
assumes "x \<in> X ` space M" and "distribution X {x} = 1" 

588 
shows "\<H>(X) = 0" 

36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

589 
proof  
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

590 
interpret X: finite_prob_space "\<lparr> space = X ` space M, 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

591 
sets = Pow (X ` space M), 
36624  592 
measure = distribution X\<rparr>" by (rule finite_prob_space_of_images) 
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

593 

0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

594 
have "distribution X (X ` space M  {x}) = distribution X (X ` space M)  distribution X {x}" 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

595 
using X.measure_compl[of "{x}"] assms by auto 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

596 
also have "\<dots> = 0" using X.prob_space assms by auto 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

597 
finally have X0: "distribution X (X ` space M  {x}) = 0" by auto 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

598 

0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

599 
{ fix y assume asm: "y \<noteq> x" "y \<in> X ` space M" 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

600 
hence "{y} \<subseteq> X ` space M  {x}" by auto 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

601 
from X.measure_mono[OF this] X0 X.positive[of "{y}"] asm 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

602 
have "distribution X {y} = 0" by auto } 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

603 

0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

604 
hence fi: "\<And> y. y \<in> X ` space M \<Longrightarrow> distribution X {y} = (if x = y then 1 else 0)" 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

605 
using assms by auto 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

606 

0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

607 
have y: "\<And>y. (if x = y then 1 else 0) * log b (if x = y then 1 else 0) = 0" by simp 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

608 

36624  609 
show ?thesis unfolding entropy_eq by (auto simp: y fi) 
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

610 
qed 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

611 
(*  upper bound on entropy for a rv  *) 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

612 

36624  613 
lemma (in finite_information_space) finite_entropy_le_card: 
614 
"\<H>(X) \<le> log b (real (card (X ` space M \<inter> {x . distribution X {x} \<noteq> 0})))" 

36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

615 
proof  
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

616 
interpret X: finite_prob_space "\<lparr>space = X ` space M, 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

617 
sets = Pow (X ` space M), 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

618 
measure = distribution X\<rparr>" 
36624  619 
using finite_prob_space_of_images by auto 
620 

36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

621 
have triv: "\<And> x. (if distribution X {x} \<noteq> 0 then distribution X {x} else 0) = distribution X {x}" 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

622 
by auto 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

623 
hence sum1: "(\<Sum> x \<in> X ` space M \<inter> {y. distribution X {y} \<noteq> 0}. distribution X {x}) = 1" 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

624 
using X.measure_finitely_additive''[of "X ` space M" "\<lambda> x. {x}", simplified] 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

625 
sets_eq_Pow inj_singleton[unfolded inj_on_def, rule_format] 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

626 
unfolding disjoint_family_on_def X.prob_space[symmetric] 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

627 
using finite_imageI[OF finite_space, of X] by (auto simp add:triv setsum_restrict_set) 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

628 
have pos: "\<And> x. x \<in> X ` space M \<inter> {y. distribution X {y} \<noteq> 0} \<Longrightarrow> inverse (distribution X {x}) > 0" 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

629 
using X.positive sets_eq_Pow unfolding inverse_positive_iff_positive less_le by auto 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

630 
{ assume asm: "X ` space M \<inter> {y. distribution X {y} \<noteq> 0} = {}" 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

631 
{ fix x assume "x \<in> X ` space M" 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

632 
hence "distribution X {x} = 0" using asm by blast } 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

633 
hence A: "(\<Sum> x \<in> X ` space M. distribution X {x}) = 0" by auto 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

634 
have B: "(\<Sum> x \<in> X ` space M. distribution X {x}) 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

635 
\<ge> (\<Sum> x \<in> X ` space M \<inter> {y. distribution X {y} \<noteq> 0}. distribution X {x})" 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

636 
using finite_imageI[OF finite_space, of X] 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

637 
by (subst setsum_mono2) auto 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

638 
from A B have "False" using sum1 by auto } note not_empty = this 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

639 
{ fix x assume asm: "x \<in> X ` space M" 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

640 
have " distribution X {x} * log b (distribution X {x}) 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

641 
=  (if distribution X {x} \<noteq> 0 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

642 
then distribution X {x} * log b (distribution X {x}) 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

643 
else 0)" 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

644 
by auto 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

645 
also have "\<dots> = (if distribution X {x} \<noteq> 0 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

646 
then distribution X {x} *  log b (distribution X {x}) 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

647 
else 0)" 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

648 
by auto 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

649 
also have "\<dots> = (if distribution X {x} \<noteq> 0 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

650 
then distribution X {x} * log b (inverse (distribution X {x})) 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

651 
else 0)" 
36624  652 
using log_inverse b_gt_1 X.positive[of "{x}"] asm by auto 
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

653 
finally have " distribution X {x} * log b (distribution X {x}) 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

654 
= (if distribution X {x} \<noteq> 0 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

655 
then distribution X {x} * log b (inverse (distribution X {x})) 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

656 
else 0)" 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

657 
by auto } note log_inv = this 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

658 
have " (\<Sum> x \<in> X ` space M. distribution X {x} * log b (distribution X {x})) 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

659 
= (\<Sum> x \<in> X ` space M. (if distribution X {x} \<noteq> 0 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

660 
then distribution X {x} * log b (inverse (distribution X {x})) 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

661 
else 0))" 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

662 
unfolding setsum_negf[symmetric] using log_inv by auto 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

663 
also have "\<dots> = (\<Sum> x \<in> X ` space M \<inter> {y. distribution X {y} \<noteq> 0}. 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

664 
distribution X {x} * log b (inverse (distribution X {x})))" 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

665 
unfolding setsum_restrict_set[OF finite_imageI[OF finite_space, of X]] by auto 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

666 
also have "\<dots> \<le> log b (\<Sum> x \<in> X ` space M \<inter> {y. distribution X {y} \<noteq> 0}. 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

667 
distribution X {x} * (inverse (distribution X {x})))" 
36624  668 
apply (subst log_setsum[OF _ _ b_gt_1 sum1, 
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

669 
unfolded greaterThan_iff, OF _ _ _]) using pos sets_eq_Pow 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

670 
X.finite_space assms X.positive not_empty by auto 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

671 
also have "\<dots> = log b (\<Sum> x \<in> X ` space M \<inter> {y. distribution X {y} \<noteq> 0}. 1)" 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

672 
by auto 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

673 
also have "\<dots> \<le> log b (real_of_nat (card (X ` space M \<inter> {y. distribution X {y} \<noteq> 0})))" 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

674 
by auto 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

675 
finally have " (\<Sum>x\<in>X ` space M. distribution X {x} * log b (distribution X {x})) 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

676 
\<le> log b (real_of_nat (card (X ` space M \<inter> {y. distribution X {y} \<noteq> 0})))" by simp 
36624  677 
thus ?thesis unfolding entropy_eq real_eq_of_nat by auto 
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

678 
qed 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

679 

0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

680 
(*  entropy is maximal for a uniform rv  *) 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

681 

0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

682 
lemma (in finite_prob_space) uniform_prob: 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

683 
assumes "x \<in> space M" 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

684 
assumes "\<And> x y. \<lbrakk>x \<in> space M ; y \<in> space M\<rbrakk> \<Longrightarrow> prob {x} = prob {y}" 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

685 
shows "prob {x} = 1 / real (card (space M))" 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

686 
proof  
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

687 
have prob_x: "\<And> y. y \<in> space M \<Longrightarrow> prob {y} = prob {x}" 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

688 
using assms(2)[OF _ `x \<in> space M`] by blast 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

689 
have "1 = prob (space M)" 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

690 
using prob_space by auto 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

691 
also have "\<dots> = (\<Sum> x \<in> space M. prob {x})" 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

692 
using measure_finitely_additive''[of "space M" "\<lambda> x. {x}", simplified] 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

693 
sets_eq_Pow inj_singleton[unfolded inj_on_def, rule_format] 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

694 
finite_space unfolding disjoint_family_on_def prob_space[symmetric] 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

695 
by (auto simp add:setsum_restrict_set) 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

696 
also have "\<dots> = (\<Sum> y \<in> space M. prob {x})" 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

697 
using prob_x by auto 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

698 
also have "\<dots> = real_of_nat (card (space M)) * prob {x}" by simp 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

699 
finally have one: "1 = real (card (space M)) * prob {x}" 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

700 
using real_eq_of_nat by auto 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

701 
hence two: "real (card (space M)) \<noteq> 0" by fastsimp 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

702 
from one have three: "prob {x} \<noteq> 0" by fastsimp 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

703 
thus ?thesis using one two three divide_cancel_right 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

704 
by (auto simp:field_simps) 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

705 
qed 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

706 

36624  707 
lemma (in finite_information_space) finite_entropy_uniform_max: 
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

708 
assumes "\<And>x y. \<lbrakk> x \<in> X ` space M ; y \<in> X ` space M \<rbrakk> \<Longrightarrow> distribution X {x} = distribution X {y}" 
36624  709 
shows "\<H>(X) = log b (real (card (X ` space M)))" 
36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

710 
proof  
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

711 
interpret X: finite_prob_space "\<lparr>space = X ` space M, 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

712 
sets = Pow (X ` space M), 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

713 
measure = distribution X\<rparr>" 
36624  714 
using finite_prob_space_of_images by auto 
715 

36080
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

716 
{ fix x assume xasm: "x \<in> X ` space M" 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

717 
hence card_gt0: "real (card (X ` space M)) > 0" 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

718 
using card_gt_0_iff X.finite_space by auto 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

719 
from xasm have "\<And> y. y \<in> X ` space M \<Longrightarrow> distribution X {y} = distribution X {x}" 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

720 
using assms by blast 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

721 
hence " (\<Sum>x\<in>X ` space M. distribution X {x} * log b (distribution X {x})) 
36624  722 
=  real (card (X ` space M)) * distribution X {x} * log b (distribution X {x})" 
723 
unfolding real_eq_of_nat by auto 

36080
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Added Information theory and Example: dining cryptographers
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parents:
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changeset

724 
also have "\<dots> =  real (card (X ` space M)) * (1 / real (card (X ` space M))) * log b (1 / real (card (X ` space M)))" 
36624  725 
by (auto simp: X.uniform_prob[simplified, OF xasm assms]) 
36080
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Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

726 
also have "\<dots> = log b (real (card (X ` space M)))" 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
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parents:
diff
changeset

727 
unfolding inverse_eq_divide[symmetric] 
36624  728 
using card_gt0 log_inverse b_gt_1 
36080
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Added Information theory and Example: dining cryptographers
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parents:
diff
changeset

729 
by (auto simp add:field_simps card_gt0) 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
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parents:
diff
changeset

730 
finally have ?thesis 
36624  731 
unfolding entropy_eq by auto } 
36080
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Added Information theory and Example: dining cryptographers
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parents:
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changeset

732 
moreover 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
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parents:
diff
changeset

733 
{ assume "X ` space M = {}" 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
hoelzl
parents:
diff
changeset

734 
hence "distribution X (X ` space M) = 0" 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
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parents:
diff
changeset

735 
using X.empty_measure by simp 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
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diff
changeset

736 
hence "False" using X.prob_space by auto } 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
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parents:
diff
changeset

737 
ultimately show ?thesis by auto 
0d9affa4e73c
Added Information theory and Example: dining cryptographers
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parents:
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changeset

738 
qed 
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Added Information theory and Example: dining cryptographers
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parents:
diff
changeset

739 

36624  740 
definition "subvimage A f g \<longleftrightarrow> (\<forall>x \<in> A. f ` {f x} \<inter> A \<subseteq> g ` {g x} \<inter> A)" 
741 

742 
lemma subvimageI: 

743 
assumes "\<And>x y. \<lbrakk> x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y" 

744 
shows "subvimage A f g" 

745 
using assms unfolding subvimage_def by blast 

746 

747 
lemma subvimageE[consumes 1]: 

748 
assumes "subvimage A f g" 

749 
obtains "\<And>x y. \<lbrakk> x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y" 

750 
using assms unfolding subvimage_def by blast 

751 

752 
lemma subvimageD: 

753 
"\<lbrakk> subvimage A f g ; x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y" 

754 
using assms unfolding subvimage_def by blast 

755 

756 
lemma subvimage_subset: 

757 
"\<lbrakk> subvimage B f g ; A \<subseteq> B \<rbrakk> \<Longrightarrow> subvimage A f g" 

758 
unfolding subvimage_def by auto 

759 

760 
lemma subvimage_idem[intro]: "subvimage A g g" 

761 
by (safe intro!: subvimageI) 

762 

763 
lemma subvimage_comp_finer[intro]: 

764 
assumes svi: "subvimage A g h" 

765 
shows "subvimage A g (f \<circ> h)" 

766 
proof (rule subvimageI, simp) 

767 
fix x y assume "x \<in> A" "y \<in> A" "g x = g y" 

768 
from svi[THEN subvimageD, OF this] 

769 
show "f (h x) = f (h y)" by simp 

770 
qed 

771 

772 
lemma subvimage_comp_gran: 

773 
assumes svi: "subvimage A g h" 

774 
assumes inj: "inj_on f (g ` A)" 

775 
shows "subvimage A (f \<circ> g) h" 

776 
by (rule subvimageI) (auto intro!: subvimageD[OF svi] simp: inj_on_iff[OF inj]) 

777 

778 
lemma subvimage_comp: 

779 
assumes svi: "subvimage (f ` A) g h" 

780 
shows "subvimage A (g \<circ> f) (h \<circ> f)" 

781 
by (rule subvimageI) (auto intro!: svi[THEN subvimageD]) 

782 

783 
lemma subvimage_trans: 

784 
assumes fg: "subvimage A f g" 

785 
assumes gh: "subvimage A g h" 

786 
shows "subvimage A f h" 

787 
by (rule subvimageI) (auto intro!: fg[THEN subvimageD] gh[THEN subvimageD]) 

788 

789 
lemma subvimage_translator: 

790 
assumes svi: "subvimage A f g" 

791 
shows "\<exists>h. \<forall>x \<in> A. h (f x) = g x" 

792 
proof (safe intro!: exI[of _ "\<lambda>x. (THE z. z \<in> (g ` (f ` {x} \<inter> A)))"]) 

793 
fix x assume "x \<in> A" 

794 
show "(THE x'. x' \<in> (g ` (f ` {f x} \<inter> A))) = g x" 

795 
by (rule theI2[of _ "g x"]) 

796 
(insert `x \<in> A`, auto intro!: svi[THEN subvimageD]) 

797 
qed 

798 

799 
lemma subvimage_translator_image: 

800 
assumes svi: "subvimage A f g" 

801 
shows "\<exists>h. h ` f ` A = g ` A" 

802 
proof  

803 
from subvimage_translator[OF svi] 

804 
obtain h where "\<And>x. x \<in> A \<Longrightarrow> h (f x) = g x" by auto 

805 
thus ?thesis 

806 
by (auto intro!: exI[of _ h] 

807 
simp: image_compose[symmetric] comp_def cong: image_cong) 

808 
qed 

809 

810 
lemma subvimage_finite: 

811 
assumes svi: "subvimage A f g" and fin: "finite (f`A)" 

812 
shows "finite (g`A)" 

813 
proof  

814 
from subvimage_translator_image[OF svi] 

815 
obtain h where "g`A = h`f`A" by fastsimp 

816 
with fin show "finite (g`A)" by simp 

817 
qed 

818 

819 
lemma subvimage_disj: 

820 
assumes svi: "subvimage A f g" 

821 
shows "f ` {x} \<inter> A \<subseteq> g ` {y} \<inter> A \<or> 

822 
f ` {x} \<inter> g ` {y} \<inter> A = {}" (is "?sub \<or> ?dist") 

823 
proof (rule disjCI) 

824 
assume "\<not> ?dist" 

825 
then obtain z where "z \<in> A" and "x = f z" and "y = g z" by auto 

826 
thus "?sub" using svi unfolding subvimage_def by auto 

827 
qed 

828 

829 
lemma setsum_image_split: 

830 
assumes svi: "subvimage A f g" and fin: "finite (f ` A)" 

831 
shows "(\<Sum>x\<in>f`A. h x) = (\<Sum>y\<in>g`A. \<Sum>x\<in>f`(g ` {y} \<inter> A). h x)" 

832 
(is "?lhs = ?rhs") 

833 
proof  

834 
have "f ` A = 

835 
snd ` (SIGMA x : g ` A. f ` (g ` {x} \<inter> A))" 

836 
(is "_ = snd ` ?SIGMA") 

837 
unfolding image_split_eq_Sigma[symmetric] 

838 
by (simp add: image_compose[symmetric] comp_def) 

839 
moreover 

840 
have snd_inj: "inj_on snd ?SIGMA" 

841 
unfolding image_split_eq_Sigma[symmetric] 

842 
by (auto intro!: inj_onI subvimageD[OF svi]) 

843 
ultimately 

844 
have "(\<Sum>x\<in>f`A. h x) = (\<Sum>(x,y)\<in>?SIGMA. h y)" 

845 
by (auto simp: setsum_reindex intro: setsum_cong) 

846 
also have "... = ?rhs" 

847 
using subvimage_finite[OF svi fin] fin 

848 
apply (subst setsum_Sigma[symmetric]) 

849 
by (auto intro!: finite_subset[of _ "f`A"]) 

850 
finally show ?thesis . 

851 
qed 

852 

853 
lemma (in finite_information_space) entropy_partition: 

854 
assumes svi: "subvimage (space M) X P" 

855 
shows "\<H>(X) = \<H>(P) + \<H>(XP)" 

856 
proof  

857 
have "(\<Sum>x\<in>X ` space M. distribution X {x} * log b (distribution X {x})) = 

858 
(\<Sum>y\<in>P `space M. \<Sum>x\<in>X ` space M. 

859 
joint_distribution X P {(x, y)} * log b (joint_distribution X P {(x, y)}))" 

860 
proof (subst setsum_image_split[OF svi], 

861 
safe intro!: finite_imageI finite_space setsum_mono_zero_cong_left imageI) 

862 
fix p x assume in_space: "p \<in> space M" "x \<in> space M" 

863 
assume "joint_distribution X P {(X x, P p)} * log b (joint_distribution X P {(X x, P p)}) \<noteq> 0" 

864 
hence "(\<lambda>x. (X x, P x)) ` {(X x, P p)} \<inter> space M \<noteq> {}" by (auto simp: distribution_def) 

865 
with svi[unfolded subvimage_def, rule_format, OF `x \<in> space M`] 

866 
show "x \<in> P ` {P p}" by auto 

867 
next 

868 
fix p x assume in_space: "p \<in> space M" "x \<in> space M" 

869 
assume "P x = P p" 

870 
from this[symmetric] svi[unfolded subvimage_def, rule_format, OF `x \<in> space M`] 

871 
have "X ` {X x} \<inter> space M \<subseteq> P ` {P p} \<inter> space M" 

872 
by auto 

873 
hence "(\<lambda>x. (X x, P x)) ` {(X x, P p)} \<inter> space M = X ` {X x} \<inter> space M" 

874 
by auto 

875 
thus "distribution X {X x} * log b (distribution X {X x}) = 

876 
joint_distribution X P {(X x, P p)} * 

877 
log b (joint_distribution X P {(X x, P p)})" 

878 
by (auto simp: distribution_def) 

879 
qed 

880 
thus ?thesis 

881 
unfolding entropy_eq conditional_entropy_eq 

882 
by (simp add: setsum_cartesian_product' setsum_subtractf setsum_distribution 

883 
setsum_left_distrib[symmetric] setsum_commute[where B="P`space M"]) 

884 
qed 

885 

886 
corollary (in finite_information_space) entropy_data_processing: 

887 
"\<H>(f \<circ> X) \<le> \<H>(X)" 

888 
by (subst (2) entropy_partition[of _ "f \<circ> X"]) (auto intro: conditional_entropy_positive) 

889 

890 
lemma (in prob_space) distribution_cong: 

891 
assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = Y x" 

892 
shows "distribution X = distribution Y" 

893 
unfolding distribution_def expand_fun_eq 

894 
using assms by (auto intro!: arg_cong[where f=prob]) 

895 

896 
lemma (in prob_space) joint_distribution_cong: 

897 
assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x" 

898 
assumes "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x" 

899 
shows "joint_distribution X Y = joint_distribution X' Y'" 

900 
unfolding distribution_def expand_fun_eq 

901 
using assms by (auto intro!: arg_cong[where f=prob]) 

902 

903 
lemma image_cong: 

904 
"\<lbrakk> \<And>x. x \<in> S \<Longrightarrow> X x = X' x \<rbrakk> \<Longrightarrow> X ` S = X' ` S" 

905 
by (auto intro!: image_eqI) 

906 

907 
lemma (in finite_information_space) mutual_information_cong: 

908 
assumes X: "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x" 

909 
assumes Y: "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x" 

910 
shows "\<I>(X ; Y) = \<I>(X' ; Y')" 

911 
proof  

912 
have "X ` space M = X' ` space M" using X by (rule image_cong) 

913 
moreover have "Y ` space M = Y' ` space M" using Y by (rule image_cong) 

914 
ultimately show ?thesis 

915 
unfolding mutual_information_eq 

916 
using 

917 
assms[THEN distribution_cong] 

918 
joint_distribution_cong[OF assms] 

919 
by (auto intro!: setsum_cong) 

920 
qed 

921 

922 
corollary (in finite_information_space) entropy_of_inj: 

923 
assumes "inj_on f (X`space M)" 

924 
shows "\<H>(f \<circ> X) = \<H>(X)" 

925 
proof (rule antisym) 

926 
show "\<H>(f \<circ> X) \<le> \<H>(X)" using entropy_data_processing . 

927 
next 

928 
have "\<H>(X) = \<H>(the_inv_into (X`space M) f \<circ> (f \<circ> X))" 

929 
by (auto intro!: mutual_information_cong simp: entropy_def the_inv_into_f_f[OF assms]) 

930 
also have "... \<le> \<H>(f \<circ> X)" 

931 
using entropy_data_processing . 

932 
finally show "\<H>(X) \<le> \<H>(f \<circ> X)" . 

933 
qed 

934 

36080
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hoelzl
parents:
diff
changeset

935 
end 