src/HOL/Probability/Information.thy
 author hoelzl Tue May 04 18:19:24 2010 +0200 (2010-05-04) changeset 36649 bfd8c550faa6 parent 36624 25153c08655e child 38656 d5d342611edb permissions -rw-r--r--
Corrected imports; better approximation of dependencies.
```     1 theory Information
```
```     2 imports Probability_Space Product_Measure Convex
```
```     3 begin
```
```     4
```
```     5 section "Convex theory"
```
```     6
```
```     7 lemma log_setsum:
```
```     8   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
```
```    19   thus ?thesis by (auto simp add:setsum_negf le_imp_neg_le)
```
```    20 qed
```
```    21
```
```    22 lemma log_setsum':
```
```    23   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"
```
```    27           "\<And> i. \<lbrakk> i \<in> s ; 0 < a i \<rbrakk> \<Longrightarrow> 0 < y i"
```
```    28   shows "log b (\<Sum> i \<in> s. a i * y i) \<ge> (\<Sum> i \<in> s. a i * log b (y i))"
```
```    29 proof -
```
```    30   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))"
```
```    33   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+
```
```    50   ultimately show ?thesis by simp
```
```    51 qed
```
```    52
```
```    53 section "Information theory"
```
```    54
```
```    55 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))"
```
```    68
```
```    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:
```
```    76   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)}"
```
```    79   and "joint_distribution X (\<lambda>x. ((), Y x)) {(x, (), y)} = joint_distribution X Y {(x, y)}"
```
```    80   and "distribution (\<lambda>x. ()) {()} = 1"
```
```    81   unfolding prob_space[symmetric]
```
```    82   by (auto intro!: arg_cong[where f=prob] simp: distribution_def)
```
```    83
```
```    84
```
```    85 context finite_information_space
```
```    86 begin
```
```    87
```
```    88 lemma distribution_mono_gt_0:
```
```    89   assumes gt_0: "0 < distribution X x"
```
```    90   assumes *: "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y"
```
```    91   shows "0 < distribution Y y"
```
```    92   by (rule less_le_trans[OF gt_0 distribution_mono]) (rule *)
```
```    93
```
```    94 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")
```
```    98 proof -
```
```    99   have "?mult \<and> ?div"
```
```   100 proof (cases "A = 0")
```
```   101   case False
```
```   102   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
```
```   107 qed
```
```   108
```
```   109 lemma split_pairs:
```
```   110   shows
```
```   111     "((A, B) = X) \<longleftrightarrow> (fst X = A \<and> snd X = B)" and
```
```   112     "(X = (A, B)) \<longleftrightarrow> (fst X = A \<and> snd X = B)" by auto
```
```   113
```
```   114 ML {*
```
```   115
```
```   116   (* tactic to solve equations of the form @{term "W * log b (X / (Y * Z)) = W * log b X - W * log b (Y * Z)"}
```
```   117      where @{term W} is a joint distribution of @{term X}, @{term Y}, and @{term Z}. *)
```
```   118
```
```   119   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}
```
```   123     THEN' assume_tac
```
```   124     THEN' clarsimp_tac (clasimpset_of @{context} addsimps2 @{thms split_pairs}))
```
```   125
```
```   126   val distr_mult_log_eq_tac = REPEAT_ALL_NEW (CHANGED o TRY o
```
```   127     (resolve_tac (mult_log_intros @ intros)
```
```   128       ORELSE' distribution_gt_0_tac
```
```   129       ORELSE' clarsimp_tac (clasimpset_of @{context})))
```
```   130
```
```   131   fun instanciate_term thy redex intro =
```
```   132     let
```
```   133       val intro_concl = Thm.concl_of intro
```
```   134
```
```   135       val lhs = intro_concl |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> fst
```
```   136
```
```   137       val m = SOME (Pattern.match thy (lhs, redex) (Vartab.empty, Vartab.empty))
```
```   138         handle Pattern.MATCH => NONE
```
```   139
```
```   140     in
```
```   141       Option.map (fn m => Envir.subst_term m intro_concl) m
```
```   142     end
```
```   143
```
```   144   fun mult_log_simproc simpset redex =
```
```   145   let
```
```   146     val ctxt = Simplifier.the_context simpset
```
```   147     val thy = ProofContext.theory_of ctxt
```
```   148     fun prove (SOME thm) = (SOME
```
```   149           (Goal.prove ctxt [] [] thm (K (distr_mult_log_eq_tac 1))
```
```   150            |> mk_meta_eq)
```
```   151             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
```
```   158 simproc_setup mult_log ("distribution X x * log b (A * B)" |
```
```   159                         "distribution X x * log b (A / B)") = {* K mult_log_simproc *}
```
```   160
```
```   161 end
```
```   162
```
```   163 lemma KL_divergence_eq_finite:
```
```   164   assumes u: "finite_measure_space (M\<lparr>measure := u\<rparr>)"
```
```   165   assumes v: "finite_measure_space (M\<lparr>measure := v\<rparr>)"
```
```   166   assumes u_0: "\<And>x. \<lbrakk> x \<in> space M ; v {x} = 0 \<rbrakk> \<Longrightarrow> u {x} = 0"
```
```   167   shows "KL_divergence b M u v = (\<Sum>x\<in>space M. u {x} * log b (u {x} / v {x}))" (is "_ = ?sum")
```
```   168 proof (simp add: KL_divergence_def, subst finite_measure_space.integral_finite_singleton, simp_all add: u)
```
```   169   have ms_u: "measure_space (M\<lparr>measure := u\<rparr>)"
```
```   170     using u unfolding finite_measure_space_def by simp
```
```   171
```
```   172   show "(\<Sum>x \<in> space M. log b (measure_space.RN_deriv (M\<lparr>measure := v\<rparr>) u x) * u {x}) = ?sum"
```
```   173     apply (rule setsum_cong[OF refl])
```
```   174     apply simp
```
```   175     apply (safe intro!: arg_cong[where f="log b"] )
```
```   176     apply (subst finite_measure_space.RN_deriv_finite_singleton)
```
```   177     using assms ms_u by auto
```
```   178 qed
```
```   179
```
```   180 lemma log_setsum_divide:
```
```   181   assumes "finite S" and "S \<noteq> {}" and "1 < b"
```
```   182   assumes "(\<Sum>x\<in>S. g x) = 1"
```
```   183   assumes pos: "\<And>x. x \<in> S \<Longrightarrow> g x \<ge> 0" "\<And>x. x \<in> S \<Longrightarrow> f x \<ge> 0"
```
```   184   assumes g_pos: "\<And>x. \<lbrakk> x \<in> S ; 0 < g x \<rbrakk> \<Longrightarrow> 0 < f x"
```
```   185   shows "- (\<Sum>x\<in>S. g x * log b (g x / f x)) \<le> log b (\<Sum>x\<in>S. f x)"
```
```   186 proof -
```
```   187   have log_mono: "\<And>x y. 0 < x \<Longrightarrow> x \<le> y \<Longrightarrow> log b x \<le> log b y"
```
```   188     using `1 < b` by (subst log_le_cancel_iff) auto
```
```   189
```
```   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))"
```
```   191   proof (unfold setsum_negf[symmetric], rule setsum_cong)
```
```   192     fix x assume x: "x \<in> S"
```
```   193     show "- (g x * log b (g x / f x)) = g x * log b (f x / g x)"
```
```   194     proof (cases "g x = 0")
```
```   195       case False
```
```   196       with pos[OF x] g_pos[OF x] have "0 < f x" "0 < g x" by simp_all
```
```   197       thus ?thesis using `1 < b` by (simp add: log_divide field_simps)
```
```   198     qed simp
```
```   199   qed rule
```
```   200   also have "... \<le> log b (\<Sum>x\<in>S. g x * (f x / g x))"
```
```   201   proof (rule log_setsum')
```
```   202     fix x assume x: "x \<in> S" "0 < g x"
```
```   203     with g_pos[OF x] show "0 < f x / g x" by (safe intro!: divide_pos_pos)
```
```   204   qed fact+
```
```   205   also have "... = log b (\<Sum>x\<in>S - {x. g x = 0}. f x)" using `finite S`
```
```   206     by (auto intro!: setsum_mono_zero_cong_right arg_cong[where f="log b"]
```
```   207         split: split_if_asm)
```
```   208   also have "... \<le> log b (\<Sum>x\<in>S. f x)"
```
```   209   proof (rule log_mono)
```
```   210     have "0 = (\<Sum>x\<in>S - {x. g x = 0}. 0)" by simp
```
```   211     also have "... < (\<Sum>x\<in>S - {x. g x = 0}. f x)" (is "_ < ?sum")
```
```   212     proof (rule setsum_strict_mono)
```
```   213       show "finite (S - {x. g x = 0})" using `finite S` by simp
```
```   214       show "S - {x. g x = 0} \<noteq> {}"
```
```   215       proof
```
```   216         assume "S - {x. g x = 0} = {}"
```
```   217         hence "(\<Sum>x\<in>S. g x) = 0" by (subst setsum_0') auto
```
```   218         with `(\<Sum>x\<in>S. g x) = 1` show False by simp
```
```   219       qed
```
```   220       fix x assume "x \<in> S - {x. g x = 0}"
```
```   221       thus "0 < f x" using g_pos[of x] pos(1)[of x] by auto
```
```   222     qed
```
```   223     finally show "0 < ?sum" .
```
```   224     show "(\<Sum>x\<in>S - {x. g x = 0}. f x) \<le> (\<Sum>x\<in>S. f x)"
```
```   225       using `finite S` pos by (auto intro!: setsum_mono2)
```
```   226   qed
```
```   227   finally show ?thesis .
```
```   228 qed
```
```   229
```
```   230 lemma KL_divergence_positive_finite:
```
```   231   assumes u: "finite_prob_space (M\<lparr>measure := u\<rparr>)"
```
```   232   assumes v: "finite_prob_space (M\<lparr>measure := v\<rparr>)"
```
```   233   assumes u_0: "\<And>x. \<lbrakk> x \<in> space M ; v {x} = 0 \<rbrakk> \<Longrightarrow> u {x} = 0"
```
```   234   and "1 < b"
```
```   235   shows "0 \<le> KL_divergence b M u v"
```
```   236 proof -
```
```   237   interpret u: finite_prob_space "M\<lparr>measure := u\<rparr>" using u .
```
```   238   interpret v: finite_prob_space "M\<lparr>measure := v\<rparr>" using v .
```
```   239
```
```   240   have *: "space M \<noteq> {}" using u.not_empty by simp
```
```   241
```
```   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>)"
```
```   246        using u v unfolding finite_prob_space_eq by simp_all
```
```   247
```
```   248      show "finite (space M)" using u.finite_space by simp
```
```   249      show "1 < b" by fact
```
```   250      show "(\<Sum>x\<in>space M. u {x}) = 1" using u.sum_over_space_eq_1 by simp
```
```   251
```
```   252      fix x assume x: "x \<in> space M"
```
```   253      thus pos: "0 \<le> u {x}" "0 \<le> v {x}"
```
```   254        using u.positive u.sets_eq_Pow v.positive v.sets_eq_Pow by simp_all
```
```   255
```
```   256      { assume "v {x} = 0" from u_0[OF x this] show "u {x} = 0" . }
```
```   257      { 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 }
```
```   260   qed
```
```   261   thus "0 \<le> KL_divergence b M u v" using v.sum_over_space_eq_1 by simp
```
```   262 qed
```
```   263
```
```   264 definition (in prob_space)
```
```   265   "mutual_information b s1 s2 X Y \<equiv>
```
```   266     let prod_space =
```
```   267       prod_measure_space (\<lparr>space = space s1, sets = sets s1, measure = distribution X\<rparr>)
```
```   268                          (\<lparr>space = space s2, sets = sets s2, measure = distribution Y\<rparr>)
```
```   269     in
```
```   270       KL_divergence b prod_space (joint_distribution X Y) (measure prod_space)"
```
```   271
```
```   272 abbreviation (in finite_information_space)
```
```   273   finite_mutual_information ("\<I>'(_ ; _')") where
```
```   274   "\<I>(X ; Y) \<equiv> mutual_information b
```
```   275     \<lparr> space = X`space M, sets = Pow (X`space M) \<rparr>
```
```   276     \<lparr> space = Y`space M, sets = Pow (Y`space M) \<rparr> X Y"
```
```   277
```
```   278 lemma (in finite_measure_space) measure_spaceI: "measure_space M"
```
```   279   by unfold_locales
```
```   280
```
```   281 lemma prod_measure_times_finite:
```
```   282   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
```
```   287
```
```   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
```
```   293
```
```   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
```
```   297
```
```   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"
```
```   316
```
```   317   interpret X: finite_prob_space "?MX" using MX .
```
```   318   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
```
```   321
```
```   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)
```
```   330
```
```   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
```
```   343
```
```   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')
```
```   349
```
```   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)
```
```   365   qed
```
```   366 qed
```
```   367
```
```   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)
```
```   373
```
```   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)
```
```   376
```
```   377 definition (in prob_space)
```
```   378   "entropy b s X = mutual_information b s s X X"
```
```   379
```
```   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"
```
```   383
```
```   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}))"
```
```   390 proof -
```
```   391   { fix f
```
```   392   { fix x y
```
```   393     have "(\<lambda>x. (X x, X x)) -` {(x, y)} = (if x = y then X -` {x} else {})" by auto
```
```   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)"
```
```   395       unfolding distribution_def by auto }
```
```   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)
```
```   404 qed
```
```   405
```
```   406 lemma (in finite_information_space) entropy_positive: "0 \<le> \<H>(X)"
```
```   407   unfolding entropy_def using mutual_information_positive .
```
```   408
```
```   409 definition (in prob_space)
```
```   410   "conditional_mutual_information b s1 s2 s3 X Y Z \<equiv>
```
```   411     let prod_space =
```
```   412       prod_measure_space \<lparr>space = space s2, sets = sets s2, measure = distribution Y\<rparr>
```
```   413                          \<lparr>space = space s3, sets = sets s3, measure = distribution Z\<rparr>
```
```   414     in
```
```   415       mutual_information b s1 prod_space X (\<lambda>x. (Y x, Z x)) -
```
```   416       mutual_information b s1 s3 X Z"
```
```   417
```
```   418 abbreviation (in finite_information_space)
```
```   419   finite_conditional_mutual_information ("\<I>'( _ ; _ | _ ')") where
```
```   420   "\<I>(X ; Y | Z) \<equiv> conditional_mutual_information b
```
```   421     \<lparr> space = X`space M, sets = Pow (X`space M) \<rparr>
```
```   422     \<lparr> space = Y`space M, sets = Pow (Y`space M) \<rparr>
```
```   423     \<lparr> space = Z`space M, sets = Pow (Z`space M) \<rparr>
```
```   424     X Y Z"
```
```   425
```
```   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)
```
```   443
```
```   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)
```
```   478 qed
```
```   479
```
```   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.
```
```   482              distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} *
```
```   483              log b (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)}/
```
```   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)"
```
```   505 proof -
```
```   506   let ?dXYZ = "distribution (\<lambda>x. (X x, Y x, Z x))"
```
```   507   let ?dXZ = "joint_distribution X Z"
```
```   508   let ?dYZ = "joint_distribution Y Z"
```
```   509   let ?dX = "distribution X"
```
```   510   let ?dZ = "distribution Z"
```
```   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)
```
```   514
```
```   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 .
```
```   522
```
```   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)
```
```   527
```
```   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
```
```   541 qed
```
```   542
```
```   543
```
```   544 definition (in prob_space)
```
```   545   "conditional_entropy b S T X Y = conditional_mutual_information b S S T X X Y"
```
```   546
```
```   547 abbreviation (in finite_information_space)
```
```   548   finite_conditional_entropy ("\<H>'(_ | _')") where
```
```   549   "\<H>(X | Y) \<equiv> conditional_entropy b
```
```   550     \<lparr> space = X`space M, sets = Pow (X`space M) \<rparr>
```
```   551     \<lparr> space = Y`space M, sets = Pow (Y`space M) \<rparr> X Y"
```
```   552
```
```   553 lemma (in finite_information_space) conditional_entropy_positive:
```
```   554   "0 \<le> \<H>(X | Y)" unfolding conditional_entropy_def using conditional_mutual_information_positive .
```
```   555
```
```   556 lemma (in finite_information_space) conditional_entropy_eq:
```
```   557   "\<H>(X | Z) =
```
```   558      - (\<Sum>(x, z)\<in>X ` space M \<times> Z ` space M.
```
```   559          joint_distribution X Z {(x, z)} *
```
```   560          log b (joint_distribution X Z {(x, z)} / distribution Z {z}))"
```
```   561 proof -
```
```   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
```
```   563   show ?thesis
```
```   564     unfolding conditional_mutual_information_eq_sum
```
```   565       conditional_entropy_def distribution_def *
```
```   566     by (auto intro!: setsum_0')
```
```   567 qed
```
```   568
```
```   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
```
```   572   using finite_space
```
```   573   by (auto simp add: setsum_addf setsum_subtractf setsum_cartesian_product'
```
```   574       setsum_left_distrib[symmetric] setsum_addf setsum_distribution)
```
```   575
```
```   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
```
```   582 qed
```
```   583
```
```   584 (* -------------Entropy of a RV with a certain event is zero---------------- *)
```
```   585
```
```   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"
```
```   589 proof -
```
```   590   interpret X: finite_prob_space "\<lparr> space = X ` space M,
```
```   591     sets = Pow (X ` space M),
```
```   592     measure = distribution X\<rparr>" by (rule finite_prob_space_of_images)
```
```   593
```
```   594   have "distribution X (X ` space M - {x}) = distribution X (X ` space M) - distribution X {x}"
```
```   595     using X.measure_compl[of "{x}"] assms by auto
```
```   596   also have "\<dots> = 0" using X.prob_space assms by auto
```
```   597   finally have X0: "distribution X (X ` space M - {x}) = 0" by auto
```
```   598
```
```   599   { fix y assume asm: "y \<noteq> x" "y \<in> X ` space M"
```
```   600     hence "{y} \<subseteq> X ` space M - {x}" by auto
```
```   601     from X.measure_mono[OF this] X0 X.positive[of "{y}"] asm
```
```   602     have "distribution X {y} = 0" by auto }
```
```   603
```
```   604   hence fi: "\<And> y. y \<in> X ` space M \<Longrightarrow> distribution X {y} = (if x = y then 1 else 0)"
```
```   605     using assms by auto
```
```   606
```
```   607   have y: "\<And>y. (if x = y then 1 else 0) * log b (if x = y then 1 else 0) = 0" by simp
```
```   608
```
```   609   show ?thesis unfolding entropy_eq by (auto simp: y fi)
```
```   610 qed
```
```   611 (* --------------- upper bound on entropy for a rv ------------------------- *)
```
```   612
```
```   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})))"
```
```   615 proof -
```
```   616   interpret X: finite_prob_space "\<lparr>space = X ` space M,
```
```   617                                     sets = Pow (X ` space M),
```
```   618                                  measure = distribution X\<rparr>"
```
```   619     using finite_prob_space_of_images by auto
```
```   620
```
```   621   have triv: "\<And> x. (if distribution X {x} \<noteq> 0 then distribution X {x} else 0) = distribution X {x}"
```
```   622     by auto
```
```   623   hence sum1: "(\<Sum> x \<in> X ` space M \<inter> {y. distribution X {y} \<noteq> 0}. distribution X {x}) = 1"
```
```   624     using X.measure_finitely_additive''[of "X ` space M" "\<lambda> x. {x}", simplified]
```
```   625       sets_eq_Pow inj_singleton[unfolded inj_on_def, rule_format]
```
```   626     unfolding disjoint_family_on_def  X.prob_space[symmetric]
```
```   627     using finite_imageI[OF finite_space, of X] by (auto simp add:triv setsum_restrict_set)
```
```   628   have pos: "\<And> x. x \<in> X ` space M \<inter> {y. distribution X {y} \<noteq> 0} \<Longrightarrow> inverse (distribution X {x}) > 0"
```
```   629     using X.positive sets_eq_Pow unfolding inverse_positive_iff_positive less_le by auto
```
```   630   { assume asm: "X ` space M \<inter> {y. distribution X {y} \<noteq> 0} = {}"
```
```   631     { fix x assume "x \<in> X ` space M"
```
```   632       hence "distribution X {x} = 0" using asm by blast }
```
```   633     hence A: "(\<Sum> x \<in> X ` space M. distribution X {x}) = 0" by auto
```
```   634     have B: "(\<Sum> x \<in> X ` space M. distribution X {x})
```
```   635       \<ge> (\<Sum> x \<in> X ` space M \<inter> {y. distribution X {y} \<noteq> 0}. distribution X {x})"
```
```   636       using finite_imageI[OF finite_space, of X]
```
```   637       by (subst setsum_mono2) auto
```
```   638     from A B have "False" using sum1 by auto } note not_empty = this
```
```   639   { fix x assume asm: "x \<in> X ` space M"
```
```   640     have "- distribution X {x} * log b (distribution X {x})
```
```   641        = - (if distribution X {x} \<noteq> 0
```
```   642             then distribution X {x} * log b (distribution X {x})
```
```   643             else 0)"
```
```   644       by auto
```
```   645     also have "\<dots> = (if distribution X {x} \<noteq> 0
```
```   646           then distribution X {x} * - log b (distribution X {x})
```
```   647           else 0)"
```
```   648       by auto
```
```   649     also have "\<dots> = (if distribution X {x} \<noteq> 0
```
```   650                     then distribution X {x} * log b (inverse (distribution X {x}))
```
```   651                     else 0)"
```
```   652       using log_inverse b_gt_1 X.positive[of "{x}"] asm by auto
```
```   653     finally have "- distribution X {x} * log b (distribution X {x})
```
```   654                  = (if distribution X {x} \<noteq> 0
```
```   655                     then distribution X {x} * log b (inverse (distribution X {x}))
```
```   656                     else 0)"
```
```   657       by auto } note log_inv = this
```
```   658   have "- (\<Sum> x \<in> X ` space M. distribution X {x} * log b (distribution X {x}))
```
```   659        = (\<Sum> x \<in> X ` space M. (if distribution X {x} \<noteq> 0
```
```   660           then distribution X {x} * log b (inverse (distribution X {x}))
```
```   661           else 0))"
```
```   662     unfolding setsum_negf[symmetric] using log_inv by auto
```
```   663   also have "\<dots> = (\<Sum> x \<in> X ` space M \<inter> {y. distribution X {y} \<noteq> 0}.
```
```   664                           distribution X {x} * log b (inverse (distribution X {x})))"
```
```   665     unfolding setsum_restrict_set[OF finite_imageI[OF finite_space, of X]] by auto
```
```   666   also have "\<dots> \<le> log b (\<Sum> x \<in> X ` space M \<inter> {y. distribution X {y} \<noteq> 0}.
```
```   667                           distribution X {x} * (inverse (distribution X {x})))"
```
```   668     apply (subst log_setsum[OF _ _ b_gt_1 sum1,
```
```   669      unfolded greaterThan_iff, OF _ _ _]) using pos sets_eq_Pow
```
```   670       X.finite_space assms X.positive not_empty by auto
```
```   671   also have "\<dots> = log b (\<Sum> x \<in> X ` space M \<inter> {y. distribution X {y} \<noteq> 0}. 1)"
```
```   672     by auto
```
```   673   also have "\<dots> \<le> log b (real_of_nat (card (X ` space M \<inter> {y. distribution X {y} \<noteq> 0})))"
```
```   674     by auto
```
```   675   finally have "- (\<Sum>x\<in>X ` space M. distribution X {x} * log b (distribution X {x}))
```
```   676                \<le> log b (real_of_nat (card (X ` space M \<inter> {y. distribution X {y} \<noteq> 0})))" by simp
```
```   677   thus ?thesis unfolding entropy_eq real_eq_of_nat by auto
```
```   678 qed
```
```   679
```
```   680 (* --------------- entropy is maximal for a uniform rv --------------------- *)
```
```   681
```
```   682 lemma (in finite_prob_space) uniform_prob:
```
```   683   assumes "x \<in> space M"
```
```   684   assumes "\<And> x y. \<lbrakk>x \<in> space M ; y \<in> space M\<rbrakk> \<Longrightarrow> prob {x} = prob {y}"
```
```   685   shows "prob {x} = 1 / real (card (space M))"
```
```   686 proof -
```
```   687   have prob_x: "\<And> y. y \<in> space M \<Longrightarrow> prob {y} = prob {x}"
```
```   688     using assms(2)[OF _ `x \<in> space M`] by blast
```
```   689   have "1 = prob (space M)"
```
```   690     using prob_space by auto
```
```   691   also have "\<dots> = (\<Sum> x \<in> space M. prob {x})"
```
```   692     using measure_finitely_additive''[of "space M" "\<lambda> x. {x}", simplified]
```
```   693       sets_eq_Pow inj_singleton[unfolded inj_on_def, rule_format]
```
```   694       finite_space unfolding disjoint_family_on_def  prob_space[symmetric]
```
```   695     by (auto simp add:setsum_restrict_set)
```
```   696   also have "\<dots> = (\<Sum> y \<in> space M. prob {x})"
```
```   697     using prob_x by auto
```
```   698   also have "\<dots> = real_of_nat (card (space M)) * prob {x}" by simp
```
```   699   finally have one: "1 = real (card (space M)) * prob {x}"
```
```   700     using real_eq_of_nat by auto
```
```   701   hence two: "real (card (space M)) \<noteq> 0" by fastsimp
```
```   702   from one have three: "prob {x} \<noteq> 0" by fastsimp
```
```   703   thus ?thesis using one two three divide_cancel_right
```
```   704     by (auto simp:field_simps)
```
```   705 qed
```
```   706
```
```   707 lemma (in finite_information_space) finite_entropy_uniform_max:
```
```   708   assumes "\<And>x y. \<lbrakk> x \<in> X ` space M ; y \<in> X ` space M \<rbrakk> \<Longrightarrow> distribution X {x} = distribution X {y}"
```
```   709   shows "\<H>(X) = log b (real (card (X ` space M)))"
```
```   710 proof -
```
```   711   interpret X: finite_prob_space "\<lparr>space = X ` space M,
```
```   712                                     sets = Pow (X ` space M),
```
```   713                                  measure = distribution X\<rparr>"
```
```   714     using finite_prob_space_of_images by auto
```
```   715
```
```   716   { fix x assume xasm: "x \<in> X ` space M"
```
```   717     hence card_gt0: "real (card (X ` space M)) > 0"
```
```   718       using card_gt_0_iff X.finite_space by auto
```
```   719     from xasm have "\<And> y. y \<in> X ` space M \<Longrightarrow> distribution X {y} = distribution X {x}"
```
```   720       using assms by blast
```
```   721     hence "- (\<Sum>x\<in>X ` space M. distribution X {x} * log b (distribution X {x}))
```
```   722          = - real (card (X ` space M)) * distribution X {x} * log b (distribution X {x})"
```
```   723       unfolding real_eq_of_nat by auto
```
```   724     also have "\<dots> = - real (card (X ` space M)) * (1 / real (card (X ` space M))) * log b (1 / real (card (X ` space M)))"
```
```   725       by (auto simp: X.uniform_prob[simplified, OF xasm assms])
```
```   726     also have "\<dots> = log b (real (card (X ` space M)))"
```
```   727       unfolding inverse_eq_divide[symmetric]
```
```   728       using card_gt0 log_inverse b_gt_1
```
```   729       by (auto simp add:field_simps card_gt0)
```
```   730     finally have ?thesis
```
```   731       unfolding entropy_eq by auto }
```
```   732   moreover
```
```   733   { assume "X ` space M = {}"
```
```   734     hence "distribution X (X ` space M) = 0"
```
```   735       using X.empty_measure by simp
```
```   736     hence "False" using X.prob_space by auto }
```
```   737   ultimately show ?thesis by auto
```
```   738 qed
```
```   739
```
```   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>(X|P)"
```
```   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
```
```   935 end
```