# Theory Information

```(*  Title:      HOL/Probability/Information.thy
Author:     Johannes Hölzl, TU München
Author:     Armin Heller, TU München
*)

section ‹Information theory›

theory Information
imports
Independent_Family
begin

subsection "Information theory"

locale information_space = prob_space +
fixes b :: real assumes b_gt_1: "1 < b"

context information_space
begin

text ‹Introduce some simplification rules for logarithm of base \<^term>‹b›.›

lemma log_neg_const:
assumes "x ≤ 0"
shows "log b x = log b 0"
proof -
{ fix u :: real
have "x ≤ 0" by fact
also have "0 < exp u"
using exp_gt_zero .
finally have "exp u ≠ x"
by auto }
then show "log b x = log b 0"
qed

lemma log_mult_eq:
"log b (A * B) = (if 0 < A * B then log b ¦A¦ + log b ¦B¦ else log b 0)"
using log_mult[of "¦A¦" "¦B¦"] b_gt_1 log_neg_const[of "A * B"]
by (auto simp: zero_less_mult_iff mult_le_0_iff)

lemma log_inverse_eq:
"log b (inverse B) = (if 0 < B then - log b B else log b 0)"
using ln_inverse log_def log_neg_const by force

lemma log_divide_eq:
"log b (A / B) = (if 0 < A * B then (log b ¦A¦) - log b ¦B¦ else log b 0)"
unfolding divide_inverse log_mult_eq log_inverse_eq abs_inverse
by (auto simp: zero_less_mult_iff mult_le_0_iff)

lemmas log_simps = log_mult_eq log_inverse_eq log_divide_eq

end

subsection "Kullback\$-\$Leibler divergence"

text ‹The Kullback\$-\$Leibler divergence is also known as relative entropy or
Kullback\$-\$Leibler distance.›

definition
"entropy_density b M N = log b ∘ enn2real ∘ RN_deriv M N"

definition
"KL_divergence b M N = integral⇧L N (entropy_density b M N)"

lemma measurable_entropy_density[measurable]: "entropy_density b M N ∈ borel_measurable M"
unfolding entropy_density_def by auto

lemma (in sigma_finite_measure) KL_density:
fixes f :: "'a ⇒ real"
assumes "1 < b"
assumes f[measurable]: "f ∈ borel_measurable M" and nn: "AE x in M. 0 ≤ f x"
shows "KL_divergence b M (density M f) = (∫x. f x * log b (f x) ∂M)"
unfolding KL_divergence_def
proof (subst integral_real_density)
show [measurable]: "entropy_density b M (density M (λx. ennreal (f x))) ∈ borel_measurable M"
using f
by (auto simp: comp_def entropy_density_def)
have "density M (RN_deriv M (density M f)) = density M f"
using f nn by (intro density_RN_deriv_density) auto
then have eq: "AE x in M. RN_deriv M (density M f) x = f x"
using f nn by (intro density_unique) auto
have "AE x in M. f x * entropy_density b M (density M (λx. ennreal (f x))) x =
f x * log b (f x)"
using eq nn by (auto simp: entropy_density_def)
then show "(∫x. f x * entropy_density b M (density M (λx. ennreal (f x))) x ∂M) = (∫x. f x * log b (f x) ∂M)"
by (intro integral_cong_AE) measurable
qed fact+

lemma (in sigma_finite_measure) KL_density_density:
fixes f g :: "'a ⇒ real"
assumes "1 < b"
assumes f: "f ∈ borel_measurable M" "AE x in M. 0 ≤ f x"
assumes g: "g ∈ borel_measurable M" "AE x in M. 0 ≤ g x"
assumes ac: "AE x in M. f x = 0 ⟶ g x = 0"
shows "KL_divergence b (density M f) (density M g) = (∫x. g x * log b (g x / f x) ∂M)"
proof -
interpret Mf: sigma_finite_measure "density M f"
using f by (subst sigma_finite_iff_density_finite) auto
have "KL_divergence b (density M f) (density M g) =
KL_divergence b (density M f) (density (density M f) (λx. g x / f x))"
using f g ac by (subst density_density_divide) simp_all
also have "… = (∫x. (g x / f x) * log b (g x / f x) ∂density M f)"
using f g ‹1 < b› by (intro Mf.KL_density) (auto simp: AE_density)
also have "… = (∫x. g x * log b (g x / f x) ∂M)"
using ac f g ‹1 < b› by (subst integral_density) (auto intro!: integral_cong_AE)
finally show ?thesis .
qed

lemma (in information_space) KL_gt_0:
fixes D :: "'a ⇒ real"
assumes "prob_space (density M D)"
assumes D: "D ∈ borel_measurable M" "AE x in M. 0 ≤ D x"
assumes int: "integrable M (λx. D x * log b (D x))"
assumes A: "density M D ≠ M"
shows "0 < KL_divergence b M (density M D)"
proof -
interpret N: prob_space "density M D" by fact

obtain A where "A ∈ sets M" "emeasure (density M D) A ≠ emeasure M A"
using measure_eqI[of "density M D" M] ‹density M D ≠ M› by auto

let ?D_set = "{x∈space M. D x ≠ 0}"
have [simp, intro]: "?D_set ∈ sets M"
using D by auto

have D_neg: "(∫⇧+ x. ennreal (- D x) ∂M) = 0"
using D by (subst nn_integral_0_iff_AE) (auto simp: ennreal_neg)

have "(∫⇧+ x. ennreal (D x) ∂M) = emeasure (density M D) (space M)"
using D by (simp add: emeasure_density cong: nn_integral_cong)
then have D_pos: "(∫⇧+ x. ennreal (D x) ∂M) = 1"
using N.emeasure_space_1 by simp

have "integrable M D"
using D D_pos D_neg unfolding real_integrable_def real_lebesgue_integral_def by simp_all
then have "integral⇧L M D = 1"
using D D_pos D_neg by (simp add: real_lebesgue_integral_def)

have "0 ≤ 1 - measure M ?D_set"
using prob_le_1 by (auto simp: field_simps)
also have "… = (∫ x. D x - indicator ?D_set x ∂M)"
using ‹integrable M D› ‹integral⇧L M D = 1›
also have "… < (∫ x. D x * (ln b * log b (D x)) ∂M)"
proof (rule integral_less_AE)
show "integrable M (λx. D x - indicator ?D_set x)"
using ‹integrable M D› by (auto simp: less_top[symmetric])
next
from integrable_mult_left(1)[OF int, of "ln b"]
show "integrable M (λx. D x * (ln b * log b (D x)))"
next
show "emeasure M {x∈space M. D x ≠ 1 ∧ D x ≠ 0} ≠ 0"
proof
assume eq_0: "emeasure M {x∈space M. D x ≠ 1 ∧ D x ≠ 0} = 0"
then have disj: "AE x in M. D x = 1 ∨ D x = 0"
using D(1) by (auto intro!: AE_I[OF subset_refl] sets.sets_Collect)

have "emeasure M {x∈space M. D x = 1} = (∫⇧+ x. indicator {x∈space M. D x = 1} x ∂M)"
using D(1) by auto
also have "… = (∫⇧+ x. ennreal (D x) ∂M)"
using disj by (auto intro!: nn_integral_cong_AE simp: indicator_def one_ennreal_def)
finally have "AE x in M. D x = 1"
using D D_pos by (intro AE_I_eq_1) auto
then have "(∫⇧+x. indicator A x∂M) = (∫⇧+x. ennreal (D x) * indicator A x∂M)"
by (intro nn_integral_cong_AE) (auto simp: one_ennreal_def[symmetric])
also have "… = density M D A"
using ‹A ∈ sets M› D by (simp add: emeasure_density)
finally show False using ‹A ∈ sets M› ‹emeasure (density M D) A ≠ emeasure M A› by simp
qed
show "{x∈space M. D x ≠ 1 ∧ D x ≠ 0} ∈ sets M"
using D(1) by (auto intro: sets.sets_Collect_conj)

show "AE t in M. t ∈ {x∈space M. D x ≠ 1 ∧ D x ≠ 0} ⟶
D t - indicator ?D_set t ≠ D t * (ln b * log b (D t))"
using D(2)
proof (eventually_elim, safe)
fix t assume Dt: "t ∈ space M" "D t ≠ 1" "D t ≠ 0" "0 ≤ D t"
and eq: "D t - indicator ?D_set t = D t * (ln b * log b (D t))"

have "D t - 1 = D t - indicator ?D_set t"
using Dt by simp
also note eq
also have "D t * (ln b * log b (D t)) = - D t * ln (1 / D t)"
using b_gt_1 ‹D t ≠ 0› ‹0 ≤ D t›
by (simp add: log_def ln_div less_le)
finally have "ln (1 / D t) = 1 / D t - 1"
using ‹D t ≠ 0› by (auto simp: field_simps)
from ln_eq_minus_one[OF _ this] ‹D t ≠ 0› ‹0 ≤ D t› ‹D t ≠ 1›
show False by auto
qed

show "AE t in M. D t - indicator ?D_set t ≤ D t * (ln b * log b (D t))"
using D(2) AE_space
proof eventually_elim
fix t assume "t ∈ space M" "0 ≤ D t"
show "D t - indicator ?D_set t ≤ D t * (ln b * log b (D t))"
proof cases
assume asm: "D t ≠ 0"
then have "0 < D t" using ‹0 ≤ D t› by auto
then have "0 < 1 / D t" by auto
have "D t - indicator ?D_set t ≤ - D t * (1 / D t - 1)"
using asm ‹t ∈ space M› by (simp add: field_simps)
also have "- D t * (1 / D t - 1) ≤ - D t * ln (1 / D t)"
using ln_le_minus_one ‹0 < 1 / D t› by (intro mult_left_mono_neg) auto
also have "… = D t * (ln b * log b (D t))"
using ‹0 < D t› b_gt_1
finally show ?thesis by simp
qed simp
qed
qed
also have "… = (∫ x. ln b * (D x * log b (D x)) ∂M)"
also have "… = ln b * (∫ x. D x * log b (D x) ∂M)"
using int by simp
finally show ?thesis
using b_gt_1 D by (subst KL_density) (auto simp: zero_less_mult_iff)
qed

lemma (in sigma_finite_measure) KL_same_eq_0: "KL_divergence b M M = 0"
proof -
have "AE x in M. 1 = RN_deriv M M x"
proof (rule RN_deriv_unique)
show "density M (λx. 1) = M"
qed auto
then have "AE x in M. log b (enn2real (RN_deriv M M x)) = 0"
by (elim AE_mp) simp
from integral_cong_AE[OF _ _ this]
have "integral⇧L M (entropy_density b M M) = 0"
then show "KL_divergence b M M = 0"
unfolding KL_divergence_def
by auto
qed

lemma (in information_space) KL_eq_0_iff_eq:
fixes D :: "'a ⇒ real"
assumes "prob_space (density M D)"
assumes D: "D ∈ borel_measurable M" "AE x in M. 0 ≤ D x"
assumes int: "integrable M (λx. D x * log b (D x))"
shows "KL_divergence b M (density M D) = 0 ⟷ density M D = M"
using KL_same_eq_0[of b] KL_gt_0[OF assms]
by (auto simp: less_le)

lemma (in information_space) KL_eq_0_iff_eq_ac:
fixes D :: "'a ⇒ real"
assumes "prob_space N"
assumes ac: "absolutely_continuous M N" "sets N = sets M"
assumes int: "integrable N (entropy_density b M N)"
shows "KL_divergence b M N = 0 ⟷ N = M"
proof -
interpret N: prob_space N by fact
have "finite_measure N" by unfold_locales
from real_RN_deriv[OF this ac] obtain D
where D:
"random_variable borel D"
"AE x in M. RN_deriv M N x = ennreal (D x)"
"AE x in N. 0 < D x"
"⋀x. 0 ≤ D x"
by this auto

have "N = density M (RN_deriv M N)"
using ac by (rule density_RN_deriv[symmetric])
also have "… = density M D"
using D by (auto intro!: density_cong)
finally have N: "N = density M D" .

from absolutely_continuous_AE[OF ac(2,1) D(2)] D b_gt_1 ac measurable_entropy_density
have "integrable N (λx. log b (D x))"
by (intro integrable_cong_AE[THEN iffD2, OF _ _ _ int])
(auto simp: N entropy_density_def)
with D b_gt_1 have "integrable M (λx. D x * log b (D x))"
by (subst integrable_real_density[symmetric]) (auto simp: N[symmetric] comp_def)
with ‹prob_space N› D show ?thesis
unfolding N
by (intro KL_eq_0_iff_eq) auto
qed

lemma (in information_space) KL_nonneg:
assumes "prob_space (density M D)"
assumes D: "D ∈ borel_measurable M" "AE x in M. 0 ≤ D x"
assumes int: "integrable M (λx. D x * log b (D x))"
shows "0 ≤ KL_divergence b M (density M D)"
using KL_gt_0[OF assms] by (cases "density M D = M") (auto simp: KL_same_eq_0)

lemma (in sigma_finite_measure) KL_density_density_nonneg:
fixes f g :: "'a ⇒ real"
assumes "1 < b"
assumes f: "f ∈ borel_measurable M" "AE x in M. 0 ≤ f x" "prob_space (density M f)"
assumes g: "g ∈ borel_measurable M" "AE x in M. 0 ≤ g x" "prob_space (density M g)"
assumes ac: "AE x in M. f x = 0 ⟶ g x = 0"
assumes int: "integrable M (λx. g x * log b (g x / f x))"
shows "0 ≤ KL_divergence b (density M f) (density M g)"
proof -
interpret Mf: prob_space "density M f" by fact
interpret Mf: information_space "density M f" b by standard fact
have eq: "density (density M f) (λx. g x / f x) = density M g" (is "?DD = _")
using f g ac by (subst density_density_divide) simp_all

have "0 ≤ KL_divergence b (density M f) (density (density M f) (λx. g x / f x))"
proof (rule Mf.KL_nonneg)
show "prob_space ?DD" unfolding eq by fact
from f g show "(λx. g x / f x) ∈ borel_measurable (density M f)"
by auto
show "AE x in density M f. 0 ≤ g x / f x"
using f g by (auto simp: AE_density)
show "integrable (density M f) (λx. g x / f x * log b (g x / f x))"
using ‹1 < b› f g ac
by (subst integrable_density)
(auto intro!: integrable_cong_AE[THEN iffD2, OF _ _ _ int] measurable_If)
qed
also have "… = KL_divergence b (density M f) (density M g)"
using f g ac by (subst density_density_divide) simp_all
finally show ?thesis .
qed

subsection ‹Finite Entropy›

definition (in information_space) finite_entropy :: "'b measure ⇒ ('a ⇒ 'b) ⇒ ('b ⇒ real) ⇒ bool"
where
"finite_entropy S X f ⟷
distributed M S X f ∧
integrable S (λx. f x * log b (f x)) ∧
(∀x∈space S. 0 ≤ f x)"

lemma (in information_space) finite_entropy_simple_function:
assumes X: "simple_function M X"
shows "finite_entropy (count_space (X`space M)) X (λa. measure M {x ∈ space M. X x = a})"
unfolding finite_entropy_def
proof safe
have [simp]: "finite (X ` space M)"
using X by (auto simp: simple_function_def)
then show "integrable (count_space (X ` space M))
(λx. prob {xa ∈ space M. X xa = x} * log b (prob {xa ∈ space M. X xa = x}))"
by (rule integrable_count_space)
have d: "distributed M (count_space (X ` space M)) X (λx. ennreal (if x ∈ X`space M then prob {xa ∈ space M. X xa = x} else 0))"
by (rule distributed_simple_function_superset[OF X]) (auto intro!: arg_cong[where f=prob])
show "distributed M (count_space (X ` space M)) X (λx. ennreal (prob {xa ∈ space M. X xa = x}))"
by (rule distributed_cong_density[THEN iffD1, OF _ _ _ d]) auto
qed (rule measure_nonneg)

lemma ac_fst:
assumes "sigma_finite_measure T"
shows "absolutely_continuous S (distr (S ⨂⇩M T) S fst)"
proof -
interpret sigma_finite_measure T by fact
{ fix A assume A: "A ∈ sets S" "emeasure S A = 0"
then have "fst -` A ∩ space (S ⨂⇩M T) = A × space T"
by (auto simp: space_pair_measure dest!: sets.sets_into_space)
with A have "emeasure (S ⨂⇩M T) (fst -` A ∩ space (S ⨂⇩M T)) = 0"
then show ?thesis
unfolding absolutely_continuous_def
by (metis emeasure_distr measurable_fst null_setsD1 null_setsD2 null_setsI sets_distr subsetI)
qed

lemma ac_snd:
assumes "sigma_finite_measure T"
shows "absolutely_continuous T (distr (S ⨂⇩M T) T snd)"
proof -
interpret sigma_finite_measure T by fact
{ fix A assume A: "A ∈ sets T" "emeasure T A = 0"
then have "snd -` A ∩ space (S ⨂⇩M T) = space S × A"
by (auto simp: space_pair_measure dest!: sets.sets_into_space)
with A have "emeasure (S ⨂⇩M T) (snd -` A ∩ space (S ⨂⇩M T)) = 0"
then show ?thesis
unfolding absolutely_continuous_def
by (metis emeasure_distr measurable_snd null_setsD1 null_setsD2 null_setsI sets_distr subsetI)
qed

lemma (in information_space) finite_entropy_integrable:
"finite_entropy S X Px ⟹ integrable S (λx. Px x * log b (Px x))"
unfolding finite_entropy_def by auto

lemma (in information_space) finite_entropy_distributed:
"finite_entropy S X Px ⟹ distributed M S X Px"
unfolding finite_entropy_def by auto

lemma (in information_space) finite_entropy_nn:
"finite_entropy S X Px ⟹ x ∈ space S ⟹ 0 ≤ Px x"
by (auto simp: finite_entropy_def)

lemma (in information_space) finite_entropy_measurable:
"finite_entropy S X Px ⟹ Px ∈ S →⇩M borel"
using distributed_real_measurable[of S Px M X]
finite_entropy_nn[of S X Px] finite_entropy_distributed[of S X Px] by auto

lemma (in information_space) subdensity_finite_entropy:
fixes g :: "'b ⇒ real" and f :: "'c ⇒ real"
assumes T: "T ∈ measurable P Q"
assumes f: "finite_entropy P X f"
assumes g: "finite_entropy Q Y g"
assumes Y: "Y = T ∘ X"
shows "AE x in P. g (T x) = 0 ⟶ f x = 0"
using subdensity[OF T, of M X "λx. ennreal (f x)" Y "λx. ennreal (g x)"]
finite_entropy_distributed[OF f] finite_entropy_distributed[OF g]
finite_entropy_nn[OF f] finite_entropy_nn[OF g]
assms
by auto

lemma (in information_space) finite_entropy_integrable_transform:
"finite_entropy S X Px ⟹ distributed M T Y Py ⟹ (⋀x. x ∈ space T ⟹ 0 ≤ Py x) ⟹
X = (λx. f (Y x)) ⟹ f ∈ measurable T S ⟹ integrable T (λx. Py x * log b (Px (f x)))"
using distributed_transform_integrable[of M T Y Py S X Px f "λx. log b (Px x)"]
using distributed_real_measurable[of S Px M X]
by (auto simp: finite_entropy_def)

subsection ‹Mutual Information›

definition (in prob_space)
"mutual_information b S T X Y =
KL_divergence b (distr M S X ⨂⇩M distr M T Y) (distr M (S ⨂⇩M T) (λx. (X x, Y x)))"

lemma (in information_space) mutual_information_indep_vars:
fixes S T X Y
defines "P ≡ distr M S X ⨂⇩M distr M T Y"
defines "Q ≡ distr M (S ⨂⇩M T) (λx. (X x, Y x))"
shows "indep_var S X T Y ⟷
(random_variable S X ∧ random_variable T Y ∧
absolutely_continuous P Q ∧ integrable Q (entropy_density b P Q) ∧
mutual_information b S T X Y = 0)"
unfolding indep_var_distribution_eq
proof safe
assume rv[measurable]: "random_variable S X" "random_variable T Y"

interpret X: prob_space "distr M S X"
by (rule prob_space_distr) fact
interpret Y: prob_space "distr M T Y"
by (rule prob_space_distr) fact
interpret XY: pair_prob_space "distr M S X" "distr M T Y" by standard
interpret P: information_space P b unfolding P_def by standard (rule b_gt_1)

interpret Q: prob_space Q unfolding Q_def
by (rule prob_space_distr) simp

{ assume "distr M S X ⨂⇩M distr M T Y = distr M (S ⨂⇩M T) (λx. (X x, Y x))"
then have [simp]: "Q = P"  unfolding Q_def P_def by simp

show ac: "absolutely_continuous P Q" by (simp add: absolutely_continuous_def)
then have ed: "entropy_density b P Q ∈ borel_measurable P"
by simp

have "AE x in P. 1 = RN_deriv P Q x"
proof (rule P.RN_deriv_unique)
show "density P (λx. 1) = Q"
unfolding ‹Q = P› by (intro measure_eqI) (auto simp: emeasure_density)
qed auto
then have ae_0: "AE x in P. entropy_density b P Q x = 0"
by (auto simp: entropy_density_def)
then have "integrable P (entropy_density b P Q) ⟷ integrable Q (λx. 0::real)"
using ed unfolding ‹Q = P› by (intro integrable_cong_AE) auto
then show "integrable Q (entropy_density b P Q)" by simp

from ae_0 have "mutual_information b S T X Y = (∫x. 0 ∂P)"
unfolding mutual_information_def KL_divergence_def P_def[symmetric] Q_def[symmetric] ‹Q = P›
by (intro integral_cong_AE) auto
then show "mutual_information b S T X Y = 0"
by simp }

{ assume ac: "absolutely_continuous P Q"
assume int: "integrable Q (entropy_density b P Q)"
assume I_eq_0: "mutual_information b S T X Y = 0"

have eq: "Q = P"
proof (rule P.KL_eq_0_iff_eq_ac[THEN iffD1])
show "prob_space Q" by unfold_locales
show "absolutely_continuous P Q" by fact
show "integrable Q (entropy_density b P Q)" by fact
show "sets Q = sets P" by (simp add: P_def Q_def sets_pair_measure)
show "KL_divergence b P Q = 0"
using I_eq_0 unfolding mutual_information_def by (simp add: P_def Q_def)
qed
then show "distr M S X ⨂⇩M distr M T Y = distr M (S ⨂⇩M T) (λx. (X x, Y x))"
unfolding P_def Q_def .. }
qed

abbreviation (in information_space)
mutual_information_Pow ("ℐ'(_ ; _')") where
"ℐ(X ; Y) ≡ mutual_information b (count_space (X`space M)) (count_space (Y`space M)) X Y"

lemma (in information_space)
fixes Pxy :: "'b × 'c ⇒ real" and Px :: "'b ⇒ real" and Py :: "'c ⇒ real"
assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
assumes Fx: "finite_entropy S X Px" and Fy: "finite_entropy T Y Py"
assumes Fxy: "finite_entropy (S ⨂⇩M T) (λx. (X x, Y x)) Pxy"
defines "f ≡ λx. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x)))"
shows mutual_information_distr': "mutual_information b S T X Y = integral⇧L (S ⨂⇩M T) f" (is "?M = ?R")
and mutual_information_nonneg': "0 ≤ mutual_information b S T X Y"
proof -
have Px: "distributed M S X Px" and Px_nn: "⋀x. x ∈ space S ⟹ 0 ≤ Px x"
using Fx by (auto simp: finite_entropy_def)
have Py: "distributed M T Y Py" and Py_nn: "⋀x. x ∈ space T ⟹ 0 ≤ Py x"
using Fy by (auto simp: finite_entropy_def)
have Pxy: "distributed M (S ⨂⇩M T) (λx. (X x, Y x)) Pxy"
and Pxy_nn: "⋀x. x ∈ space (S ⨂⇩M T) ⟹ 0 ≤ Pxy x"
"⋀x y. x ∈ space S ⟹ y ∈ space T ⟹ 0 ≤ Pxy (x, y)"
using Fxy by (auto simp: finite_entropy_def space_pair_measure)

have [measurable]: "Px ∈ S →⇩M borel"
using Px Px_nn by (intro distributed_real_measurable)
have [measurable]: "Py ∈ T →⇩M borel"
using Py Py_nn by (intro distributed_real_measurable)
have measurable_Pxy[measurable]: "Pxy ∈ (S ⨂⇩M T) →⇩M borel"
using Pxy Pxy_nn by (intro distributed_real_measurable) (auto simp: space_pair_measure)

have X[measurable]: "random_variable S X"
using Px by auto
have Y[measurable]: "random_variable T Y"
using Py by auto
interpret S: sigma_finite_measure S by fact
interpret T: sigma_finite_measure T by fact
interpret ST: pair_sigma_finite S T ..
interpret X: prob_space "distr M S X" using X by (rule prob_space_distr)
interpret Y: prob_space "distr M T Y" using Y by (rule prob_space_distr)
interpret XY: pair_prob_space "distr M S X" "distr M T Y" ..
let ?P = "S ⨂⇩M T"
let ?D = "distr M ?P (λx. (X x, Y x))"

{ fix A assume "A ∈ sets S"
with X[THEN measurable_space] Y[THEN measurable_space]
have "emeasure (distr M S X) A = emeasure ?D (A × space T)"
by (auto simp: emeasure_distr intro!: arg_cong[where f="emeasure M"]) }
note marginal_eq1 = this
{ fix A assume "A ∈ sets T"
with X[THEN measurable_space] Y[THEN measurable_space]
have "emeasure (distr M T Y) A = emeasure ?D (space S × A)"
by (auto simp: emeasure_distr intro!: arg_cong[where f="emeasure M"]) }
note marginal_eq2 = this

have distr_eq: "distr M S X ⨂⇩M distr M T Y = density ?P (λx. ennreal (Px (fst x) * Py (snd x)))"
unfolding Px(1)[THEN distributed_distr_eq_density] Py(1)[THEN distributed_distr_eq_density]
proof (subst pair_measure_density)
show "(λx. ennreal (Px x)) ∈ borel_measurable S" "(λy. ennreal (Py y)) ∈ borel_measurable T"
using Px Py by (auto simp: distributed_def)
show "sigma_finite_measure (density T Py)" unfolding Py(1)[THEN distributed_distr_eq_density, symmetric] ..
show "density (S ⨂⇩M T) (λ(x, y). ennreal (Px x) * ennreal (Py y)) =
density (S ⨂⇩M T) (λx. ennreal (Px (fst x) * Py (snd x)))"
using Px_nn Py_nn by (auto intro!: density_cong simp: distributed_def ennreal_mult space_pair_measure)
qed fact

have M: "?M = KL_divergence b (density ?P (λx. ennreal (Px (fst x) * Py (snd x)))) (density ?P (λx. ennreal (Pxy x)))"
unfolding mutual_information_def distr_eq Pxy(1)[THEN distributed_distr_eq_density] ..

from Px Py have f: "(λx. Px (fst x) * Py (snd x)) ∈ borel_measurable ?P"
by (intro borel_measurable_times) (auto intro: distributed_real_measurable measurable_fst'' measurable_snd'')
have PxPy_nonneg: "AE x in ?P. 0 ≤ Px (fst x) * Py (snd x)"
using Px_nn Py_nn by (auto simp: space_pair_measure)

have A: "(AE x in ?P. Px (fst x) = 0 ⟶ Pxy x = 0)"
by (rule subdensity_real[OF measurable_fst Pxy Px]) (insert Px_nn Pxy_nn, auto simp: space_pair_measure)
moreover
have B: "(AE x in ?P. Py (snd x) = 0 ⟶ Pxy x = 0)"
by (rule subdensity_real[OF measurable_snd Pxy Py]) (insert Py_nn Pxy_nn, auto simp: space_pair_measure)
ultimately have ac: "AE x in ?P. Px (fst x) * Py (snd x) = 0 ⟶ Pxy x = 0"
by auto

show "?M = ?R"
unfolding M f_def using Pxy_nn Px_nn Py_nn
by (intro ST.KL_density_density b_gt_1 f PxPy_nonneg ac) (auto simp: space_pair_measure)

have X: "X = fst ∘ (λx. (X x, Y x))" and Y: "Y = snd ∘ (λx. (X x, Y x))"
by auto

have "integrable (S ⨂⇩M T) (λx. Pxy x * log b (Pxy x) - Pxy x * log b (Px (fst x)) - Pxy x * log b (Py (snd x)))"
using finite_entropy_integrable[OF Fxy]
using finite_entropy_integrable_transform[OF Fx Pxy, of fst]
using finite_entropy_integrable_transform[OF Fy Pxy, of snd]
moreover have "f ∈ borel_measurable (S ⨂⇩M T)"
unfolding f_def using Px Py Pxy
by (auto intro: distributed_real_measurable measurable_fst'' measurable_snd''
intro!: borel_measurable_times borel_measurable_log borel_measurable_divide)
ultimately have int: "integrable (S ⨂⇩M T) f"
apply (rule integrable_cong_AE_imp)
using A B
by  (auto simp: f_def log_divide_eq log_mult_eq field_simps space_pair_measure Px_nn Py_nn Pxy_nn
less_le)

show "0 ≤ ?M" unfolding M
proof (intro ST.KL_density_density_nonneg)
show "prob_space (density (S ⨂⇩M T) (λx. ennreal (Pxy x))) "
unfolding distributed_distr_eq_density[OF Pxy, symmetric]
using distributed_measurable[OF Pxy] by (rule prob_space_distr)
show "prob_space (density (S ⨂⇩M T) (λx. ennreal (Px (fst x) * Py (snd x))))"
unfolding distr_eq[symmetric] by unfold_locales
show "integrable (S ⨂⇩M T) (λx. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x))))"
using int unfolding f_def .
qed (insert ac, auto simp: b_gt_1 Px_nn Py_nn Pxy_nn space_pair_measure)
qed

lemma (in information_space)
fixes Pxy :: "'b × 'c ⇒ real" and Px :: "'b ⇒ real" and Py :: "'c ⇒ real"
assumes "sigma_finite_measure S" "sigma_finite_measure T"
assumes Px: "distributed M S X Px" and Px_nn: "⋀x. x ∈ space S ⟹ 0 ≤ Px x"
and Py: "distributed M T Y Py" and Py_nn: "⋀y. y ∈ space T ⟹ 0 ≤ Py y"
and Pxy: "distributed M (S ⨂⇩M T) (λx. (X x, Y x)) Pxy"
and Pxy_nn: "⋀x y. x ∈ space S ⟹ y ∈ space T ⟹ 0 ≤ Pxy (x, y)"
defines "f ≡ λx. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x)))"
shows mutual_information_distr: "mutual_information b S T X Y = integral⇧L (S ⨂⇩M T) f" (is "?M = ?R")
and mutual_information_nonneg: "integrable (S ⨂⇩M T) f ⟹ 0 ≤ mutual_information b S T X Y"
proof -
have X[measurable]: "random_variable S X"
using Px by (auto simp: distributed_def)
have Y[measurable]: "random_variable T Y"
using Py by (auto simp: distributed_def)
have [measurable]: "Px ∈ S →⇩M borel"
using Px Px_nn by (intro distributed_real_measurable)
have [measurable]: "Py ∈ T →⇩M borel"
using Py Py_nn by (intro distributed_real_measurable)
have measurable_Pxy[measurable]: "Pxy ∈ (S ⨂⇩M T) →⇩M borel"
using Pxy Pxy_nn by (intro distributed_real_measurable) (auto simp: space_pair_measure)

interpret S: sigma_finite_measure S by fact
interpret T: sigma_finite_measure T by fact
interpret ST: pair_sigma_finite S T ..
interpret X: prob_space "distr M S X" using X by (rule prob_space_distr)
interpret Y: prob_space "distr M T Y" using Y by (rule prob_space_distr)
interpret XY: pair_prob_space "distr M S X" "distr M T Y" ..
let ?P = "S ⨂⇩M T"
let ?D = "distr M ?P (λx. (X x, Y x))"

{ fix A assume "A ∈ sets S"
with X[THEN measurable_space] Y[THEN measurable_space]
have "emeasure (distr M S X) A = emeasure ?D (A × space T)"
by (auto simp: emeasure_distr intro!: arg_cong[where f="emeasure M"]) }
note marginal_eq1 = this
{ fix A assume "A ∈ sets T"
with X[THEN measurable_space] Y[THEN measurable_space]
have "emeasure (distr M T Y) A = emeasure ?D (space S × A)"
by (auto simp: emeasure_distr intro!: arg_cong[where f="emeasure M"]) }
note marginal_eq2 = this

have distr_eq: "distr M S X ⨂⇩M distr M T Y = density ?P (λx. ennreal (Px (fst x) * Py (snd x)))"
unfolding Px(1)[THEN distributed_distr_eq_density] Py(1)[THEN distributed_distr_eq_density]
proof (subst pair_measure_density)
show "(λx. ennreal (Px x)) ∈ borel_measurable S" "(λy. ennreal (Py y)) ∈ borel_measurable T"
using Px Py by (auto simp: distributed_def)
show "sigma_finite_measure (density T Py)" unfolding Py(1)[THEN distributed_distr_eq_density, symmetric] ..
show "density (S ⨂⇩M T) (λ(x, y). ennreal (Px x) * ennreal (Py y)) =
density (S ⨂⇩M T) (λx. ennreal (Px (fst x) * Py (snd x)))"
using Px_nn Py_nn by (auto intro!: density_cong simp: distributed_def ennreal_mult space_pair_measure)
qed fact

have M: "?M = KL_divergence b (density ?P (λx. ennreal (Px (fst x) * Py (snd x)))) (density ?P (λx. ennreal (Pxy x)))"
unfolding mutual_information_def distr_eq Pxy(1)[THEN distributed_distr_eq_density] ..

from Px Py have f: "(λx. Px (fst x) * Py (snd x)) ∈ borel_measurable ?P"
by (intro borel_measurable_times) (auto intro: distributed_real_measurable measurable_fst'' measurable_snd'')
have PxPy_nonneg: "AE x in ?P. 0 ≤ Px (fst x) * Py (snd x)"
using Px_nn Py_nn by (auto simp: space_pair_measure)

have "(AE x in ?P. Px (fst x) = 0 ⟶ Pxy x = 0)"
by (rule subdensity_real[OF measurable_fst Pxy Px]) (insert Px_nn Pxy_nn, auto simp: space_pair_measure)
moreover
have "(AE x in ?P. Py (snd x) = 0 ⟶ Pxy x = 0)"
by (rule subdensity_real[OF measurable_snd Pxy Py]) (insert Py_nn Pxy_nn, auto simp: space_pair_measure)
ultimately have ac: "AE x in ?P. Px (fst x) * Py (snd x) = 0 ⟶ Pxy x = 0"
by auto

show "?M = ?R"
unfolding M f_def
using b_gt_1 f PxPy_nonneg ac Pxy_nn
by (intro ST.KL_density_density) (auto simp: space_pair_measure)

assume int: "integrable (S ⨂⇩M T) f"
show "0 ≤ ?M" unfolding M
proof (intro ST.KL_density_density_nonneg)
show "prob_space (density (S ⨂⇩M T) (λx. ennreal (Pxy x))) "
unfolding distributed_distr_eq_density[OF Pxy, symmetric]
using distributed_measurable[OF Pxy] by (rule prob_space_distr)
show "prob_space (density (S ⨂⇩M T) (λx. ennreal (Px (fst x) * Py (snd x))))"
unfolding distr_eq[symmetric] by unfold_locales
show "integrable (S ⨂⇩M T) (λx. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x))))"
using int unfolding f_def .
qed (insert ac, auto simp: b_gt_1 Px_nn Py_nn Pxy_nn space_pair_measure)
qed

lemma (in information_space)
fixes Pxy :: "'b × 'c ⇒ real" and Px :: "'b ⇒ real" and Py :: "'c ⇒ real"
assumes "sigma_finite_measure S" "sigma_finite_measure T"
assumes Px[measurable]: "distributed M S X Px" and Px_nn: "⋀x. x ∈ space S ⟹ 0 ≤ Px x"
and Py[measurable]: "distributed M T Y Py" and Py_nn:  "⋀x. x ∈ space T ⟹ 0 ≤ Py x"
and Pxy[measurable]: "distributed M (S ⨂⇩M T) (λx. (X x, Y x)) Pxy"
and Pxy_nn: "⋀x. x ∈ space (S ⨂⇩M T) ⟹ 0 ≤ Pxy x"
assumes ae: "AE x in S. AE y in T. Pxy (x, y) = Px x * Py y"
shows mutual_information_eq_0: "mutual_information b S T X Y = 0"
proof -
interpret S: sigma_finite_measure S by fact
interpret T: sigma_finite_measure T by fact
interpret ST: pair_sigma_finite S T ..
note
distributed_real_measurable[OF Px_nn Px, measurable]
distributed_real_measurable[OF Py_nn Py, measurable]
distributed_real_measurable[OF Pxy_nn Pxy, measurable]

have "AE x in S ⨂⇩M T. Px (fst x) = 0 ⟶ Pxy x = 0"
by (rule subdensity_real[OF measurable_fst Pxy Px]) (auto simp: Px_nn Pxy_nn space_pair_measure)
moreover
have "AE x in S ⨂⇩M T. Py (snd x) = 0 ⟶ Pxy x = 0"
by (rule subdensity_real[OF measurable_snd Pxy Py]) (auto simp: Py_nn Pxy_nn space_pair_measure)
moreover
have "AE x in S ⨂⇩M T. Pxy x = Px (fst x) * Py (snd x)"
by (intro ST.AE_pair_measure) (auto simp: ae intro!: measurable_snd'' measurable_fst'')
ultimately have "AE x in S ⨂⇩M T. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x))) = 0"
by eventually_elim simp
then have "(∫x. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x))) ∂(S ⨂⇩M T)) = (∫x. 0 ∂(S ⨂⇩M T))"
by (intro integral_cong_AE) auto
then show ?thesis
by (subst mutual_information_distr[OF assms(1-8)]) (auto simp add: space_pair_measure)
qed

lemma (in information_space) mutual_information_simple_distributed:
assumes X: "simple_distributed M X Px" and Y: "simple_distributed M Y Py"
assumes XY: "simple_distributed M (λx. (X x, Y x)) Pxy"
shows "ℐ(X ; Y) = (∑(x, y)∈(λx. (X x, Y x))`space M. Pxy (x, y) * log b (Pxy (x, y) / (Px x * Py y)))"
proof (subst mutual_information_distr[OF _ _ simple_distributed[OF X] _ simple_distributed[OF Y] _ simple_distributed_joint[OF XY]])
note fin = simple_distributed_joint_finite[OF XY, simp]
show "sigma_finite_measure (count_space (X ` space M))"
show "sigma_finite_measure (count_space (Y ` space M))"
let ?Pxy = "λx. (if x ∈ (λx. (X x, Y x)) ` space M then Pxy x else 0)"
let ?f = "λx. ?Pxy x * log b (?Pxy x / (Px (fst x) * Py (snd x)))"
have "⋀x. ?f x = (if x ∈ (λx. (X x, Y x)) ` space M then Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x))) else 0)"
by auto
with fin show "(∫ x. ?f x ∂(count_space (X ` space M) ⨂⇩M count_space (Y ` space M))) =
(∑(x, y)∈(λx. (X x, Y x)) ` space M. Pxy (x, y) * log b (Pxy (x, y) / (Px x * Py y)))"
by (auto simp add: pair_measure_count_space lebesgue_integral_count_space_finite sum.If_cases split_beta'
intro!: sum.cong)
qed (insert X Y XY, auto simp: simple_distributed_def)

lemma (in information_space)
fixes Pxy :: "'b × 'c ⇒ real" and Px :: "'b ⇒ real" and Py :: "'c ⇒ real"
assumes Px: "simple_distributed M X Px" and Py: "simple_distributed M Y Py"
assumes Pxy: "simple_distributed M (λx. (X x, Y x)) Pxy"
assumes ae: "∀x∈space M. Pxy (X x, Y x) = Px (X x) * Py (Y x)"
shows mutual_information_eq_0_simple: "ℐ(X ; Y) = 0"
proof (subst mutual_information_simple_distributed[OF Px Py Pxy])
have "(∑(x, y)∈(λx. (X x, Y x)) ` space M. Pxy (x, y) * log b (Pxy (x, y) / (Px x * Py y))) =
(∑(x, y)∈(λx. (X x, Y x)) ` space M. 0)"
by (intro sum.cong) (auto simp: ae)
then show "(∑(x, y)∈(λx. (X x, Y x)) ` space M.
Pxy (x, y) * log b (Pxy (x, y) / (Px x * Py y))) = 0" by simp
qed

subsection ‹Entropy›

definition (in prob_space) entropy :: "real ⇒ 'b measure ⇒ ('a ⇒ 'b) ⇒ real" where
"entropy b S X = - KL_divergence b S (distr M S X)"

abbreviation (in information_space)
entropy_Pow ("ℋ'(_')") where
"ℋ(X) ≡ entropy b (count_space (X`space M)) X"

lemma (in prob_space) distributed_RN_deriv:
assumes X: "distributed M S X Px"
shows "AE x in S. RN_deriv S (density S Px) x = Px x"
proof -
have "distributed M S X (RN_deriv S (density S Px))"
by (metis RN_derivI assms borel_measurable_RN_deriv distributed_def)
then show ?thesis
using assms distributed_unique by blast
qed

lemma (in information_space)
fixes X :: "'a ⇒ 'b"
assumes X[measurable]: "distributed M MX X f" and nn: "⋀x. x ∈ space MX ⟹ 0 ≤ f x"
shows entropy_distr: "entropy b MX X = - (∫x. f x * log b (f x) ∂MX)" (is ?eq)
proof -
note D = distributed_measurable[OF X] distributed_borel_measurable[OF X]
note ae = distributed_RN_deriv[OF X]
note distributed_real_measurable[OF nn X, measurable]

have ae_eq: "AE x in distr M MX X. log b (enn2real (RN_deriv MX (distr M MX X) x)) = log b (f x)"
unfolding distributed_distr_eq_density[OF X]
using D ae by (auto simp: AE_density)

have int_eq: "(∫ x. f x * log b (f x) ∂MX) = (∫ x. log b (f x) ∂distr M MX X)"
unfolding distributed_distr_eq_density[OF X]
using D
by (subst integral_density) (auto simp: nn)

show ?eq
unfolding entropy_def KL_divergence_def entropy_density_def comp_def int_eq neg_equal_iff_equal
using ae_eq by (intro integral_cong_AE) (auto simp: nn)
qed

lemma (in information_space) entropy_le:
fixes Px :: "'b ⇒ real" and MX :: "'b measure"
assumes X[measurable]: "distributed M MX X Px" and Px_nn[simp]: "⋀x. x ∈ space MX ⟹ 0 ≤ Px x"
and fin: "emeasure MX {x ∈ space MX. Px x ≠ 0} ≠ top"
and int: "integrable MX (λx. - Px x * log b (Px x))"
shows "entropy b MX X ≤ log b (measure MX {x ∈ space MX. Px x ≠ 0})"
proof -
note Px = distributed_borel_measurable[OF X]
interpret X: prob_space "distr M MX X"
using distributed_measurable[OF X] by (rule prob_space_distr)

have " - log b (measure MX {x ∈ space MX. Px x ≠ 0}) =
- log b (∫ x. indicator {x ∈ space MX. Px x ≠ 0} x ∂MX)"
using Px Px_nn fin by (auto simp: measure_def)
also have "- log b (∫ x. indicator {x ∈ space MX. Px x ≠ 0} x ∂MX) = - log b (∫ x. 1 / Px x ∂distr M MX X)"
proof -
have "integral⇧L MX (indicator {x ∈ space MX. Px x ≠ 0}) = LINT x|MX. Px x *⇩R (1 / Px x)"
by (rule Bochner_Integration.integral_cong) auto
also have "... = LINT x|density MX (λx. ennreal (Px x)). 1 / Px x"
by (rule integral_density [symmetric]) (use Px Px_nn in auto)
finally show ?thesis
unfolding distributed_distr_eq_density[OF X] by simp
qed
also have "… ≤ (∫ x. - log b (1 / Px x) ∂distr M MX X)"
proof (rule X.jensens_inequality[of "λx. 1 / Px x" "{0<..}" 0 1 "λx. - log b x"])
show "AE x in distr M MX X. 1 / Px x ∈ {0<..}"
unfolding distributed_distr_eq_density[OF X]
using Px by (auto simp: AE_density)
have [simp]: "⋀x. x ∈ space MX ⟹ ennreal (if Px x = 0 then 0 else 1) = indicator {x ∈ space MX. Px x ≠ 0} x"
by (auto simp: one_ennreal_def)
have "(∫⇧+ x. ennreal (- (if Px x = 0 then 0 else 1)) ∂MX) = (∫⇧+ x. 0 ∂MX)"
by (intro nn_integral_cong) (auto simp: ennreal_neg)
then show "integrable (distr M MX X) (λx. 1 / Px x)"
unfolding distributed_distr_eq_density[OF X] using Px
by (auto simp: nn_integral_density real_integrable_def fin ennreal_neg ennreal_mult[symmetric]
cong: nn_integral_cong)
have "integrable MX (λx. Px x * log b (1 / Px x)) =
integrable MX (λx. - Px x * log b (Px x))"
using Px
by (intro integrable_cong_AE) (auto simp: log_divide_eq less_le)
then show "integrable (distr M MX X) (λx. - log b (1 / Px x))"
unfolding distributed_distr_eq_density[OF X]
using Px int
by (subst integrable_real_density) auto
qed (auto simp: minus_log_convex[OF b_gt_1])
also have "… = (∫ x. log b (Px x) ∂distr M MX X)"
unfolding distributed_distr_eq_density[OF X] using Px
by (intro integral_cong_AE) (auto simp: AE_density log_divide_eq)
also have "… = - entropy b MX X"
unfolding distributed_distr_eq_density[OF X] using Px
by (subst entropy_distr[OF X]) (auto simp: integral_density)
finally show ?thesis
by simp
qed

lemma (in information_space) entropy_le_space:
fixes Px :: "'b ⇒ real" and MX :: "'b measure"
assumes X: "distributed M MX X Px" and Px_nn[simp]: "⋀x. x ∈ space MX ⟹ 0 ≤ Px x"
and fin: "finite_measure MX"
and int: "integrable MX (λx. - Px x * log b (Px x))"
shows "entropy b MX X ≤ log b (measure MX (space MX))"
proof -
note Px = distributed_borel_measurable[OF X]
interpret finite_measure MX by fact
have "entropy b MX X ≤ log b (measure MX {x ∈ space MX. Px x ≠ 0})"
using int X by (intro entropy_le) auto
also have "… ≤ log b (measure MX (space MX))"
using Px distributed_imp_emeasure_nonzero[OF X]
by (intro Transcendental.log_mono)
(auto intro!: finite_measure_mono b_gt_1 less_le[THEN iffD2]
simp: emeasure_eq_measure cong: conj_cong)
finally show ?thesis .
qed

lemma (in information_space) entropy_uniform:
assumes X: "distributed M MX X (λx. indicator A x / measure MX A)" (is "distributed _ _ _ ?f")
shows "entropy b MX X = log b (measure MX A)"
proof (subst entropy_distr[OF X])
have [simp]: "emeasure MX A ≠ ∞"
using uniform_distributed_params[OF X] by (auto simp add: measure_def)
have eq: "(∫ x. indicator A x / measure MX A * log b (indicator A x / measure MX A) ∂MX) =
(∫ x. (- log b (measure MX A) / measure MX A) * indicator A x ∂MX)"
using uniform_distributed_params[OF X]
by (intro Bochner_Integration.integral_cong) (auto split: split_indicator simp: log_divide_eq zero_less_measure_iff)
show "- (∫ x. indicator A x / measure MX A * log b (indicator A x / measure MX A) ∂MX) =
log b (measure MX A)"
unfolding eq using uniform_distributed_params[OF X]
by (subst Bochner_Integration.integral_mult_right) (auto simp: measure_def less_top[symmetric] intro!: integrable_real_indicator)
qed simp

lemma (in information_space) entropy_simple_distributed:
"simple_distributed M X f ⟹ ℋ(X) = - (∑x∈X`space M. f x * log b (f x))"
by (subst entropy_distr[OF simple_distributed])

lemma (in information_space) entropy_le_card_not_0:
assumes X: "simple_distributed M X f"
shows "ℋ(X) ≤ log b (card (X ` space M ∩ {x. f x ≠ 0}))"
proof -
let ?X = "count_space (X`space M)"
have "ℋ(X) ≤ log b (measure ?X {x ∈ space ?X. f x ≠ 0})"
by (rule entropy_le[OF simple_distributed[OF X]])
(insert X, auto simp: simple_distributed_finite[OF X] subset_eq integrable_count_space emeasure_count_space)
also have "measure ?X {x ∈ space ?X. f x ≠ 0} = card (X ` space M ∩ {x. f x ≠ 0})"
by (simp_all add: simple_distributed_finite[OF X] subset_eq emeasure_count_space measure_def Int_def)
finally show ?thesis .
qed

lemma (in information_space) entropy_le_card:
assumes X: "simple_distributed M X f"
shows "ℋ(X) ≤ log b (real (card (X ` space M)))"
proof -
let ?X = "count_space (X`space M)"
have "ℋ(X) ≤ log b (measure ?X (space ?X))"
by (rule entropy_le_space[OF simple_distributed[OF X]])
(insert X, auto simp: simple_distributed_finite[OF X] subset_eq integrable_count_space emeasure_count_space finite_measure_count_space)
also have "measure ?X (space ?X) = card (X ` space M)"
by (simp_all add: simple_distributed_finite[OF X] subset_eq emeasure_count_space measure_def)
finally show ?thesis .
qed

subsection ‹Conditional Mutual Information›

definition (in prob_space)
"conditional_mutual_information b MX MY MZ X Y Z ≡
mutual_information b MX (MY ⨂⇩M MZ) X (λx. (Y x, Z x)) -
mutual_information b MX MZ X Z"

abbreviation (in information_space)
conditional_mutual_information_Pow ("ℐ'( _ ; _ | _ ')") where
"ℐ(X ; Y | Z) ≡ conditional_mutual_information b
(count_space (X ` space M)) (count_space (Y ` space M)) (count_space (Z ` space M)) X Y Z"

lemma (in information_space)
assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T" and P: "sigma_finite_measure P"
assumes Px[measurable]: "distributed M S X Px"
and Px_nn[simp]: "⋀x. x ∈ space S ⟹ 0 ≤ Px x"
assumes Pz[measurable]: "distributed M P Z Pz"
and Pz_nn[simp]: "⋀z. z ∈ space P ⟹ 0 ≤ Pz z"
assumes Pyz[measurable]: "distributed M (T ⨂⇩M P) (λx. (Y x, Z x)) Pyz"
and Pyz_nn[simp]: "⋀y z. y ∈ space T ⟹ z ∈ space P ⟹ 0 ≤ Pyz (y, z)"
assumes Pxz[measurable]: "distributed M (S ⨂⇩M P) (λx. (X x, Z x)) Pxz"
and Pxz_nn[simp]: "⋀x z. x ∈ space S ⟹ z ∈ space P ⟹ 0 ≤ Pxz (x, z)"
assumes Pxyz[measurable]: "distributed M (S ⨂⇩M T ⨂⇩M P) (λx. (X x, Y x, Z x)) Pxyz"
and Pxyz_nn[simp]: "⋀x y z. x ∈ space S ⟹ y ∈ space T ⟹ z ∈ space P ⟹ 0 ≤ Pxyz (x, y, z)"
assumes I1: "integrable (S ⨂⇩M T ⨂⇩M P) (λ(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Px x * Pyz (y, z))))"
assumes I2: "integrable (S ⨂⇩M T ⨂⇩M P) (λ(x, y, z). Pxyz (x, y, z) * log b (Pxz (x, z) / (Px x * Pz z)))"
shows conditional_mutual_information_generic_eq: "conditional_mutual_information b S T P X Y Z
= (∫(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z))) ∂(S ⨂⇩M T ⨂⇩M P))" (is "?eq")
and conditional_mutual_information_generic_nonneg: "0 ≤ conditional_mutual_information b S T P X Y Z" (is "?nonneg")
proof -
have [measurable]: "Px ∈ S →⇩M borel"
using Px Px_nn by (intro distributed_real_measurable)
have [measurable]: "Pz ∈ P →⇩M borel"
using Pz Pz_nn by (intro distributed_real_measurable)
have measurable_Pyz[measurable]: "Pyz ∈ (T ⨂⇩M P) →⇩M borel"
using Pyz Pyz_nn by (intro distributed_real_measurable) (auto simp: space_pair_measure)
have measurable_Pxz[measurable]: "Pxz ∈ (S ⨂⇩M P) →⇩M borel"
using Pxz Pxz_nn by (intro distributed_real_measurable) (auto simp: space_pair_measure)
have measurable_Pxyz[measurable]: "Pxyz ∈ (S ⨂⇩M T ⨂⇩M P) →⇩M borel"
using Pxyz Pxyz_nn by (intro distributed_real_measurable) (auto simp: space_pair_measure)

interpret S: sigma_finite_measure S by fact
interpret T: sigma_finite_measure T by fact
interpret P: sigma_finite_measure P by fact
interpret TP: pair_sigma_finite T P ..
interpret SP: pair_sigma_finite S P ..
interpret ST: pair_sigma_finite S T ..
interpret SPT: pair_sigma_finite "S ⨂⇩M P" T ..
interpret STP: pair_sigma_finite S "T ⨂⇩M P" ..
interpret TPS: pair_sigma_finite "T ⨂⇩M P" S ..
have TP: "sigma_finite_measure (T ⨂⇩M P)" ..
have SP: "sigma_finite_measure (S ⨂⇩M P)" ..
have YZ: "random_variable (T ⨂⇩M P) (λx. (Y x, Z x))"
using Pyz by (simp add: distributed_measurable)

from Pxz Pxyz have distr_eq: "distr M (S ⨂⇩M P) (λx. (X x, Z x)) =
distr (distr M (S ⨂⇩M T ⨂⇩M P) (λx. (X x, Y x, Z x))) (S ⨂⇩M P) (λ(x, y, z). (x, z))"

have "mutual_information b S P X Z =
(∫x. Pxz x * log b (Pxz x / (Px (fst x) * Pz (snd x))) ∂(S ⨂⇩M P))"
by (rule mutual_information_distr[OF S P Px Px_nn Pz Pz_nn Pxz Pxz_nn])
also have "… = (∫(x,y,z). Pxyz (x,y,z) * log b (Pxz (x,z) / (Px x * Pz z)) ∂(S ⨂⇩M T ⨂⇩M P))"
using b_gt_1 Pxz Px Pz
by (subst distributed_transform_integral[OF Pxyz _ Pxz _, where T="λ(x, y, z). (x, z)"])
(auto simp: split_beta' space_pair_measure)
finally have mi_eq:
"mutual_information b S P X Z = (∫(x,y,z). Pxyz (x,y,z) * log b (Pxz (x,z) / (Px x * Pz z)) ∂(S ⨂⇩M T ⨂⇩M P))" .

have ae1: "AE x in S ⨂⇩M T ⨂⇩M P. Px (fst x) = 0 ⟶ Pxyz x = 0"
by (intro subdensity_real[of fst, OF _ Pxyz Px]) (auto simp: space_pair_measure)
moreover have ae2: "AE x in S ⨂⇩M T ⨂⇩M P. Pz (snd (snd x)) = 0 ⟶ Pxyz x = 0"
by (intro subdensity_real[of "λx. snd (snd x)", OF _ Pxyz Pz]) (auto simp: space_pair_measure)
moreover have ae3: "AE x in S ⨂⇩M T ⨂⇩M P. Pxz (fst x, snd (snd x)) = 0 ⟶ Pxyz x = 0"
by (intro subdensity_real[of "λx. (fst x, snd (snd x))", OF _ Pxyz Pxz]) (auto simp: space_pair_measure)
moreover have ae4: "AE x in S ⨂⇩M T ⨂⇩M P. Pyz (snd x) = 0 ⟶ Pxyz x = 0"
by (intro subdensity_real[of snd, OF _ Pxyz Pyz]) (auto simp: space_pair_measure)
ultimately have ae: "AE x in S ⨂⇩M T ⨂⇩M P.
Pxyz x * log b (Pxyz x / (Px (fst x) * Pyz (snd x))) -
Pxyz x * log b (Pxz (fst x, snd (snd x)) / (Px (fst x) * Pz (snd (snd x)))) =
Pxyz x * log b (Pxyz x * Pz (snd (snd x)) / (Pxz (fst x, snd (snd x)) * Pyz (snd x))) "
using AE_space
proof eventually_elim
case (elim x)
show ?case
proof cases
assume "Pxyz x ≠ 0"
with elim have "0 < Px (fst x)" "0 < Pz (snd (snd x))" "0 < Pxz (fst x, snd (snd x))"
"0 < Pyz (snd x)" "0 < Pxyz x"
by (auto simp: space_pair_measure less_le)
then show ?thesis
using b_gt_1 by (simp add: log_simps less_imp_le field_simps)
qed simp
qed
with I1 I2 show ?eq
unfolding conditional_mutual_information_def
apply (subst mi_eq)
apply (subst mutual_information_distr[OF S TP Px Px_nn Pyz _ Pxyz])
apply (auto simp: space_pair_measure)
apply (subst Bochner_Integration.integral_diff[symmetric])
apply (auto intro!: integral_cong_AE simp: split_beta' simp del: Bochner_Integration.integral_diff)
done

let ?P = "density (S ⨂⇩M T ⨂⇩M P) Pxyz"
interpret P: prob_space ?P
unfolding distributed_distr_eq_density[OF Pxyz, symmetric]
by (rule prob_space_distr) simp

let ?Q = "density (T ⨂⇩M P) Pyz"
interpret Q: prob_space ?Q
unfolding distributed_distr_eq_density[OF Pyz, symmetric]
by (rule prob_space_distr) simp

let ?f = "λ(x, y, z). Pxz (x, z) * (Pyz (y, z) / Pz z) / Pxyz (x, y, z)"

from subdensity_real[of snd, OF _ Pyz Pz _ AE_I2 AE_I2]
have aeX1: "AE x in T ⨂⇩M P. Pz (snd x) = 0 ⟶ Pyz x = 0"
by (auto simp: comp_def space_pair_measure)
have aeX2: "AE x in T ⨂⇩M P. 0 ≤ Pz (snd x)"
using Pz by (intro TP.AE_pair_measure) (auto simp: comp_def)

have aeX3: "AE y in T ⨂⇩M P. (∫⇧+ x. ennreal (Pxz (x, snd y)) ∂S) = ennreal (Pz (snd y))"
using Pz distributed_marginal_eq_joint2[OF P S Pz Pxz]
by (intro TP.AE_pair_measure) auto

have "(∫⇧+ x. ?f x ∂?P) ≤ (∫⇧+ (x, y, z). Pxz (x, z) * (Pyz (y, z) / Pz z) ∂(S ⨂⇩M T ⨂⇩M P))"
by (subst nn_integral_density)
(auto intro!: nn_integral_mono simp: space_pair_measure ennreal_mult[symmetric])
also have "… = (∫⇧+(y, z). (∫⇧+ x. ennreal (Pxz (x, z)) * ennreal (Pyz (y, z) / Pz z) ∂S) ∂(T ⨂⇩M P))"
by (subst STP.nn_integral_snd[symmetric])
(auto simp add: split_beta' ennreal_mult[symmetric] space_pair_measure intro!: nn_integral_cong)
also have "… = (∫⇧+x. ennreal (Pyz x) * 1 ∂T ⨂⇩M P)"
proof -
have D: "(∫⇧+ x. ennreal (Pxz (x, b)) * ennreal (Pyz (a, b) / Pz b) ∂S) = ennreal (Pyz (a, b))"
if "Pz b = 0 ⟶ Pyz (a, b) = 0" "a ∈ space T ∧ b ∈ space P"
"(∫⇧+ x. ennreal (Pxz (x, b)) ∂S) = ennreal (Pz b)"
for a b
using that by (subst nn_integral_multc) (auto split: prod.split simp: ennreal_mult[symmetric])
show ?thesis
apply (rule nn_integral_cong_AE)
using aeX1 aeX2 aeX3
by (force simp add: space_pair_measure D)
qed
also have "… = 1"
using Q.emeasure_space_1 distributed_distr_eq_density[OF Pyz]
by (subst nn_integral_density[symmetric]) auto
finally have le1: "(∫⇧+ x. ?f x ∂?P) ≤ 1" .
also have "… < ∞" by simp
finally have fin: "(∫⇧+ x. ?f x ∂?P) ≠ ∞" by simp

have pos: "(∫⇧+x. ?f x ∂?P) ≠ 0"
apply (subst nn_integral_density)
proof
let ?g = "λx. ennreal (Pxyz x) * (Pxz (fst x, snd (snd x)) * Pyz (snd x) / (Pz (snd (snd x)) * Pxyz x))"
assume "(∫⇧+x. ?g x ∂(S ⨂⇩M T ⨂⇩M P)) = 0"
then have "AE x in S ⨂⇩M T ⨂⇩M P. ?g x = 0"
by (intro nn_integral_0_iff_AE[THEN iffD1]) auto
then have "AE x in S ⨂⇩M T ⨂⇩M P. Pxyz x = 0"
using ae1 ae2 ae3 ae4
by (auto split: if_split_asm simp: mult_le_0_iff divide_le_0_iff space_pair_measure)
then have "(∫⇧+ x. ennreal (Pxyz x) ∂S ⨂⇩M T ⨂⇩M P) = 0"
by (subst nn_integral_cong_AE[of _ "λx. 0"]) auto
with P.emeasure_space_1 show False
by (subst (asm) emeasure_density) (auto cong: nn_integral_cong)
qed

have neg: "(∫⇧+ x. - ?f x ∂?P) = 0"
by (subst nn_integral_0_iff_AE) (auto simp: space_pair_measure ennreal_neg)

have I3: "integrable (S ⨂⇩M T ⨂⇩M P) (λ(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z))))"
apply (rule integrable_cong_AE[THEN iffD1, OF _ _ _ Bochner_Integration.integrable_diff[OF I1 I2]])
using ae
apply (auto simp: split_beta')
done

have "- log b 1 ≤ - log b (integral⇧L ?P ?f)"
proof (intro le_imp_neg_le Transcendental.log_mono[OF b_gt_1])
have If: "integrable ?P ?f"
unfolding real_integrable_def
proof (intro conjI)
from neg show "(∫⇧+ x. - ?f x ∂?P) ≠ ∞"
by simp
from fin show "(∫⇧+ x. ?f x ∂?P) ≠ ∞"
by simp
qed simp
then have "(∫⇧+ x. ?f x ∂?P) = (∫x. ?f x ∂?P)"
apply (rule nn_integral_eq_integral)
apply (auto simp: space_pair_measure ennreal_neg)
done
with pos le1
show "0 < (∫x. ?f x ∂?P)" "(∫x. ?f x ∂?P) ≤ 1"
qed
also have "- log b (integral⇧L ?P ?f) ≤ (∫ x. - log b (?f x) ∂?P)"
proof (rule P.jensens_inequality[where a=0 and b=1 and I="{0<..}"])
show "AE x in ?P. ?f x ∈ {0<..}"
unfolding AE_density[OF distributed_borel_measurable[OF Pxyz]]
using ae1 ae2 ae3 ae4
by (auto simp: space_pair_measure less_le)
show "integrable ?P ?f"
unfolding real_integrable_def
using fin neg by (auto simp: split_beta')
have "integrable (S ⨂⇩M T ⨂⇩M P) (λx. Pxyz x * - log b (?f x))"
apply (rule integrable_cong_AE[THEN iffD1, OF _ _ _ I3])
using ae1 ae2 ae3 ae4
apply (auto simp: log_divide_eq log_mult_eq zero_le_mult_iff zero_less_mult_iff zero_less_divide_iff field_simps
less_le space_pair_measure)
done
then
show "integrable ?P (λx. - log b (?f x))"
by (subst integrable_real_density) (auto simp: space_pair_measure)
qed (auto simp: b_gt_1 minus_log_convex)
also have "… = conditional_mutual_information b S T P X Y Z"
unfolding ‹?eq›
apply (subst integral_real_density)
apply simp
apply (force simp: space_pair_measure)
apply simp
apply (intro integral_cong_AE)
using ae1 ae2 ae3 ae4
apply (auto simp: ```