theory Information
imports Probability_Space Product_Measure Convex
begin
section "Convex theory"
lemma log_setsum:
assumes "finite s" "s \<noteq> {}"
assumes "b > 1"
assumes "(\<Sum> i \<in> s. a i) = 1"
assumes "\<And> i. i \<in> s \<Longrightarrow> a i \<ge> 0"
assumes "\<And> i. i \<in> s \<Longrightarrow> y i \<in> {0 <..}"
shows "log b (\<Sum> i \<in> s. a i * y i) \<ge> (\<Sum> i \<in> s. a i * log b (y i))"
proof -
have "convex_on {0 <..} (\<lambda> x. - log b x)"
by (rule minus_log_convex[OF `b > 1`])
hence "- log b (\<Sum> i \<in> s. a i * y i) \<le> (\<Sum> i \<in> s. a i * - log b (y i))"
using convex_on_setsum[of _ _ "\<lambda> x. - log b x"] assms pos_is_convex by fastsimp
thus ?thesis by (auto simp add:setsum_negf le_imp_neg_le)
qed
lemma log_setsum':
assumes "finite s" "s \<noteq> {}"
assumes "b > 1"
assumes "(\<Sum> i \<in> s. a i) = 1"
assumes pos: "\<And> i. i \<in> s \<Longrightarrow> 0 \<le> a i"
"\<And> i. \<lbrakk> i \<in> s ; 0 < a i \<rbrakk> \<Longrightarrow> 0 < y i"
shows "log b (\<Sum> i \<in> s. a i * y i) \<ge> (\<Sum> i \<in> s. a i * log b (y i))"
proof -
have "\<And>y. (\<Sum> i \<in> s - {i. a i = 0}. a i * y i) = (\<Sum> i \<in> s. a i * y i)"
using assms by (auto intro!: setsum_mono_zero_cong_left)
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))"
proof (rule log_setsum)
have "setsum a (s - {i. a i = 0}) = setsum a s"
using assms(1) by (rule setsum_mono_zero_cong_left) auto
thus sum_1: "setsum a (s - {i. a i = 0}) = 1"
"finite (s - {i. a i = 0})" using assms by simp_all
show "s - {i. a i = 0} \<noteq> {}"
proof
assume *: "s - {i. a i = 0} = {}"
hence "setsum a (s - {i. a i = 0}) = 0" by (simp add: * setsum_empty)
with sum_1 show False by simp
qed
fix i assume "i \<in> s - {i. a i = 0}"
hence "i \<in> s" "a i \<noteq> 0" by simp_all
thus "0 \<le> a i" "y i \<in> {0<..}" using pos[of i] by auto
qed fact+
ultimately show ?thesis by simp
qed
section "Information theory"
lemma (in finite_prob_space) sum_over_space_distrib:
"(\<Sum>x\<in>X`space M. distribution X {x}) = 1"
unfolding distribution_def prob_space[symmetric] using finite_space
by (subst measure_finitely_additive'')
(auto simp add: disjoint_family_on_def sets_eq_Pow intro!: arg_cong[where f=prob])
locale finite_information_space = finite_prob_space +
fixes b :: real assumes b_gt_1: "1 < b"
definition
"KL_divergence b M X Y =
measure_space.integral (M\<lparr>measure := X\<rparr>)
(\<lambda>x. log b ((measure_space.RN_deriv (M \<lparr>measure := Y\<rparr> ) X) x))"
lemma (in finite_prob_space) distribution_mono:
assumes "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y"
shows "distribution X x \<le> distribution Y y"
unfolding distribution_def
using assms by (auto simp: sets_eq_Pow intro!: measure_mono)
lemma (in prob_space) distribution_remove_const:
shows "joint_distribution X (\<lambda>x. ()) {(x, ())} = distribution X {x}"
and "joint_distribution (\<lambda>x. ()) X {((), x)} = distribution X {x}"
and "joint_distribution X (\<lambda>x. (Y x, ())) {(x, y, ())} = joint_distribution X Y {(x, y)}"
and "joint_distribution X (\<lambda>x. ((), Y x)) {(x, (), y)} = joint_distribution X Y {(x, y)}"
and "distribution (\<lambda>x. ()) {()} = 1"
unfolding prob_space[symmetric]
by (auto intro!: arg_cong[where f=prob] simp: distribution_def)
context finite_information_space
begin
lemma distribution_mono_gt_0:
assumes gt_0: "0 < distribution X x"
assumes *: "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y"
shows "0 < distribution Y y"
by (rule less_le_trans[OF gt_0 distribution_mono]) (rule *)
lemma
assumes "0 \<le> A" and pos: "0 < A \<Longrightarrow> 0 < B" "0 < A \<Longrightarrow> 0 < C"
shows mult_log_mult: "A * log b (B * C) = A * log b B + A * log b C" (is "?mult")
and mult_log_divide: "A * log b (B / C) = A * log b B - A * log b C" (is "?div")
proof -
have "?mult \<and> ?div"
proof (cases "A = 0")
case False
hence "0 < A" using `0 \<le> A` by auto
with pos[OF this] show "?mult \<and> ?div" using b_gt_1
by (auto simp: log_divide log_mult field_simps)
qed simp
thus ?mult and ?div by auto
qed
lemma split_pairs:
shows
"((A, B) = X) \<longleftrightarrow> (fst X = A \<and> snd X = B)" and
"(X = (A, B)) \<longleftrightarrow> (fst X = A \<and> snd X = B)" by auto
ML {*
(* tactic to solve equations of the form @{term "W * log b (X / (Y * Z)) = W * log b X - W * log b (Y * Z)"}
where @{term W} is a joint distribution of @{term X}, @{term Y}, and @{term Z}. *)
val mult_log_intros = [@{thm mult_log_divide}, @{thm mult_log_mult}]
val intros = [@{thm divide_pos_pos}, @{thm mult_pos_pos}, @{thm positive_distribution}]
val distribution_gt_0_tac = (rtac @{thm distribution_mono_gt_0}
THEN' assume_tac
THEN' clarsimp_tac (clasimpset_of @{context} addsimps2 @{thms split_pairs}))
val distr_mult_log_eq_tac = REPEAT_ALL_NEW (CHANGED o TRY o
(resolve_tac (mult_log_intros @ intros)
ORELSE' distribution_gt_0_tac
ORELSE' clarsimp_tac (clasimpset_of @{context})))
fun instanciate_term thy redex intro =
let
val intro_concl = Thm.concl_of intro
val lhs = intro_concl |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> fst
val m = SOME (Pattern.match thy (lhs, redex) (Vartab.empty, Vartab.empty))
handle Pattern.MATCH => NONE
in
Option.map (fn m => Envir.subst_term m intro_concl) m
end
fun mult_log_simproc simpset redex =
let
val ctxt = Simplifier.the_context simpset
val thy = ProofContext.theory_of ctxt
fun prove (SOME thm) = (SOME
(Goal.prove ctxt [] [] thm (K (distr_mult_log_eq_tac 1))
|> mk_meta_eq)
handle THM _ => NONE)
| prove NONE = NONE
in
get_first (instanciate_term thy (term_of redex) #> prove) mult_log_intros
end
*}
simproc_setup mult_log ("distribution X x * log b (A * B)" |
"distribution X x * log b (A / B)") = {* K mult_log_simproc *}
end
lemma KL_divergence_eq_finite:
assumes u: "finite_measure_space (M\<lparr>measure := u\<rparr>)"
assumes v: "finite_measure_space (M\<lparr>measure := v\<rparr>)"
assumes u_0: "\<And>x. \<lbrakk> x \<in> space M ; v {x} = 0 \<rbrakk> \<Longrightarrow> u {x} = 0"
shows "KL_divergence b M u v = (\<Sum>x\<in>space M. u {x} * log b (u {x} / v {x}))" (is "_ = ?sum")
proof (simp add: KL_divergence_def, subst finite_measure_space.integral_finite_singleton, simp_all add: u)
have ms_u: "measure_space (M\<lparr>measure := u\<rparr>)"
using u unfolding finite_measure_space_def by simp
show "(\<Sum>x \<in> space M. log b (measure_space.RN_deriv (M\<lparr>measure := v\<rparr>) u x) * u {x}) = ?sum"
apply (rule setsum_cong[OF refl])
apply simp
apply (safe intro!: arg_cong[where f="log b"] )
apply (subst finite_measure_space.RN_deriv_finite_singleton)
using assms ms_u by auto
qed
lemma log_setsum_divide:
assumes "finite S" and "S \<noteq> {}" and "1 < b"
assumes "(\<Sum>x\<in>S. g x) = 1"
assumes pos: "\<And>x. x \<in> S \<Longrightarrow> g x \<ge> 0" "\<And>x. x \<in> S \<Longrightarrow> f x \<ge> 0"
assumes g_pos: "\<And>x. \<lbrakk> x \<in> S ; 0 < g x \<rbrakk> \<Longrightarrow> 0 < f x"
shows "- (\<Sum>x\<in>S. g x * log b (g x / f x)) \<le> log b (\<Sum>x\<in>S. f x)"
proof -
have log_mono: "\<And>x y. 0 < x \<Longrightarrow> x \<le> y \<Longrightarrow> log b x \<le> log b y"
using `1 < b` by (subst log_le_cancel_iff) auto
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))"
proof (unfold setsum_negf[symmetric], rule setsum_cong)
fix x assume x: "x \<in> S"
show "- (g x * log b (g x / f x)) = g x * log b (f x / g x)"
proof (cases "g x = 0")
case False
with pos[OF x] g_pos[OF x] have "0 < f x" "0 < g x" by simp_all
thus ?thesis using `1 < b` by (simp add: log_divide field_simps)
qed simp
qed rule
also have "... \<le> log b (\<Sum>x\<in>S. g x * (f x / g x))"
proof (rule log_setsum')
fix x assume x: "x \<in> S" "0 < g x"
with g_pos[OF x] show "0 < f x / g x" by (safe intro!: divide_pos_pos)
qed fact+
also have "... = log b (\<Sum>x\<in>S - {x. g x = 0}. f x)" using `finite S`
by (auto intro!: setsum_mono_zero_cong_right arg_cong[where f="log b"]
split: split_if_asm)
also have "... \<le> log b (\<Sum>x\<in>S. f x)"
proof (rule log_mono)
have "0 = (\<Sum>x\<in>S - {x. g x = 0}. 0)" by simp
also have "... < (\<Sum>x\<in>S - {x. g x = 0}. f x)" (is "_ < ?sum")
proof (rule setsum_strict_mono)
show "finite (S - {x. g x = 0})" using `finite S` by simp
show "S - {x. g x = 0} \<noteq> {}"
proof
assume "S - {x. g x = 0} = {}"
hence "(\<Sum>x\<in>S. g x) = 0" by (subst setsum_0') auto
with `(\<Sum>x\<in>S. g x) = 1` show False by simp
qed
fix x assume "x \<in> S - {x. g x = 0}"
thus "0 < f x" using g_pos[of x] pos(1)[of x] by auto
qed
finally show "0 < ?sum" .
show "(\<Sum>x\<in>S - {x. g x = 0}. f x) \<le> (\<Sum>x\<in>S. f x)"
using `finite S` pos by (auto intro!: setsum_mono2)
qed
finally show ?thesis .
qed
lemma KL_divergence_positive_finite:
assumes u: "finite_prob_space (M\<lparr>measure := u\<rparr>)"
assumes v: "finite_prob_space (M\<lparr>measure := v\<rparr>)"
assumes u_0: "\<And>x. \<lbrakk> x \<in> space M ; v {x} = 0 \<rbrakk> \<Longrightarrow> u {x} = 0"
and "1 < b"
shows "0 \<le> KL_divergence b M u v"
proof -
interpret u: finite_prob_space "M\<lparr>measure := u\<rparr>" using u .
interpret v: finite_prob_space "M\<lparr>measure := v\<rparr>" using v .
have *: "space M \<noteq> {}" using u.not_empty by simp
have "- (KL_divergence b M u v) \<le> log b (\<Sum>x\<in>space M. v {x})"
proof (subst KL_divergence_eq_finite, safe intro!: log_setsum_divide *)
show "finite_measure_space (M\<lparr>measure := u\<rparr>)"
"finite_measure_space (M\<lparr>measure := v\<rparr>)"
using u v unfolding finite_prob_space_eq by simp_all
show "finite (space M)" using u.finite_space by simp
show "1 < b" by fact
show "(\<Sum>x\<in>space M. u {x}) = 1" using u.sum_over_space_eq_1 by simp
fix x assume x: "x \<in> space M"
thus pos: "0 \<le> u {x}" "0 \<le> v {x}"
using u.positive u.sets_eq_Pow v.positive v.sets_eq_Pow by simp_all
{ assume "v {x} = 0" from u_0[OF x this] show "u {x} = 0" . }
{ assume "0 < u {x}"
hence "v {x} \<noteq> 0" using u_0[OF x] by auto
with pos show "0 < v {x}" by simp }
qed
thus "0 \<le> KL_divergence b M u v" using v.sum_over_space_eq_1 by simp
qed
definition (in prob_space)
"mutual_information b s1 s2 X Y \<equiv>
let prod_space =
prod_measure_space (\<lparr>space = space s1, sets = sets s1, measure = distribution X\<rparr>)
(\<lparr>space = space s2, sets = sets s2, measure = distribution Y\<rparr>)
in
KL_divergence b prod_space (joint_distribution X Y) (measure prod_space)"
abbreviation (in finite_information_space)
finite_mutual_information ("\<I>'(_ ; _')") where
"\<I>(X ; Y) \<equiv> mutual_information b
\<lparr> space = X`space M, sets = Pow (X`space M) \<rparr>
\<lparr> space = Y`space M, sets = Pow (Y`space M) \<rparr> X Y"
lemma (in finite_measure_space) measure_spaceI: "measure_space M"
by unfold_locales
lemma prod_measure_times_finite:
assumes fms: "finite_measure_space M" "finite_measure_space M'" and a: "a \<in> space M \<times> space M'"
shows "prod_measure M M' {a} = measure M {fst a} * measure M' {snd a}"
proof (cases a)
case (Pair b c)
hence a_eq: "{a} = {b} \<times> {c}" by simp
with fms[THEN finite_measure_space.measure_spaceI]
fms[THEN finite_measure_space.sets_eq_Pow] a Pair
show ?thesis unfolding a_eq
by (subst prod_measure_times) simp_all
qed
lemma setsum_cartesian_product':
"(\<Sum>x\<in>A \<times> B. f x) = (\<Sum>x\<in>A. setsum (\<lambda>y. f (x, y)) B)"
unfolding setsum_cartesian_product by simp
lemma (in finite_information_space)
assumes MX: "finite_prob_space \<lparr> space = space MX, sets = sets MX, measure = distribution X\<rparr>"
(is "finite_prob_space ?MX")
assumes MY: "finite_prob_space \<lparr> space = space MY, sets = sets MY, measure = distribution Y\<rparr>"
(is "finite_prob_space ?MY")
and X_space: "X ` space M \<subseteq> space MX" and Y_space: "Y ` space M \<subseteq> space MY"
shows mutual_information_eq_generic:
"mutual_information b MX MY X Y = (\<Sum> (x,y) \<in> space MX \<times> space MY.
joint_distribution X Y {(x,y)} *
log b (joint_distribution X Y {(x,y)} /
(distribution X {x} * distribution Y {y})))"
(is "?equality")
and mutual_information_positive_generic:
"0 \<le> mutual_information b MX MY X Y" (is "?positive")
proof -
let ?P = "prod_measure_space ?MX ?MY"
let ?measure = "joint_distribution X Y"
let ?P' = "measure_update (\<lambda>_. ?measure) ?P"
interpret X: finite_prob_space "?MX" using MX .
moreover interpret Y: finite_prob_space "?MY" using MY .
ultimately have ms_X: "measure_space ?MX"
and ms_Y: "measure_space ?MY" by unfold_locales
have fms_P: "finite_measure_space ?P"
by (rule finite_measure_space_finite_prod_measure) fact+
have fms_P': "finite_measure_space ?P'"
using finite_product_measure_space[of "space MX" "space MY"]
X.finite_space Y.finite_space sigma_prod_sets_finite[OF X.finite_space Y.finite_space]
X.sets_eq_Pow Y.sets_eq_Pow
by (simp add: prod_measure_space_def)
{ fix x assume "x \<in> space ?P"
hence x_in_MX: "{fst x} \<in> sets MX" using X.sets_eq_Pow
by (auto simp: prod_measure_space_def)
assume "measure ?P {x} = 0"
with prod_measure_times[OF ms_X ms_Y, of "{fst x}" "{snd x}"] x_in_MX
have "distribution X {fst x} = 0 \<or> distribution Y {snd x} = 0"
by (simp add: prod_measure_space_def)
hence "joint_distribution X Y {x} = 0"
by (cases x) (auto simp: distribution_order) }
note measure_0 = this
show ?equality
unfolding Let_def mutual_information_def using fms_P fms_P' measure_0 MX MY
by (subst KL_divergence_eq_finite)
(simp_all add: prod_measure_space_def prod_measure_times_finite
finite_prob_space_eq setsum_cartesian_product')
show ?positive
unfolding Let_def mutual_information_def using measure_0 b_gt_1
proof (safe intro!: KL_divergence_positive_finite, simp_all)
from ms_X ms_Y X.top Y.top X.prob_space Y.prob_space
have "measure ?P (space ?P) = 1"
by (simp add: prod_measure_space_def, subst prod_measure_times, simp_all)
with fms_P show "finite_prob_space ?P"
by (simp add: finite_prob_space_eq)
from ms_X ms_Y X.top Y.top X.prob_space Y.prob_space Y.not_empty X_space Y_space
have "measure ?P' (space ?P') = 1" unfolding prob_space[symmetric]
by (auto simp add: prod_measure_space_def distribution_def vimage_Times comp_def
intro!: arg_cong[where f=prob])
with fms_P' show "finite_prob_space ?P'"
by (simp add: finite_prob_space_eq)
qed
qed
lemma (in finite_information_space) mutual_information_eq:
"\<I>(X;Y) = (\<Sum> (x,y) \<in> X ` space M \<times> Y ` space M.
distribution (\<lambda>x. (X x, Y x)) {(x,y)} * log b (distribution (\<lambda>x. (X x, Y x)) {(x,y)} /
(distribution X {x} * distribution Y {y})))"
by (subst mutual_information_eq_generic) (simp_all add: finite_prob_space_of_images)
lemma (in finite_information_space) mutual_information_positive: "0 \<le> \<I>(X;Y)"
by (subst mutual_information_positive_generic) (simp_all add: finite_prob_space_of_images)
definition (in prob_space)
"entropy b s X = mutual_information b s s X X"
abbreviation (in finite_information_space)
finite_entropy ("\<H>'(_')") where
"\<H>(X) \<equiv> entropy b \<lparr> space = X`space M, sets = Pow (X`space M) \<rparr> X"
lemma (in finite_information_space) joint_distribution_remove[simp]:
"joint_distribution X X {(x, x)} = distribution X {x}"
unfolding distribution_def by (auto intro!: arg_cong[where f=prob])
lemma (in finite_information_space) entropy_eq:
"\<H>(X) = -(\<Sum> x \<in> X ` space M. distribution X {x} * log b (distribution X {x}))"
proof -
{ fix f
{ fix x y
have "(\<lambda>x. (X x, X x)) -` {(x, y)} = (if x = y then X -` {x} else {})" by auto
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)"
unfolding distribution_def by auto }
hence "(\<Sum>(x, y) \<in> X ` space M \<times> X ` space M. joint_distribution X X {(x, y)} * f x y) =
(\<Sum>x \<in> X ` space M. distribution X {x} * f x x)"
unfolding setsum_cartesian_product' by (simp add: setsum_cases finite_space) }
note remove_cartesian_product = this
show ?thesis
unfolding entropy_def mutual_information_eq setsum_negf[symmetric] remove_cartesian_product
by (auto intro!: setsum_cong)
qed
lemma (in finite_information_space) entropy_positive: "0 \<le> \<H>(X)"
unfolding entropy_def using mutual_information_positive .
definition (in prob_space)
"conditional_mutual_information b s1 s2 s3 X Y Z \<equiv>
let prod_space =
prod_measure_space \<lparr>space = space s2, sets = sets s2, measure = distribution Y\<rparr>
\<lparr>space = space s3, sets = sets s3, measure = distribution Z\<rparr>
in
mutual_information b s1 prod_space X (\<lambda>x. (Y x, Z x)) -
mutual_information b s1 s3 X Z"
abbreviation (in finite_information_space)
finite_conditional_mutual_information ("\<I>'( _ ; _ | _ ')") where
"\<I>(X ; Y | Z) \<equiv> conditional_mutual_information b
\<lparr> space = X`space M, sets = Pow (X`space M) \<rparr>
\<lparr> space = Y`space M, sets = Pow (Y`space M) \<rparr>
\<lparr> space = Z`space M, sets = Pow (Z`space M) \<rparr>
X Y Z"
lemma (in finite_information_space) setsum_distribution_gen:
assumes "Z -` {c} \<inter> space M = (\<Union>x \<in> X`space M. Y -` {f x}) \<inter> space M"
and "inj_on f (X`space M)"
shows "(\<Sum>x \<in> X`space M. distribution Y {f x}) = distribution Z {c}"
unfolding distribution_def assms
using finite_space assms
by (subst measure_finitely_additive'')
(auto simp add: disjoint_family_on_def sets_eq_Pow inj_on_def
intro!: arg_cong[where f=prob])
lemma (in finite_information_space) setsum_distribution:
"(\<Sum>x \<in> X`space M. joint_distribution X Y {(x, y)}) = distribution Y {y}"
"(\<Sum>y \<in> Y`space M. joint_distribution X Y {(x, y)}) = distribution X {x}"
"(\<Sum>x \<in> X`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution Y Z {(y, z)}"
"(\<Sum>y \<in> Y`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Z {(x, z)}"
"(\<Sum>z \<in> Z`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Y {(x, y)}"
by (auto intro!: inj_onI setsum_distribution_gen)
lemma (in finite_information_space) conditional_mutual_information_eq_sum:
"\<I>(X ; Y | Z) =
(\<Sum>(x, y, z)\<in>X ` space M \<times> (\<lambda>x. (Y x, Z x)) ` space M.
distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} *
log b (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)}/
distribution (\<lambda>x. (Y x, Z x)) {(y, z)})) -
(\<Sum>(x, z)\<in>X ` space M \<times> Z ` space M.
distribution (\<lambda>x. (X x, Z x)) {(x,z)} * log b (distribution (\<lambda>x. (X x, Z x)) {(x,z)} / distribution Z {z}))"
(is "_ = ?rhs")
proof -
have setsum_product:
"\<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)
= (\<Sum>v\<in>Y ` space M \<times> Z ` space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x,v)} * f v)"
proof (safe intro!: setsum_mono_zero_cong_left imageI)
fix x y z f
assume *: "(Y y, Z z) \<notin> (\<lambda>x. (Y x, Z x)) ` space M" and "y \<in> space M" "z \<in> space M"
hence "(\<lambda>x. (X x, Y x, Z x)) -` {(x, Y y, Z z)} \<inter> space M = {}"
proof safe
fix x' assume x': "x' \<in> space M" and eq: "Y x' = Y y" "Z x' = Z z"
have "(Y y, Z z) \<in> (\<lambda>x. (Y x, Z x)) ` space M" using eq[symmetric] x' by auto
thus "x' \<in> {}" using * by auto
qed
thus "joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, Y y, Z z)} * f (Y y) (Z z) = 0"
unfolding distribution_def by simp
qed (simp add: finite_space)
thus ?thesis
unfolding conditional_mutual_information_def Let_def mutual_information_eq
apply (subst mutual_information_eq_generic)
by (auto simp add: prod_measure_space_def sigma_prod_sets_finite finite_space
finite_prob_space_of_images finite_product_prob_space_of_images
setsum_cartesian_product' setsum_product setsum_subtractf setsum_addf
setsum_left_distrib[symmetric] setsum_distribution
cong: setsum_cong)
qed
lemma (in finite_information_space) conditional_mutual_information_eq:
"\<I>(X ; Y | Z) = (\<Sum>(x, y, z) \<in> X ` space M \<times> Y ` space M \<times> Z ` space M.
distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} *
log b (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)}/
(joint_distribution X Z {(x, z)} * joint_distribution Y Z {(y,z)} / distribution Z {z})))"
unfolding conditional_mutual_information_def Let_def mutual_information_eq
apply (subst mutual_information_eq_generic)
by (auto simp add: prod_measure_space_def sigma_prod_sets_finite finite_space
finite_prob_space_of_images finite_product_prob_space_of_images
setsum_cartesian_product' setsum_product setsum_subtractf setsum_addf
setsum_left_distrib[symmetric] setsum_distribution setsum_commute[where A="Y`space M"]
cong: setsum_cong)
lemma (in finite_information_space) conditional_mutual_information_eq_mutual_information:
"\<I>(X ; Y) = \<I>(X ; Y | (\<lambda>x. ()))"
proof -
have [simp]: "(\<lambda>x. ()) ` space M = {()}" using not_empty by auto
show ?thesis
unfolding conditional_mutual_information_eq mutual_information_eq
by (simp add: setsum_cartesian_product' distribution_remove_const)
qed
lemma (in finite_information_space) conditional_mutual_information_positive:
"0 \<le> \<I>(X ; Y | Z)"
proof -
let ?dXYZ = "distribution (\<lambda>x. (X x, Y x, Z x))"
let ?dXZ = "joint_distribution X Z"
let ?dYZ = "joint_distribution Y Z"
let ?dX = "distribution X"
let ?dZ = "distribution Z"
let ?M = "X ` space M \<times> Y ` space M \<times> Z ` space M"
have split_beta: "\<And>f. split f = (\<lambda>x. f (fst x) (snd x))" by (simp add: expand_fun_eq)
have "- (\<Sum>(x, y, z) \<in> ?M. ?dXYZ {(x, y, z)} *
log b (?dXYZ {(x, y, z)} / (?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z})))
\<le> log b (\<Sum>(x, y, z) \<in> ?M. ?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z})"
unfolding split_beta
proof (rule log_setsum_divide)
show "?M \<noteq> {}" using not_empty by simp
show "1 < b" using b_gt_1 .
fix x assume "x \<in> ?M"
show "0 \<le> ?dXYZ {(fst x, fst (snd x), snd (snd x))}" using positive_distribution .
show "0 \<le> ?dXZ {(fst x, snd (snd x))} * ?dYZ {(fst (snd x), snd (snd x))} / ?dZ {snd (snd x)}"
by (auto intro!: mult_nonneg_nonneg positive_distribution simp: zero_le_divide_iff)
assume *: "0 < ?dXYZ {(fst x, fst (snd x), snd (snd x))}"
thus "0 < ?dXZ {(fst x, snd (snd x))} * ?dYZ {(fst (snd x), snd (snd x))} / ?dZ {snd (snd x)}"
by (auto intro!: divide_pos_pos mult_pos_pos
intro: distribution_order(6) distribution_mono_gt_0)
qed (simp_all add: setsum_cartesian_product' sum_over_space_distrib setsum_distribution finite_space)
also have "(\<Sum>(x, y, z) \<in> ?M. ?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z}) = (\<Sum>z\<in>Z`space M. ?dZ {z})"
apply (simp add: setsum_cartesian_product')
apply (subst setsum_commute)
apply (subst (2) setsum_commute)
by (auto simp: setsum_divide_distrib[symmetric] setsum_product[symmetric] setsum_distribution
intro!: setsum_cong)
finally show ?thesis
unfolding conditional_mutual_information_eq sum_over_space_distrib by simp
qed
definition (in prob_space)
"conditional_entropy b S T X Y = conditional_mutual_information b S S T X X Y"
abbreviation (in finite_information_space)
finite_conditional_entropy ("\<H>'(_ | _')") where
"\<H>(X | Y) \<equiv> conditional_entropy b
\<lparr> space = X`space M, sets = Pow (X`space M) \<rparr>
\<lparr> space = Y`space M, sets = Pow (Y`space M) \<rparr> X Y"
lemma (in finite_information_space) conditional_entropy_positive:
"0 \<le> \<H>(X | Y)" unfolding conditional_entropy_def using conditional_mutual_information_positive .
lemma (in finite_information_space) conditional_entropy_eq:
"\<H>(X | Z) =
- (\<Sum>(x, z)\<in>X ` space M \<times> Z ` space M.
joint_distribution X Z {(x, z)} *
log b (joint_distribution X Z {(x, z)} / distribution Z {z}))"
proof -
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
show ?thesis
unfolding conditional_mutual_information_eq_sum
conditional_entropy_def distribution_def *
by (auto intro!: setsum_0')
qed
lemma (in finite_information_space) mutual_information_eq_entropy_conditional_entropy:
"\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)"
unfolding mutual_information_eq entropy_eq conditional_entropy_eq
using finite_space
by (auto simp add: setsum_addf setsum_subtractf setsum_cartesian_product'
setsum_left_distrib[symmetric] setsum_addf setsum_distribution)
lemma (in finite_information_space) conditional_entropy_less_eq_entropy:
"\<H>(X | Z) \<le> \<H>(X)"
proof -
have "\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)" using mutual_information_eq_entropy_conditional_entropy .
with mutual_information_positive[of X Z] entropy_positive[of X]
show ?thesis by auto
qed
(* -------------Entropy of a RV with a certain event is zero---------------- *)
lemma (in finite_information_space) finite_entropy_certainty_eq_0:
assumes "x \<in> X ` space M" and "distribution X {x} = 1"
shows "\<H>(X) = 0"
proof -
interpret X: finite_prob_space "\<lparr> space = X ` space M,
sets = Pow (X ` space M),
measure = distribution X\<rparr>" by (rule finite_prob_space_of_images)
have "distribution X (X ` space M - {x}) = distribution X (X ` space M) - distribution X {x}"
using X.measure_compl[of "{x}"] assms by auto
also have "\<dots> = 0" using X.prob_space assms by auto
finally have X0: "distribution X (X ` space M - {x}) = 0" by auto
{ fix y assume asm: "y \<noteq> x" "y \<in> X ` space M"
hence "{y} \<subseteq> X ` space M - {x}" by auto
from X.measure_mono[OF this] X0 X.positive[of "{y}"] asm
have "distribution X {y} = 0" by auto }
hence fi: "\<And> y. y \<in> X ` space M \<Longrightarrow> distribution X {y} = (if x = y then 1 else 0)"
using assms by auto
have y: "\<And>y. (if x = y then 1 else 0) * log b (if x = y then 1 else 0) = 0" by simp
show ?thesis unfolding entropy_eq by (auto simp: y fi)
qed
(* --------------- upper bound on entropy for a rv ------------------------- *)
lemma (in finite_information_space) finite_entropy_le_card:
"\<H>(X) \<le> log b (real (card (X ` space M \<inter> {x . distribution X {x} \<noteq> 0})))"
proof -
interpret X: finite_prob_space "\<lparr>space = X ` space M,
sets = Pow (X ` space M),
measure = distribution X\<rparr>"
using finite_prob_space_of_images by auto
have triv: "\<And> x. (if distribution X {x} \<noteq> 0 then distribution X {x} else 0) = distribution X {x}"
by auto
hence sum1: "(\<Sum> x \<in> X ` space M \<inter> {y. distribution X {y} \<noteq> 0}. distribution X {x}) = 1"
using X.measure_finitely_additive''[of "X ` space M" "\<lambda> x. {x}", simplified]
sets_eq_Pow inj_singleton[unfolded inj_on_def, rule_format]
unfolding disjoint_family_on_def X.prob_space[symmetric]
using finite_imageI[OF finite_space, of X] by (auto simp add:triv setsum_restrict_set)
have pos: "\<And> x. x \<in> X ` space M \<inter> {y. distribution X {y} \<noteq> 0} \<Longrightarrow> inverse (distribution X {x}) > 0"
using X.positive sets_eq_Pow unfolding inverse_positive_iff_positive less_le by auto
{ assume asm: "X ` space M \<inter> {y. distribution X {y} \<noteq> 0} = {}"
{ fix x assume "x \<in> X ` space M"
hence "distribution X {x} = 0" using asm by blast }
hence A: "(\<Sum> x \<in> X ` space M. distribution X {x}) = 0" by auto
have B: "(\<Sum> x \<in> X ` space M. distribution X {x})
\<ge> (\<Sum> x \<in> X ` space M \<inter> {y. distribution X {y} \<noteq> 0}. distribution X {x})"
using finite_imageI[OF finite_space, of X]
by (subst setsum_mono2) auto
from A B have "False" using sum1 by auto } note not_empty = this
{ fix x assume asm: "x \<in> X ` space M"
have "- distribution X {x} * log b (distribution X {x})
= - (if distribution X {x} \<noteq> 0
then distribution X {x} * log b (distribution X {x})
else 0)"
by auto
also have "\<dots> = (if distribution X {x} \<noteq> 0
then distribution X {x} * - log b (distribution X {x})
else 0)"
by auto
also have "\<dots> = (if distribution X {x} \<noteq> 0
then distribution X {x} * log b (inverse (distribution X {x}))
else 0)"
using log_inverse b_gt_1 X.positive[of "{x}"] asm by auto
finally have "- distribution X {x} * log b (distribution X {x})
= (if distribution X {x} \<noteq> 0
then distribution X {x} * log b (inverse (distribution X {x}))
else 0)"
by auto } note log_inv = this
have "- (\<Sum> x \<in> X ` space M. distribution X {x} * log b (distribution X {x}))
= (\<Sum> x \<in> X ` space M. (if distribution X {x} \<noteq> 0
then distribution X {x} * log b (inverse (distribution X {x}))
else 0))"
unfolding setsum_negf[symmetric] using log_inv by auto
also have "\<dots> = (\<Sum> x \<in> X ` space M \<inter> {y. distribution X {y} \<noteq> 0}.
distribution X {x} * log b (inverse (distribution X {x})))"
unfolding setsum_restrict_set[OF finite_imageI[OF finite_space, of X]] by auto
also have "\<dots> \<le> log b (\<Sum> x \<in> X ` space M \<inter> {y. distribution X {y} \<noteq> 0}.
distribution X {x} * (inverse (distribution X {x})))"
apply (subst log_setsum[OF _ _ b_gt_1 sum1,
unfolded greaterThan_iff, OF _ _ _]) using pos sets_eq_Pow
X.finite_space assms X.positive not_empty by auto
also have "\<dots> = log b (\<Sum> x \<in> X ` space M \<inter> {y. distribution X {y} \<noteq> 0}. 1)"
by auto
also have "\<dots> \<le> log b (real_of_nat (card (X ` space M \<inter> {y. distribution X {y} \<noteq> 0})))"
by auto
finally have "- (\<Sum>x\<in>X ` space M. distribution X {x} * log b (distribution X {x}))
\<le> log b (real_of_nat (card (X ` space M \<inter> {y. distribution X {y} \<noteq> 0})))" by simp
thus ?thesis unfolding entropy_eq real_eq_of_nat by auto
qed
(* --------------- entropy is maximal for a uniform rv --------------------- *)
lemma (in finite_prob_space) uniform_prob:
assumes "x \<in> space M"
assumes "\<And> x y. \<lbrakk>x \<in> space M ; y \<in> space M\<rbrakk> \<Longrightarrow> prob {x} = prob {y}"
shows "prob {x} = 1 / real (card (space M))"
proof -
have prob_x: "\<And> y. y \<in> space M \<Longrightarrow> prob {y} = prob {x}"
using assms(2)[OF _ `x \<in> space M`] by blast
have "1 = prob (space M)"
using prob_space by auto
also have "\<dots> = (\<Sum> x \<in> space M. prob {x})"
using measure_finitely_additive''[of "space M" "\<lambda> x. {x}", simplified]
sets_eq_Pow inj_singleton[unfolded inj_on_def, rule_format]
finite_space unfolding disjoint_family_on_def prob_space[symmetric]
by (auto simp add:setsum_restrict_set)
also have "\<dots> = (\<Sum> y \<in> space M. prob {x})"
using prob_x by auto
also have "\<dots> = real_of_nat (card (space M)) * prob {x}" by simp
finally have one: "1 = real (card (space M)) * prob {x}"
using real_eq_of_nat by auto
hence two: "real (card (space M)) \<noteq> 0" by fastsimp
from one have three: "prob {x} \<noteq> 0" by fastsimp
thus ?thesis using one two three divide_cancel_right
by (auto simp:field_simps)
qed
lemma (in finite_information_space) finite_entropy_uniform_max:
assumes "\<And>x y. \<lbrakk> x \<in> X ` space M ; y \<in> X ` space M \<rbrakk> \<Longrightarrow> distribution X {x} = distribution X {y}"
shows "\<H>(X) = log b (real (card (X ` space M)))"
proof -
interpret X: finite_prob_space "\<lparr>space = X ` space M,
sets = Pow (X ` space M),
measure = distribution X\<rparr>"
using finite_prob_space_of_images by auto
{ fix x assume xasm: "x \<in> X ` space M"
hence card_gt0: "real (card (X ` space M)) > 0"
using card_gt_0_iff X.finite_space by auto
from xasm have "\<And> y. y \<in> X ` space M \<Longrightarrow> distribution X {y} = distribution X {x}"
using assms by blast
hence "- (\<Sum>x\<in>X ` space M. distribution X {x} * log b (distribution X {x}))
= - real (card (X ` space M)) * distribution X {x} * log b (distribution X {x})"
unfolding real_eq_of_nat by auto
also have "\<dots> = - real (card (X ` space M)) * (1 / real (card (X ` space M))) * log b (1 / real (card (X ` space M)))"
by (auto simp: X.uniform_prob[simplified, OF xasm assms])
also have "\<dots> = log b (real (card (X ` space M)))"
unfolding inverse_eq_divide[symmetric]
using card_gt0 log_inverse b_gt_1
by (auto simp add:field_simps card_gt0)
finally have ?thesis
unfolding entropy_eq by auto }
moreover
{ assume "X ` space M = {}"
hence "distribution X (X ` space M) = 0"
using X.empty_measure by simp
hence "False" using X.prob_space by auto }
ultimately show ?thesis by auto
qed
definition "subvimage A f g \<longleftrightarrow> (\<forall>x \<in> A. f -` {f x} \<inter> A \<subseteq> g -` {g x} \<inter> A)"
lemma subvimageI:
assumes "\<And>x y. \<lbrakk> x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y"
shows "subvimage A f g"
using assms unfolding subvimage_def by blast
lemma subvimageE[consumes 1]:
assumes "subvimage A f g"
obtains "\<And>x y. \<lbrakk> x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y"
using assms unfolding subvimage_def by blast
lemma subvimageD:
"\<lbrakk> subvimage A f g ; x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y"
using assms unfolding subvimage_def by blast
lemma subvimage_subset:
"\<lbrakk> subvimage B f g ; A \<subseteq> B \<rbrakk> \<Longrightarrow> subvimage A f g"
unfolding subvimage_def by auto
lemma subvimage_idem[intro]: "subvimage A g g"
by (safe intro!: subvimageI)
lemma subvimage_comp_finer[intro]:
assumes svi: "subvimage A g h"
shows "subvimage A g (f \<circ> h)"
proof (rule subvimageI, simp)
fix x y assume "x \<in> A" "y \<in> A" "g x = g y"
from svi[THEN subvimageD, OF this]
show "f (h x) = f (h y)" by simp
qed
lemma subvimage_comp_gran:
assumes svi: "subvimage A g h"
assumes inj: "inj_on f (g ` A)"
shows "subvimage A (f \<circ> g) h"
by (rule subvimageI) (auto intro!: subvimageD[OF svi] simp: inj_on_iff[OF inj])
lemma subvimage_comp:
assumes svi: "subvimage (f ` A) g h"
shows "subvimage A (g \<circ> f) (h \<circ> f)"
by (rule subvimageI) (auto intro!: svi[THEN subvimageD])
lemma subvimage_trans:
assumes fg: "subvimage A f g"
assumes gh: "subvimage A g h"
shows "subvimage A f h"
by (rule subvimageI) (auto intro!: fg[THEN subvimageD] gh[THEN subvimageD])
lemma subvimage_translator:
assumes svi: "subvimage A f g"
shows "\<exists>h. \<forall>x \<in> A. h (f x) = g x"
proof (safe intro!: exI[of _ "\<lambda>x. (THE z. z \<in> (g ` (f -` {x} \<inter> A)))"])
fix x assume "x \<in> A"
show "(THE x'. x' \<in> (g ` (f -` {f x} \<inter> A))) = g x"
by (rule theI2[of _ "g x"])
(insert `x \<in> A`, auto intro!: svi[THEN subvimageD])
qed
lemma subvimage_translator_image:
assumes svi: "subvimage A f g"
shows "\<exists>h. h ` f ` A = g ` A"
proof -
from subvimage_translator[OF svi]
obtain h where "\<And>x. x \<in> A \<Longrightarrow> h (f x) = g x" by auto
thus ?thesis
by (auto intro!: exI[of _ h]
simp: image_compose[symmetric] comp_def cong: image_cong)
qed
lemma subvimage_finite:
assumes svi: "subvimage A f g" and fin: "finite (f`A)"
shows "finite (g`A)"
proof -
from subvimage_translator_image[OF svi]
obtain h where "g`A = h`f`A" by fastsimp
with fin show "finite (g`A)" by simp
qed
lemma subvimage_disj:
assumes svi: "subvimage A f g"
shows "f -` {x} \<inter> A \<subseteq> g -` {y} \<inter> A \<or>
f -` {x} \<inter> g -` {y} \<inter> A = {}" (is "?sub \<or> ?dist")
proof (rule disjCI)
assume "\<not> ?dist"
then obtain z where "z \<in> A" and "x = f z" and "y = g z" by auto
thus "?sub" using svi unfolding subvimage_def by auto
qed
lemma setsum_image_split:
assumes svi: "subvimage A f g" and fin: "finite (f ` A)"
shows "(\<Sum>x\<in>f`A. h x) = (\<Sum>y\<in>g`A. \<Sum>x\<in>f`(g -` {y} \<inter> A). h x)"
(is "?lhs = ?rhs")
proof -
have "f ` A =
snd ` (SIGMA x : g ` A. f ` (g -` {x} \<inter> A))"
(is "_ = snd ` ?SIGMA")
unfolding image_split_eq_Sigma[symmetric]
by (simp add: image_compose[symmetric] comp_def)
moreover
have snd_inj: "inj_on snd ?SIGMA"
unfolding image_split_eq_Sigma[symmetric]
by (auto intro!: inj_onI subvimageD[OF svi])
ultimately
have "(\<Sum>x\<in>f`A. h x) = (\<Sum>(x,y)\<in>?SIGMA. h y)"
by (auto simp: setsum_reindex intro: setsum_cong)
also have "... = ?rhs"
using subvimage_finite[OF svi fin] fin
apply (subst setsum_Sigma[symmetric])
by (auto intro!: finite_subset[of _ "f`A"])
finally show ?thesis .
qed
lemma (in finite_information_space) entropy_partition:
assumes svi: "subvimage (space M) X P"
shows "\<H>(X) = \<H>(P) + \<H>(X|P)"
proof -
have "(\<Sum>x\<in>X ` space M. distribution X {x} * log b (distribution X {x})) =
(\<Sum>y\<in>P `space M. \<Sum>x\<in>X ` space M.
joint_distribution X P {(x, y)} * log b (joint_distribution X P {(x, y)}))"
proof (subst setsum_image_split[OF svi],
safe intro!: finite_imageI finite_space setsum_mono_zero_cong_left imageI)
fix p x assume in_space: "p \<in> space M" "x \<in> space M"
assume "joint_distribution X P {(X x, P p)} * log b (joint_distribution X P {(X x, P p)}) \<noteq> 0"
hence "(\<lambda>x. (X x, P x)) -` {(X x, P p)} \<inter> space M \<noteq> {}" by (auto simp: distribution_def)
with svi[unfolded subvimage_def, rule_format, OF `x \<in> space M`]
show "x \<in> P -` {P p}" by auto
next
fix p x assume in_space: "p \<in> space M" "x \<in> space M"
assume "P x = P p"
from this[symmetric] svi[unfolded subvimage_def, rule_format, OF `x \<in> space M`]
have "X -` {X x} \<inter> space M \<subseteq> P -` {P p} \<inter> space M"
by auto
hence "(\<lambda>x. (X x, P x)) -` {(X x, P p)} \<inter> space M = X -` {X x} \<inter> space M"
by auto
thus "distribution X {X x} * log b (distribution X {X x}) =
joint_distribution X P {(X x, P p)} *
log b (joint_distribution X P {(X x, P p)})"
by (auto simp: distribution_def)
qed
thus ?thesis
unfolding entropy_eq conditional_entropy_eq
by (simp add: setsum_cartesian_product' setsum_subtractf setsum_distribution
setsum_left_distrib[symmetric] setsum_commute[where B="P`space M"])
qed
corollary (in finite_information_space) entropy_data_processing:
"\<H>(f \<circ> X) \<le> \<H>(X)"
by (subst (2) entropy_partition[of _ "f \<circ> X"]) (auto intro: conditional_entropy_positive)
lemma (in prob_space) distribution_cong:
assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = Y x"
shows "distribution X = distribution Y"
unfolding distribution_def expand_fun_eq
using assms by (auto intro!: arg_cong[where f=prob])
lemma (in prob_space) joint_distribution_cong:
assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x"
assumes "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x"
shows "joint_distribution X Y = joint_distribution X' Y'"
unfolding distribution_def expand_fun_eq
using assms by (auto intro!: arg_cong[where f=prob])
lemma image_cong:
"\<lbrakk> \<And>x. x \<in> S \<Longrightarrow> X x = X' x \<rbrakk> \<Longrightarrow> X ` S = X' ` S"
by (auto intro!: image_eqI)
lemma (in finite_information_space) mutual_information_cong:
assumes X: "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x"
assumes Y: "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x"
shows "\<I>(X ; Y) = \<I>(X' ; Y')"
proof -
have "X ` space M = X' ` space M" using X by (rule image_cong)
moreover have "Y ` space M = Y' ` space M" using Y by (rule image_cong)
ultimately show ?thesis
unfolding mutual_information_eq
using
assms[THEN distribution_cong]
joint_distribution_cong[OF assms]
by (auto intro!: setsum_cong)
qed
corollary (in finite_information_space) entropy_of_inj:
assumes "inj_on f (X`space M)"
shows "\<H>(f \<circ> X) = \<H>(X)"
proof (rule antisym)
show "\<H>(f \<circ> X) \<le> \<H>(X)" using entropy_data_processing .
next
have "\<H>(X) = \<H>(the_inv_into (X`space M) f \<circ> (f \<circ> X))"
by (auto intro!: mutual_information_cong simp: entropy_def the_inv_into_f_f[OF assms])
also have "... \<le> \<H>(f \<circ> X)"
using entropy_data_processing .
finally show "\<H>(X) \<le> \<H>(f \<circ> X)" .
qed
end