src/HOL/Probability/Information.thy
 author hoelzl Tue Mar 22 18:53:05 2011 +0100 (2011-03-22) changeset 42066 6db76c88907a parent 41981 cdf7693bbe08 child 42067 66c8281349ec permissions -rw-r--r--
generalized Caratheodory from algebra to ring_of_sets
1 theory Information
2 imports
3   Probability_Space
4   "~~/src/HOL/Library/Convex"
5 begin
7 lemma (in prob_space) not_zero_less_distribution[simp]:
8   "(\<not> 0 < distribution X A) \<longleftrightarrow> distribution X A = 0"
9   using distribution_positive[of X A] by arith
11 lemma log_le: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> x \<le> y \<Longrightarrow> log a x \<le> log a y"
12   by (subst log_le_cancel_iff) auto
14 lemma log_less: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> x < y \<Longrightarrow> log a x < log a y"
15   by (subst log_less_cancel_iff) auto
17 lemma setsum_cartesian_product':
18   "(\<Sum>x\<in>A \<times> B. f x) = (\<Sum>x\<in>A. setsum (\<lambda>y. f (x, y)) B)"
19   unfolding setsum_cartesian_product by simp
21 section "Convex theory"
23 lemma log_setsum:
24   assumes "finite s" "s \<noteq> {}"
25   assumes "b > 1"
26   assumes "(\<Sum> i \<in> s. a i) = 1"
27   assumes "\<And> i. i \<in> s \<Longrightarrow> a i \<ge> 0"
28   assumes "\<And> i. i \<in> s \<Longrightarrow> y i \<in> {0 <..}"
29   shows "log b (\<Sum> i \<in> s. a i * y i) \<ge> (\<Sum> i \<in> s. a i * log b (y i))"
30 proof -
31   have "convex_on {0 <..} (\<lambda> x. - log b x)"
32     by (rule minus_log_convex[OF `b > 1`])
33   hence "- log b (\<Sum> i \<in> s. a i * y i) \<le> (\<Sum> i \<in> s. a i * - log b (y i))"
34     using convex_on_setsum[of _ _ "\<lambda> x. - log b x"] assms pos_is_convex by fastsimp
35   thus ?thesis by (auto simp add:setsum_negf le_imp_neg_le)
36 qed
38 lemma log_setsum':
39   assumes "finite s" "s \<noteq> {}"
40   assumes "b > 1"
41   assumes "(\<Sum> i \<in> s. a i) = 1"
42   assumes pos: "\<And> i. i \<in> s \<Longrightarrow> 0 \<le> a i"
43           "\<And> i. \<lbrakk> i \<in> s ; 0 < a i \<rbrakk> \<Longrightarrow> 0 < y i"
44   shows "log b (\<Sum> i \<in> s. a i * y i) \<ge> (\<Sum> i \<in> s. a i * log b (y i))"
45 proof -
46   have "\<And>y. (\<Sum> i \<in> s - {i. a i = 0}. a i * y i) = (\<Sum> i \<in> s. a i * y i)"
47     using assms by (auto intro!: setsum_mono_zero_cong_left)
48   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))"
49   proof (rule log_setsum)
50     have "setsum a (s - {i. a i = 0}) = setsum a s"
51       using assms(1) by (rule setsum_mono_zero_cong_left) auto
52     thus sum_1: "setsum a (s - {i. a i = 0}) = 1"
53       "finite (s - {i. a i = 0})" using assms by simp_all
55     show "s - {i. a i = 0} \<noteq> {}"
56     proof
57       assume *: "s - {i. a i = 0} = {}"
58       hence "setsum a (s - {i. a i = 0}) = 0" by (simp add: * setsum_empty)
59       with sum_1 show False by simp
60     qed
62     fix i assume "i \<in> s - {i. a i = 0}"
63     hence "i \<in> s" "a i \<noteq> 0" by simp_all
64     thus "0 \<le> a i" "y i \<in> {0<..}" using pos[of i] by auto
65   qed fact+
66   ultimately show ?thesis by simp
67 qed
69 lemma log_setsum_divide:
70   assumes "finite S" and "S \<noteq> {}" and "1 < b"
71   assumes "(\<Sum>x\<in>S. g x) = 1"
72   assumes pos: "\<And>x. x \<in> S \<Longrightarrow> g x \<ge> 0" "\<And>x. x \<in> S \<Longrightarrow> f x \<ge> 0"
73   assumes g_pos: "\<And>x. \<lbrakk> x \<in> S ; 0 < g x \<rbrakk> \<Longrightarrow> 0 < f x"
74   shows "- (\<Sum>x\<in>S. g x * log b (g x / f x)) \<le> log b (\<Sum>x\<in>S. f x)"
75 proof -
76   have log_mono: "\<And>x y. 0 < x \<Longrightarrow> x \<le> y \<Longrightarrow> log b x \<le> log b y"
77     using `1 < b` by (subst log_le_cancel_iff) auto
79   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))"
80   proof (unfold setsum_negf[symmetric], rule setsum_cong)
81     fix x assume x: "x \<in> S"
82     show "- (g x * log b (g x / f x)) = g x * log b (f x / g x)"
83     proof (cases "g x = 0")
84       case False
85       with pos[OF x] g_pos[OF x] have "0 < f x" "0 < g x" by simp_all
86       thus ?thesis using `1 < b` by (simp add: log_divide field_simps)
87     qed simp
88   qed rule
89   also have "... \<le> log b (\<Sum>x\<in>S. g x * (f x / g x))"
90   proof (rule log_setsum')
91     fix x assume x: "x \<in> S" "0 < g x"
92     with g_pos[OF x] show "0 < f x / g x" by (safe intro!: divide_pos_pos)
93   qed fact+
94   also have "... = log b (\<Sum>x\<in>S - {x. g x = 0}. f x)" using `finite S`
95     by (auto intro!: setsum_mono_zero_cong_right arg_cong[where f="log b"]
96         split: split_if_asm)
97   also have "... \<le> log b (\<Sum>x\<in>S. f x)"
98   proof (rule log_mono)
99     have "0 = (\<Sum>x\<in>S - {x. g x = 0}. 0)" by simp
100     also have "... < (\<Sum>x\<in>S - {x. g x = 0}. f x)" (is "_ < ?sum")
101     proof (rule setsum_strict_mono)
102       show "finite (S - {x. g x = 0})" using `finite S` by simp
103       show "S - {x. g x = 0} \<noteq> {}"
104       proof
105         assume "S - {x. g x = 0} = {}"
106         hence "(\<Sum>x\<in>S. g x) = 0" by (subst setsum_0') auto
107         with `(\<Sum>x\<in>S. g x) = 1` show False by simp
108       qed
109       fix x assume "x \<in> S - {x. g x = 0}"
110       thus "0 < f x" using g_pos[of x] pos(1)[of x] by auto
111     qed
112     finally show "0 < ?sum" .
113     show "(\<Sum>x\<in>S - {x. g x = 0}. f x) \<le> (\<Sum>x\<in>S. f x)"
114       using `finite S` pos by (auto intro!: setsum_mono2)
115   qed
116   finally show ?thesis .
117 qed
119 lemma split_pairs:
120   "((A, B) = X) \<longleftrightarrow> (fst X = A \<and> snd X = B)" and
121   "(X = (A, B)) \<longleftrightarrow> (fst X = A \<and> snd X = B)" by auto
123 section "Information theory"
125 locale information_space = prob_space +
126   fixes b :: real assumes b_gt_1: "1 < b"
128 context information_space
129 begin
131 text {* Introduce some simplification rules for logarithm of base @{term b}. *}
133 lemma log_neg_const:
134   assumes "x \<le> 0"
135   shows "log b x = log b 0"
136 proof -
137   { fix u :: real
138     have "x \<le> 0" by fact
139     also have "0 < exp u"
140       using exp_gt_zero .
141     finally have "exp u \<noteq> x"
142       by auto }
143   then show "log b x = log b 0"
144     by (simp add: log_def ln_def)
145 qed
147 lemma log_mult_eq:
148   "log b (A * B) = (if 0 < A * B then log b \<bar>A\<bar> + log b \<bar>B\<bar> else log b 0)"
149   using log_mult[of b "\<bar>A\<bar>" "\<bar>B\<bar>"] b_gt_1 log_neg_const[of "A * B"]
150   by (auto simp: zero_less_mult_iff mult_le_0_iff)
152 lemma log_inverse_eq:
153   "log b (inverse B) = (if 0 < B then - log b B else log b 0)"
154   using log_inverse[of b B] log_neg_const[of "inverse B"] b_gt_1 by simp
156 lemma log_divide_eq:
157   "log b (A / B) = (if 0 < A * B then log b \<bar>A\<bar> - log b \<bar>B\<bar> else log b 0)"
158   unfolding divide_inverse log_mult_eq log_inverse_eq abs_inverse
159   by (auto simp: zero_less_mult_iff mult_le_0_iff)
161 lemmas log_simps = log_mult_eq log_inverse_eq log_divide_eq
163 end
165 subsection "Kullback\$-\$Leibler divergence"
167 text {* The Kullback\$-\$Leibler divergence is also known as relative entropy or
168 Kullback\$-\$Leibler distance. *}
170 definition
171   "KL_divergence b M \<nu> = \<integral>x. log b (real (RN_deriv M \<nu> x)) \<partial>M\<lparr>measure := \<nu>\<rparr>"
173 lemma (in sigma_finite_measure) KL_divergence_vimage:
174   assumes T: "T \<in> measure_preserving M M'"
175     and T': "T' \<in> measure_preserving (M'\<lparr> measure := \<nu>' \<rparr>) (M\<lparr> measure := \<nu> \<rparr>)"
176     and inv: "\<And>x. x \<in> space M \<Longrightarrow> T' (T x) = x"
177     and inv': "\<And>x. x \<in> space M' \<Longrightarrow> T (T' x) = x"
178   and \<nu>': "measure_space (M'\<lparr>measure := \<nu>'\<rparr>)" "measure_space.absolutely_continuous M' \<nu>'"
179   and "1 < b"
180   shows "KL_divergence b M' \<nu>' = KL_divergence b M \<nu>"
181 proof -
182   interpret \<nu>': measure_space "M'\<lparr>measure := \<nu>'\<rparr>" by fact
183   have M: "measure_space (M\<lparr> measure := \<nu>\<rparr>)"
184     by (rule \<nu>'.measure_space_vimage[OF _ T'], simp) default
185   have "sigma_algebra (M'\<lparr> measure := \<nu>'\<rparr>)" by default
186   then have saM': "sigma_algebra M'" by simp
187   then interpret M': measure_space M' by (rule measure_space_vimage) fact
188   have ac: "absolutely_continuous \<nu>" unfolding absolutely_continuous_def
189   proof safe
190     fix N assume N: "N \<in> sets M" and N_0: "\<mu> N = 0"
191     then have N': "T' -` N \<inter> space M' \<in> sets M'"
192       using T' by (auto simp: measurable_def measure_preserving_def)
193     have "T -` (T' -` N \<inter> space M') \<inter> space M = N"
194       using inv T N sets_into_space[OF N] by (auto simp: measurable_def measure_preserving_def)
195     then have "measure M' (T' -` N \<inter> space M') = 0"
196       using measure_preservingD[OF T N'] N_0 by auto
197     with \<nu>'(2) N' show "\<nu> N = 0" using measure_preservingD[OF T', of N] N
198       unfolding M'.absolutely_continuous_def measurable_def by auto
199   qed
201   have sa: "sigma_algebra (M\<lparr>measure := \<nu>\<rparr>)" by simp default
202   have AE: "AE x. RN_deriv M' \<nu>' (T x) = RN_deriv M \<nu> x"
203     by (rule RN_deriv_vimage[OF T T' inv \<nu>'])
204   show ?thesis
205     unfolding KL_divergence_def
206   proof (subst \<nu>'.integral_vimage[OF sa T'])
207     show "(\<lambda>x. log b (real (RN_deriv M \<nu> x))) \<in> borel_measurable (M\<lparr>measure := \<nu>\<rparr>)"
208       by (auto intro!: RN_deriv[OF M ac] borel_measurable_log[OF _ `1 < b`])
209     have "(\<integral> x. log b (real (RN_deriv M' \<nu>' x)) \<partial>M'\<lparr>measure := \<nu>'\<rparr>) =
210       (\<integral> x. log b (real (RN_deriv M' \<nu>' (T (T' x)))) \<partial>M'\<lparr>measure := \<nu>'\<rparr>)" (is "?l = _")
211       using inv' by (auto intro!: \<nu>'.integral_cong)
212     also have "\<dots> = (\<integral> x. log b (real (RN_deriv M \<nu> (T' x))) \<partial>M'\<lparr>measure := \<nu>'\<rparr>)" (is "_ = ?r")
213       using M ac AE
214       by (intro \<nu>'.integral_cong_AE \<nu>'.almost_everywhere_vimage[OF sa T'] absolutely_continuous_AE[OF M])
215          (auto elim!: AE_mp)
216     finally show "?l = ?r" .
217   qed
218 qed
220 lemma (in sigma_finite_measure) KL_divergence_cong:
221   assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?\<nu>")
222   assumes [simp]: "sets N = sets M" "space N = space M"
223     "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A"
224     "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = \<nu>' A"
225   shows "KL_divergence b M \<nu> = KL_divergence b N \<nu>'"
226 proof -
227   interpret \<nu>: measure_space ?\<nu> by fact
228   have "KL_divergence b M \<nu> = \<integral>x. log b (real (RN_deriv N \<nu>' x)) \<partial>?\<nu>"
229     by (simp cong: RN_deriv_cong \<nu>.integral_cong add: KL_divergence_def)
230   also have "\<dots> = KL_divergence b N \<nu>'"
231     by (auto intro!: \<nu>.integral_cong_measure[symmetric] simp: KL_divergence_def)
232   finally show ?thesis .
233 qed
235 lemma (in finite_measure_space) KL_divergence_eq_finite:
236   assumes v: "finite_measure_space (M\<lparr>measure := \<nu>\<rparr>)"
237   assumes ac: "absolutely_continuous \<nu>"
238   shows "KL_divergence b M \<nu> = (\<Sum>x\<in>space M. real (\<nu> {x}) * log b (real (\<nu> {x}) / real (\<mu> {x})))" (is "_ = ?sum")
239 proof (simp add: KL_divergence_def finite_measure_space.integral_finite_singleton[OF v])
240   interpret v: finite_measure_space "M\<lparr>measure := \<nu>\<rparr>" by fact
241   have ms: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
242   show "(\<Sum>x \<in> space M. log b (real (RN_deriv M \<nu> x)) * real (\<nu> {x})) = ?sum"
243     using RN_deriv_finite_measure[OF ms ac]
244     by (auto intro!: setsum_cong simp: field_simps)
245 qed
247 lemma (in finite_prob_space) KL_divergence_positive_finite:
248   assumes v: "finite_prob_space (M\<lparr>measure := \<nu>\<rparr>)"
249   assumes ac: "absolutely_continuous \<nu>"
250   and "1 < b"
251   shows "0 \<le> KL_divergence b M \<nu>"
252 proof -
253   interpret v: finite_prob_space "M\<lparr>measure := \<nu>\<rparr>" by fact
254   have ms: "finite_measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
256   have "- (KL_divergence b M \<nu>) \<le> log b (\<Sum>x\<in>space M. real (\<mu> {x}))"
257   proof (subst KL_divergence_eq_finite[OF ms ac], safe intro!: log_setsum_divide not_empty)
258     show "finite (space M)" using finite_space by simp
259     show "1 < b" by fact
260     show "(\<Sum>x\<in>space M. real (\<nu> {x})) = 1"
261       using v.finite_sum_over_space_eq_1 by (simp add: v.\<mu>'_def)
263     fix x assume "x \<in> space M"
264     then have x: "{x} \<in> sets M" unfolding sets_eq_Pow by auto
265     { assume "0 < real (\<nu> {x})"
266       then have "\<nu> {x} \<noteq> 0" by auto
267       then have "\<mu> {x} \<noteq> 0"
268         using ac[unfolded absolutely_continuous_def, THEN bspec, of "{x}"] x by auto
269       thus "0 < real (\<mu> {x})" using real_measure[OF x] by auto }
270     show "0 \<le> real (\<mu> {x})" "0 \<le> real (\<nu> {x})"
271       using real_measure[OF x] v.real_measure[of "{x}"] x by auto
272   qed
273   thus "0 \<le> KL_divergence b M \<nu>" using finite_sum_over_space_eq_1 by (simp add: \<mu>'_def)
274 qed
276 subsection {* Mutual Information *}
278 definition (in prob_space)
279   "mutual_information b S T X Y =
280     KL_divergence b (S\<lparr>measure := extreal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := extreal\<circ>distribution Y\<rparr>)
281       (extreal\<circ>joint_distribution X Y)"
283 definition (in prob_space)
284   "entropy b s X = mutual_information b s s X X"
286 abbreviation (in information_space)
287   mutual_information_Pow ("\<I>'(_ ; _')") where
288   "\<I>(X ; Y) \<equiv> mutual_information b
289     \<lparr> space = X`space M, sets = Pow (X`space M), measure = extreal\<circ>distribution X \<rparr>
290     \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = extreal\<circ>distribution Y \<rparr> X Y"
292 lemma (in prob_space) finite_variables_absolutely_continuous:
293   assumes X: "finite_random_variable S X" and Y: "finite_random_variable T Y"
294   shows "measure_space.absolutely_continuous
295     (S\<lparr>measure := extreal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := extreal\<circ>distribution Y\<rparr>)
296     (extreal\<circ>joint_distribution X Y)"
297 proof -
298   interpret X: finite_prob_space "S\<lparr>measure := extreal\<circ>distribution X\<rparr>"
299     using X by (rule distribution_finite_prob_space)
300   interpret Y: finite_prob_space "T\<lparr>measure := extreal\<circ>distribution Y\<rparr>"
301     using Y by (rule distribution_finite_prob_space)
302   interpret XY: pair_finite_prob_space
303     "S\<lparr>measure := extreal\<circ>distribution X\<rparr>" "T\<lparr> measure := extreal\<circ>distribution Y\<rparr>" by default
304   interpret P: finite_prob_space "XY.P\<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>"
305     using assms by (auto intro!: joint_distribution_finite_prob_space)
306   note rv = assms[THEN finite_random_variableD]
307   show "XY.absolutely_continuous (extreal\<circ>joint_distribution X Y)"
308   proof (rule XY.absolutely_continuousI)
309     show "finite_measure_space (XY.P\<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>)" by default
310     fix x assume "x \<in> space XY.P" and "XY.\<mu> {x} = 0"
311     then obtain a b where "x = (a, b)"
312       and "distribution X {a} = 0 \<or> distribution Y {b} = 0"
313       by (cases x) (auto simp: space_pair_measure)
314     with finite_distribution_order(5,6)[OF X Y]
315     show "(extreal \<circ> joint_distribution X Y) {x} = 0" by auto
316   qed
317 qed
319 lemma (in information_space)
320   assumes MX: "finite_random_variable MX X"
321   assumes MY: "finite_random_variable MY Y"
322   shows mutual_information_generic_eq:
323     "mutual_information b MX MY X Y = (\<Sum> (x,y) \<in> space MX \<times> space MY.
324       joint_distribution X Y {(x,y)} *
325       log b (joint_distribution X Y {(x,y)} /
326       (distribution X {x} * distribution Y {y})))"
327     (is ?sum)
328   and mutual_information_positive_generic:
329      "0 \<le> mutual_information b MX MY X Y" (is ?positive)
330 proof -
331   interpret X: finite_prob_space "MX\<lparr>measure := extreal\<circ>distribution X\<rparr>"
332     using MX by (rule distribution_finite_prob_space)
333   interpret Y: finite_prob_space "MY\<lparr>measure := extreal\<circ>distribution Y\<rparr>"
334     using MY by (rule distribution_finite_prob_space)
335   interpret XY: pair_finite_prob_space "MX\<lparr>measure := extreal\<circ>distribution X\<rparr>" "MY\<lparr>measure := extreal\<circ>distribution Y\<rparr>" by default
336   interpret P: finite_prob_space "XY.P\<lparr>measure := extreal\<circ>joint_distribution X Y\<rparr>"
337     using assms by (auto intro!: joint_distribution_finite_prob_space)
339   have P_ms: "finite_measure_space (XY.P\<lparr>measure := extreal\<circ>joint_distribution X Y\<rparr>)" by default
340   have P_ps: "finite_prob_space (XY.P\<lparr>measure := extreal\<circ>joint_distribution X Y\<rparr>)" by default
342   show ?sum
343     unfolding Let_def mutual_information_def
344     by (subst XY.KL_divergence_eq_finite[OF P_ms finite_variables_absolutely_continuous[OF MX MY]])
345        (auto simp add: space_pair_measure setsum_cartesian_product')
347   show ?positive
348     using XY.KL_divergence_positive_finite[OF P_ps finite_variables_absolutely_continuous[OF MX MY] b_gt_1]
349     unfolding mutual_information_def .
350 qed
352 lemma (in information_space) mutual_information_commute_generic:
353   assumes X: "random_variable S X" and Y: "random_variable T Y"
354   assumes ac: "measure_space.absolutely_continuous
355     (S\<lparr>measure := extreal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := extreal\<circ>distribution Y\<rparr>) (extreal\<circ>joint_distribution X Y)"
356   shows "mutual_information b S T X Y = mutual_information b T S Y X"
357 proof -
358   let ?S = "S\<lparr>measure := extreal\<circ>distribution X\<rparr>" and ?T = "T\<lparr>measure := extreal\<circ>distribution Y\<rparr>"
359   interpret S: prob_space ?S using X by (rule distribution_prob_space)
360   interpret T: prob_space ?T using Y by (rule distribution_prob_space)
361   interpret P: pair_prob_space ?S ?T ..
362   interpret Q: pair_prob_space ?T ?S ..
363   show ?thesis
364     unfolding mutual_information_def
365   proof (intro Q.KL_divergence_vimage[OF Q.measure_preserving_swap _ _ _ _ ac b_gt_1])
366     show "(\<lambda>(x,y). (y,x)) \<in> measure_preserving
367       (P.P \<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>) (Q.P \<lparr> measure := extreal\<circ>joint_distribution Y X\<rparr>)"
368       using X Y unfolding measurable_def
369       unfolding measure_preserving_def using P.pair_sigma_algebra_swap_measurable
370       by (auto simp add: space_pair_measure distribution_def intro!: arg_cong[where f=\<mu>'])
371     have "prob_space (P.P\<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>)"
372       using X Y by (auto intro!: distribution_prob_space random_variable_pairI)
373     then show "measure_space (P.P\<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>)"
374       unfolding prob_space_def by simp
375   qed auto
376 qed
378 lemma (in information_space) mutual_information_commute:
379   assumes X: "finite_random_variable S X" and Y: "finite_random_variable T Y"
380   shows "mutual_information b S T X Y = mutual_information b T S Y X"
381   unfolding mutual_information_generic_eq[OF X Y] mutual_information_generic_eq[OF Y X]
382   unfolding joint_distribution_commute_singleton[of X Y]
383   by (auto simp add: ac_simps intro!: setsum_reindex_cong[OF swap_inj_on])
385 lemma (in information_space) mutual_information_commute_simple:
386   assumes X: "simple_function M X" and Y: "simple_function M Y"
387   shows "\<I>(X;Y) = \<I>(Y;X)"
388   by (intro mutual_information_commute X Y simple_function_imp_finite_random_variable)
390 lemma (in information_space) mutual_information_eq:
391   assumes "simple_function M X" "simple_function M Y"
392   shows "\<I>(X;Y) = (\<Sum> (x,y) \<in> X ` space M \<times> Y ` space M.
393     distribution (\<lambda>x. (X x, Y x)) {(x,y)} * log b (distribution (\<lambda>x. (X x, Y x)) {(x,y)} /
394                                                    (distribution X {x} * distribution Y {y})))"
395   using assms by (simp add: mutual_information_generic_eq)
397 lemma (in information_space) mutual_information_generic_cong:
398   assumes X: "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x"
399   assumes Y: "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x"
400   shows "mutual_information b MX MY X Y = mutual_information b MX MY X' Y'"
401   unfolding mutual_information_def using X Y
402   by (simp cong: distribution_cong)
404 lemma (in information_space) mutual_information_cong:
405   assumes X: "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x"
406   assumes Y: "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x"
407   shows "\<I>(X; Y) = \<I>(X'; Y')"
408   unfolding mutual_information_def using X Y
409   by (simp cong: distribution_cong image_cong)
411 lemma (in information_space) mutual_information_positive:
412   assumes "simple_function M X" "simple_function M Y"
413   shows "0 \<le> \<I>(X;Y)"
414   using assms by (simp add: mutual_information_positive_generic)
416 subsection {* Entropy *}
418 abbreviation (in information_space)
419   entropy_Pow ("\<H>'(_')") where
420   "\<H>(X) \<equiv> entropy b \<lparr> space = X`space M, sets = Pow (X`space M), measure = extreal\<circ>distribution X \<rparr> X"
422 lemma (in information_space) entropy_generic_eq:
423   fixes X :: "'a \<Rightarrow> 'c"
424   assumes MX: "finite_random_variable MX X"
425   shows "entropy b MX X = -(\<Sum> x \<in> space MX. distribution X {x} * log b (distribution X {x}))"
426 proof -
427   interpret MX: finite_prob_space "MX\<lparr>measure := extreal\<circ>distribution X\<rparr>"
428     using MX by (rule distribution_finite_prob_space)
429   let "?X x" = "distribution X {x}"
430   let "?XX x y" = "joint_distribution X X {(x, y)}"
432   { fix x y :: 'c
433     { assume "x \<noteq> y"
434       then have "(\<lambda>x. (X x, X x)) -` {(x,y)} \<inter> space M = {}" by auto
435       then have "joint_distribution X X {(x, y)} = 0" by (simp add: distribution_def) }
436     then have "?XX x y * log b (?XX x y / (?X x * ?X y)) =
437         (if x = y then - ?X y * log b (?X y) else 0)"
438       by (auto simp: log_simps zero_less_mult_iff) }
439   note remove_XX = this
441   show ?thesis
442     unfolding entropy_def mutual_information_generic_eq[OF MX MX]
443     unfolding setsum_cartesian_product[symmetric] setsum_negf[symmetric] remove_XX
444     using MX.finite_space by (auto simp: setsum_cases)
445 qed
447 lemma (in information_space) entropy_eq:
448   assumes "simple_function M X"
449   shows "\<H>(X) = -(\<Sum> x \<in> X ` space M. distribution X {x} * log b (distribution X {x}))"
450   using assms by (simp add: entropy_generic_eq)
452 lemma (in information_space) entropy_positive:
453   "simple_function M X \<Longrightarrow> 0 \<le> \<H>(X)"
454   unfolding entropy_def by (simp add: mutual_information_positive)
456 lemma (in information_space) entropy_certainty_eq_0:
457   assumes X: "simple_function M X" and "x \<in> X ` space M" and "distribution X {x} = 1"
458   shows "\<H>(X) = 0"
459 proof -
460   let ?X = "\<lparr> space = X ` space M, sets = Pow (X ` space M), measure = extreal\<circ>distribution X\<rparr>"
461   note simple_function_imp_finite_random_variable[OF `simple_function M X`]
462   from distribution_finite_prob_space[OF this, of "\<lparr> measure = extreal\<circ>distribution X \<rparr>"]
463   interpret X: finite_prob_space ?X by simp
464   have "distribution X (X ` space M - {x}) = distribution X (X ` space M) - distribution X {x}"
465     using X.measure_compl[of "{x}"] assms by auto
466   also have "\<dots> = 0" using X.prob_space assms by auto
467   finally have X0: "distribution X (X ` space M - {x}) = 0" by auto
468   { fix y assume *: "y \<in> X ` space M"
469     { assume asm: "y \<noteq> x"
470       with * have "{y} \<subseteq> X ` space M - {x}" by auto
471       from X.measure_mono[OF this] X0 asm *
472       have "distribution X {y} = 0"  by (auto intro: antisym) }
473     then have "distribution X {y} = (if x = y then 1 else 0)"
474       using assms by auto }
475   note fi = this
476   have y: "\<And>y. (if x = y then 1 else 0) * log b (if x = y then 1 else 0) = 0" by simp
477   show ?thesis unfolding entropy_eq[OF `simple_function M X`] by (auto simp: y fi)
478 qed
480 lemma (in information_space) entropy_le_card_not_0:
481   assumes X: "simple_function M X"
482   shows "\<H>(X) \<le> log b (card (X ` space M \<inter> {x. distribution X {x} \<noteq> 0}))"
483 proof -
484   let "?p x" = "distribution X {x}"
485   have "\<H>(X) = (\<Sum>x\<in>X`space M. ?p x * log b (1 / ?p x))"
486     unfolding entropy_eq[OF X] setsum_negf[symmetric]
487     by (auto intro!: setsum_cong simp: log_simps)
488   also have "\<dots> \<le> log b (\<Sum>x\<in>X`space M. ?p x * (1 / ?p x))"
489     using not_empty b_gt_1 `simple_function M X` sum_over_space_real_distribution[OF X]
490     by (intro log_setsum') (auto simp: simple_function_def)
491   also have "\<dots> = log b (\<Sum>x\<in>X`space M. if ?p x \<noteq> 0 then 1 else 0)"
492     by (intro arg_cong[where f="\<lambda>X. log b X"] setsum_cong) auto
493   finally show ?thesis
494     using `simple_function M X` by (auto simp: setsum_cases real_eq_of_nat simple_function_def)
495 qed
497 lemma (in prob_space) measure'_translate:
498   assumes X: "random_variable S X" and A: "A \<in> sets S"
499   shows "finite_measure.\<mu>' (S\<lparr> measure := extreal\<circ>distribution X \<rparr>) A = distribution X A"
500 proof -
501   interpret S: prob_space "S\<lparr> measure := extreal\<circ>distribution X \<rparr>"
502     using distribution_prob_space[OF X] .
503   from A show "S.\<mu>' A = distribution X A"
504     unfolding S.\<mu>'_def by (simp add: distribution_def_raw \<mu>'_def)
505 qed
507 lemma (in information_space) entropy_uniform_max:
508   assumes X: "simple_function M X"
509   assumes "\<And>x y. \<lbrakk> x \<in> X ` space M ; y \<in> X ` space M \<rbrakk> \<Longrightarrow> distribution X {x} = distribution X {y}"
510   shows "\<H>(X) = log b (real (card (X ` space M)))"
511 proof -
512   let ?X = "\<lparr> space = X ` space M, sets = Pow (X ` space M), measure = undefined\<rparr>\<lparr> measure := extreal\<circ>distribution X\<rparr>"
513   note frv = simple_function_imp_finite_random_variable[OF X]
514   from distribution_finite_prob_space[OF this, of "\<lparr> measure = extreal\<circ>distribution X \<rparr>"]
515   interpret X: finite_prob_space ?X by simp
516   note rv = finite_random_variableD[OF frv]
517   have card_gt0: "0 < card (X ` space M)" unfolding card_gt_0_iff
518     using `simple_function M X` not_empty by (auto simp: simple_function_def)
519   { fix x assume "x \<in> space ?X"
520     moreover then have "X.\<mu>' {x} = 1 / card (space ?X)"
521     proof (rule X.uniform_prob)
522       fix x y assume "x \<in> space ?X" "y \<in> space ?X"
523       with assms(2)[of x y] show "X.\<mu>' {x} = X.\<mu>' {y}"
524         by (subst (1 2) measure'_translate[OF rv]) auto
525     qed
526     ultimately have "distribution X {x} = 1 / card (space ?X)"
527       by (subst (asm) measure'_translate[OF rv]) auto }
528   thus ?thesis
529     using not_empty X.finite_space b_gt_1 card_gt0
530     by (simp add: entropy_eq[OF `simple_function M X`] real_eq_of_nat[symmetric] log_simps)
531 qed
533 lemma (in information_space) entropy_le_card:
534   assumes "simple_function M X"
535   shows "\<H>(X) \<le> log b (real (card (X ` space M)))"
536 proof cases
537   assume "X ` space M \<inter> {x. distribution X {x} \<noteq> 0} = {}"
538   then have "\<And>x. x\<in>X`space M \<Longrightarrow> distribution X {x} = 0" by auto
539   moreover
540   have "0 < card (X`space M)"
541     using `simple_function M X` not_empty
542     by (auto simp: card_gt_0_iff simple_function_def)
543   then have "log b 1 \<le> log b (real (card (X`space M)))"
544     using b_gt_1 by (intro log_le) auto
545   ultimately show ?thesis using assms by (simp add: entropy_eq)
546 next
547   assume False: "X ` space M \<inter> {x. distribution X {x} \<noteq> 0} \<noteq> {}"
548   have "card (X ` space M \<inter> {x. distribution X {x} \<noteq> 0}) \<le> card (X ` space M)"
549     (is "?A \<le> ?B") using assms not_empty by (auto intro!: card_mono simp: simple_function_def)
550   note entropy_le_card_not_0[OF assms]
551   also have "log b (real ?A) \<le> log b (real ?B)"
552     using b_gt_1 False not_empty `?A \<le> ?B` assms
553     by (auto intro!: log_le simp: card_gt_0_iff simp: simple_function_def)
554   finally show ?thesis .
555 qed
557 lemma (in information_space) entropy_commute:
558   assumes "simple_function M X" "simple_function M Y"
559   shows "\<H>(\<lambda>x. (X x, Y x)) = \<H>(\<lambda>x. (Y x, X x))"
560 proof -
561   have sf: "simple_function M (\<lambda>x. (X x, Y x))" "simple_function M (\<lambda>x. (Y x, X x))"
562     using assms by (auto intro: simple_function_Pair)
563   have *: "(\<lambda>x. (Y x, X x))`space M = (\<lambda>(a,b). (b,a))`(\<lambda>x. (X x, Y x))`space M"
564     by auto
565   have inj: "\<And>X. inj_on (\<lambda>(a,b). (b,a)) X"
566     by (auto intro!: inj_onI)
567   show ?thesis
568     unfolding sf[THEN entropy_eq] unfolding * setsum_reindex[OF inj]
569     by (simp add: joint_distribution_commute[of Y X] split_beta)
570 qed
572 lemma (in information_space) entropy_eq_cartesian_product:
573   assumes "simple_function M X" "simple_function M Y"
574   shows "\<H>(\<lambda>x. (X x, Y x)) = -(\<Sum>x\<in>X`space M. \<Sum>y\<in>Y`space M.
575     joint_distribution X Y {(x,y)} * log b (joint_distribution X Y {(x,y)}))"
576 proof -
577   have sf: "simple_function M (\<lambda>x. (X x, Y x))"
578     using assms by (auto intro: simple_function_Pair)
579   { fix x assume "x\<notin>(\<lambda>x. (X x, Y x))`space M"
580     then have "(\<lambda>x. (X x, Y x)) -` {x} \<inter> space M = {}" by auto
581     then have "joint_distribution X Y {x} = 0"
582       unfolding distribution_def by auto }
583   then show ?thesis using sf assms
584     unfolding entropy_eq[OF sf] neg_equal_iff_equal setsum_cartesian_product
585     by (auto intro!: setsum_mono_zero_cong_left simp: simple_function_def)
586 qed
588 subsection {* Conditional Mutual Information *}
590 definition (in prob_space)
591   "conditional_mutual_information b MX MY MZ X Y Z \<equiv>
592     mutual_information b MX (MY \<Otimes>\<^isub>M MZ) X (\<lambda>x. (Y x, Z x)) -
593     mutual_information b MX MZ X Z"
595 abbreviation (in information_space)
596   conditional_mutual_information_Pow ("\<I>'( _ ; _ | _ ')") where
597   "\<I>(X ; Y | Z) \<equiv> conditional_mutual_information b
598     \<lparr> space = X`space M, sets = Pow (X`space M), measure = extreal\<circ>distribution X \<rparr>
599     \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = extreal\<circ>distribution Y \<rparr>
600     \<lparr> space = Z`space M, sets = Pow (Z`space M), measure = extreal\<circ>distribution Z \<rparr>
601     X Y Z"
603 lemma (in information_space) conditional_mutual_information_generic_eq:
604   assumes MX: "finite_random_variable MX X"
605     and MY: "finite_random_variable MY Y"
606     and MZ: "finite_random_variable MZ Z"
607   shows "conditional_mutual_information b MX MY MZ X Y Z = (\<Sum>(x, y, z) \<in> space MX \<times> space MY \<times> space MZ.
608              distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} *
609              log b (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} /
610     (joint_distribution X Z {(x, z)} * (joint_distribution Y Z {(y,z)} / distribution Z {z}))))"
611   (is "_ = (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?XZ x z * (?YZ y z / ?Z z))))")
612 proof -
613   let ?X = "\<lambda>x. distribution X {x}"
614   note finite_var = MX MY MZ
615   note YZ = finite_random_variable_pairI[OF finite_var(2,3)]
616   note XYZ = finite_random_variable_pairI[OF MX YZ]
617   note XZ = finite_random_variable_pairI[OF finite_var(1,3)]
618   note ZX = finite_random_variable_pairI[OF finite_var(3,1)]
619   note YZX = finite_random_variable_pairI[OF finite_var(2) ZX]
620   note order1 =
621     finite_distribution_order(5,6)[OF finite_var(1) YZ]
622     finite_distribution_order(5,6)[OF finite_var(1,3)]
624   note random_var = finite_var[THEN finite_random_variableD]
625   note finite = finite_var(1) YZ finite_var(3) XZ YZX
627   have order2: "\<And>x y z. \<lbrakk>x \<in> space MX; y \<in> space MY; z \<in> space MZ; joint_distribution X Z {(x, z)} = 0\<rbrakk>
628           \<Longrightarrow> joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} = 0"
629     unfolding joint_distribution_commute_singleton[of X]
630     unfolding joint_distribution_assoc_singleton[symmetric]
631     using finite_distribution_order(6)[OF finite_var(2) ZX]
632     by auto
634   have "(\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?XZ x z * (?YZ y z / ?Z z)))) =
635     (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * (log b (?XYZ x y z / (?X x * ?YZ y z)) - log b (?XZ x z / (?X x * ?Z z))))"
636     (is "(\<Sum>(x, y, z)\<in>?S. ?L x y z) = (\<Sum>(x, y, z)\<in>?S. ?R x y z)")
637   proof (safe intro!: setsum_cong)
638     fix x y z assume space: "x \<in> space MX" "y \<in> space MY" "z \<in> space MZ"
639     show "?L x y z = ?R x y z"
640     proof cases
641       assume "?XYZ x y z \<noteq> 0"
642       with space have "0 < ?X x" "0 < ?Z z" "0 < ?XZ x z" "0 < ?YZ y z" "0 < ?XYZ x y z"
643         using order1 order2 by (auto simp: less_le)
644       with b_gt_1 show ?thesis
645         by (simp add: log_mult log_divide zero_less_mult_iff zero_less_divide_iff)
646     qed simp
647   qed
648   also have "\<dots> = (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?X x * ?YZ y z))) -
649                   (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XZ x z / (?X x * ?Z z)))"
650     by (auto simp add: setsum_subtractf[symmetric] field_simps intro!: setsum_cong)
651   also have "(\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XZ x z / (?X x * ?Z z))) =
652              (\<Sum>(x, z)\<in>space MX \<times> space MZ. ?XZ x z * log b (?XZ x z / (?X x * ?Z z)))"
653     unfolding setsum_cartesian_product[symmetric] setsum_commute[of _ _ "space MY"]
654               setsum_left_distrib[symmetric]
655     unfolding joint_distribution_commute_singleton[of X]
656     unfolding joint_distribution_assoc_singleton[symmetric]
657     using setsum_joint_distribution_singleton[OF finite_var(2) ZX]
658     by (intro setsum_cong refl) (simp add: space_pair_measure)
659   also have "(\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?X x * ?YZ y z))) -
660              (\<Sum>(x, z)\<in>space MX \<times> space MZ. ?XZ x z * log b (?XZ x z / (?X x * ?Z z))) =
661              conditional_mutual_information b MX MY MZ X Y Z"
662     unfolding conditional_mutual_information_def
663     unfolding mutual_information_generic_eq[OF finite_var(1,3)]
664     unfolding mutual_information_generic_eq[OF finite_var(1) YZ]
665     by (simp add: space_sigma space_pair_measure setsum_cartesian_product')
666   finally show ?thesis by simp
667 qed
669 lemma (in information_space) conditional_mutual_information_eq:
670   assumes "simple_function M X" "simple_function M Y" "simple_function M Z"
671   shows "\<I>(X;Y|Z) = (\<Sum>(x, y, z) \<in> X`space M \<times> Y`space M \<times> Z`space M.
672              distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} *
673              log b (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} /
674     (joint_distribution X Z {(x, z)} * joint_distribution Y Z {(y,z)} / distribution Z {z})))"
675   by (subst conditional_mutual_information_generic_eq[OF assms[THEN simple_function_imp_finite_random_variable]])
676      simp
678 lemma (in information_space) conditional_mutual_information_eq_mutual_information:
679   assumes X: "simple_function M X" and Y: "simple_function M Y"
680   shows "\<I>(X ; Y) = \<I>(X ; Y | (\<lambda>x. ()))"
681 proof -
682   have [simp]: "(\<lambda>x. ()) ` space M = {()}" using not_empty by auto
683   have C: "simple_function M (\<lambda>x. ())" by auto
684   show ?thesis
685     unfolding conditional_mutual_information_eq[OF X Y C]
686     unfolding mutual_information_eq[OF X Y]
687     by (simp add: setsum_cartesian_product' distribution_remove_const)
688 qed
690 lemma (in prob_space) distribution_unit[simp]: "distribution (\<lambda>x. ()) {()} = 1"
691   unfolding distribution_def using prob_space by auto
693 lemma (in prob_space) joint_distribution_unit[simp]: "distribution (\<lambda>x. (X x, ())) {(a, ())} = distribution X {a}"
694   unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
696 lemma (in prob_space) setsum_distribution:
697   assumes X: "finite_random_variable MX X" shows "(\<Sum>a\<in>space MX. distribution X {a}) = 1"
698   using setsum_joint_distribution[OF assms, of "\<lparr> space = UNIV, sets = Pow UNIV \<rparr>" "\<lambda>x. ()" "{()}"]
699   using sigma_algebra_Pow[of "UNIV::unit set" "()"] by simp
701 lemma (in prob_space) setsum_real_distribution:
702   fixes MX :: "('c, 'd) measure_space_scheme"
703   assumes X: "finite_random_variable MX X" shows "(\<Sum>a\<in>space MX. distribution X {a}) = 1"
704   using setsum_joint_distribution[OF assms, of "\<lparr> space = UNIV, sets = Pow UNIV, measure = undefined \<rparr>" "\<lambda>x. ()" "{()}"]
705   using sigma_algebra_Pow[of "UNIV::unit set" "\<lparr> measure = undefined \<rparr>"]
706   by auto
708 lemma (in information_space) conditional_mutual_information_generic_positive:
709   assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y" and Z: "finite_random_variable MZ Z"
710   shows "0 \<le> conditional_mutual_information b MX MY MZ X Y Z"
711 proof (cases "space MX \<times> space MY \<times> space MZ = {}")
712   case True show ?thesis
713     unfolding conditional_mutual_information_generic_eq[OF assms] True
714     by simp
715 next
716   case False
717   let ?dXYZ = "distribution (\<lambda>x. (X x, Y x, Z x))"
718   let ?dXZ = "joint_distribution X Z"
719   let ?dYZ = "joint_distribution Y Z"
720   let ?dX = "distribution X"
721   let ?dZ = "distribution Z"
722   let ?M = "space MX \<times> space MY \<times> space MZ"
724   note YZ = finite_random_variable_pairI[OF Y Z]
725   note XZ = finite_random_variable_pairI[OF X Z]
726   note ZX = finite_random_variable_pairI[OF Z X]
727   note YZ = finite_random_variable_pairI[OF Y Z]
728   note XYZ = finite_random_variable_pairI[OF X YZ]
729   note finite = Z YZ XZ XYZ
730   have order: "\<And>x y z. \<lbrakk>x \<in> space MX; y \<in> space MY; z \<in> space MZ; joint_distribution X Z {(x, z)} = 0\<rbrakk>
731           \<Longrightarrow> joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} = 0"
732     unfolding joint_distribution_commute_singleton[of X]
733     unfolding joint_distribution_assoc_singleton[symmetric]
734     using finite_distribution_order(6)[OF Y ZX]
735     by auto
737   note order = order
738     finite_distribution_order(5,6)[OF X YZ]
739     finite_distribution_order(5,6)[OF Y Z]
741   have "- conditional_mutual_information b MX MY MZ X Y Z = - (\<Sum>(x, y, z) \<in> ?M. ?dXYZ {(x, y, z)} *
742     log b (?dXYZ {(x, y, z)} / (?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z})))"
743     unfolding conditional_mutual_information_generic_eq[OF assms] neg_equal_iff_equal by auto
744   also have "\<dots> \<le> log b (\<Sum>(x, y, z) \<in> ?M. ?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z})"
745     unfolding split_beta'
746   proof (rule log_setsum_divide)
747     show "?M \<noteq> {}" using False by simp
748     show "1 < b" using b_gt_1 .
750     show "finite ?M" using assms
751       unfolding finite_sigma_algebra_def finite_sigma_algebra_axioms_def by auto
753     show "(\<Sum>x\<in>?M. ?dXYZ {(fst x, fst (snd x), snd (snd x))}) = 1"
754       unfolding setsum_cartesian_product'
755       unfolding setsum_commute[of _ "space MY"]
756       unfolding setsum_commute[of _ "space MZ"]
757       by (simp_all add: space_pair_measure
758                         setsum_joint_distribution_singleton[OF X YZ]
759                         setsum_joint_distribution_singleton[OF Y Z]
760                         setsum_distribution[OF Z])
762     fix x assume "x \<in> ?M"
763     let ?x = "(fst x, fst (snd x), snd (snd x))"
765     show "0 \<le> ?dXYZ {?x}"
766       "0 \<le> ?dXZ {(fst x, snd (snd x))} * ?dYZ {(fst (snd x), snd (snd x))} / ?dZ {snd (snd x)}"
767      by (simp_all add: mult_nonneg_nonneg divide_nonneg_nonneg)
769     assume *: "0 < ?dXYZ {?x}"
770     with `x \<in> ?M` finite order show "0 < ?dXZ {(fst x, snd (snd x))} * ?dYZ {(fst (snd x), snd (snd x))} / ?dZ {snd (snd x)}"
771       by (cases x) (auto simp add: zero_le_mult_iff zero_le_divide_iff less_le)
772   qed
773   also have "(\<Sum>(x, y, z) \<in> ?M. ?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z}) = (\<Sum>z\<in>space MZ. ?dZ {z})"
774     apply (simp add: setsum_cartesian_product')
775     apply (subst setsum_commute)
776     apply (subst (2) setsum_commute)
777     by (auto simp: setsum_divide_distrib[symmetric] setsum_product[symmetric]
778                    setsum_joint_distribution_singleton[OF X Z]
779                    setsum_joint_distribution_singleton[OF Y Z]
780           intro!: setsum_cong)
781   also have "log b (\<Sum>z\<in>space MZ. ?dZ {z}) = 0"
782     unfolding setsum_real_distribution[OF Z] by simp
783   finally show ?thesis by simp
784 qed
786 lemma (in information_space) conditional_mutual_information_positive:
787   assumes "simple_function M X" and "simple_function M Y" and "simple_function M Z"
788   shows "0 \<le> \<I>(X;Y|Z)"
789   by (rule conditional_mutual_information_generic_positive[OF assms[THEN simple_function_imp_finite_random_variable]])
791 subsection {* Conditional Entropy *}
793 definition (in prob_space)
794   "conditional_entropy b S T X Y = conditional_mutual_information b S S T X X Y"
796 abbreviation (in information_space)
797   conditional_entropy_Pow ("\<H>'(_ | _')") where
798   "\<H>(X | Y) \<equiv> conditional_entropy b
799     \<lparr> space = X`space M, sets = Pow (X`space M), measure = extreal\<circ>distribution X \<rparr>
800     \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = extreal\<circ>distribution Y \<rparr> X Y"
802 lemma (in information_space) conditional_entropy_positive:
803   "simple_function M X \<Longrightarrow> simple_function M Y \<Longrightarrow> 0 \<le> \<H>(X | Y)"
804   unfolding conditional_entropy_def by (auto intro!: conditional_mutual_information_positive)
806 lemma (in information_space) conditional_entropy_generic_eq:
807   fixes MX :: "('c, 'd) measure_space_scheme" and MY :: "('e, 'f) measure_space_scheme"
808   assumes MX: "finite_random_variable MX X"
809   assumes MZ: "finite_random_variable MZ Z"
810   shows "conditional_entropy b MX MZ X Z =
811      - (\<Sum>(x, z)\<in>space MX \<times> space MZ.
812          joint_distribution X Z {(x, z)} * log b (joint_distribution X Z {(x, z)} / distribution Z {z}))"
813 proof -
814   interpret MX: finite_sigma_algebra MX using MX by simp
815   interpret MZ: finite_sigma_algebra MZ using MZ by simp
816   let "?XXZ x y z" = "joint_distribution X (\<lambda>x. (X x, Z x)) {(x, y, z)}"
817   let "?XZ x z" = "joint_distribution X Z {(x, z)}"
818   let "?Z z" = "distribution Z {z}"
819   let "?f x y z" = "log b (?XXZ x y z * ?Z z / (?XZ x z * ?XZ y z))"
820   { fix x z have "?XXZ x x z = ?XZ x z"
821       unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) }
822   note this[simp]
823   { fix x x' :: 'c and z assume "x' \<noteq> x"
824     then have "?XXZ x x' z = 0"
825       by (auto simp: distribution_def empty_measure'[symmetric]
826                simp del: empty_measure' intro!: arg_cong[where f=\<mu>']) }
827   note this[simp]
828   { fix x x' z assume *: "x \<in> space MX" "z \<in> space MZ"
829     then have "(\<Sum>x'\<in>space MX. ?XXZ x x' z * ?f x x' z)
830       = (\<Sum>x'\<in>space MX. if x = x' then ?XZ x z * ?f x x z else 0)"
831       by (auto intro!: setsum_cong)
832     also have "\<dots> = ?XZ x z * ?f x x z"
833       using `x \<in> space MX` by (simp add: setsum_cases[OF MX.finite_space])
834     also have "\<dots> = ?XZ x z * log b (?Z z / ?XZ x z)" by auto
835     also have "\<dots> = - ?XZ x z * log b (?XZ x z / ?Z z)"
836       using finite_distribution_order(6)[OF MX MZ]
837       by (auto simp: log_simps field_simps zero_less_mult_iff)
838     finally have "(\<Sum>x'\<in>space MX. ?XXZ x x' z * ?f x x' z) = - ?XZ x z * log b (?XZ x z / ?Z z)" . }
839   note * = this
840   show ?thesis
841     unfolding conditional_entropy_def
842     unfolding conditional_mutual_information_generic_eq[OF MX MX MZ]
843     by (auto simp: setsum_cartesian_product' setsum_negf[symmetric]
844                    setsum_commute[of _ "space MZ"] *
845              intro!: setsum_cong)
846 qed
848 lemma (in information_space) conditional_entropy_eq:
849   assumes "simple_function M X" "simple_function M Z"
850   shows "\<H>(X | Z) =
851      - (\<Sum>(x, z)\<in>X ` space M \<times> Z ` space M.
852          joint_distribution X Z {(x, z)} *
853          log b (joint_distribution X Z {(x, z)} / distribution Z {z}))"
854   by (subst conditional_entropy_generic_eq[OF assms[THEN simple_function_imp_finite_random_variable]])
855      simp
857 lemma (in information_space) conditional_entropy_eq_ce_with_hypothesis:
858   assumes X: "simple_function M X" and Y: "simple_function M Y"
859   shows "\<H>(X | Y) =
860     -(\<Sum>y\<in>Y`space M. distribution Y {y} *
861       (\<Sum>x\<in>X`space M. joint_distribution X Y {(x,y)} / distribution Y {(y)} *
862               log b (joint_distribution X Y {(x,y)} / distribution Y {(y)})))"
863   unfolding conditional_entropy_eq[OF assms]
864   using finite_distribution_order(5,6)[OF assms[THEN simple_function_imp_finite_random_variable]]
865   by (auto simp: setsum_cartesian_product'  setsum_commute[of _ "Y`space M"] setsum_right_distrib
866            intro!: setsum_cong)
868 lemma (in information_space) conditional_entropy_eq_cartesian_product:
869   assumes "simple_function M X" "simple_function M Y"
870   shows "\<H>(X | Y) = -(\<Sum>x\<in>X`space M. \<Sum>y\<in>Y`space M.
871     joint_distribution X Y {(x,y)} *
872     log b (joint_distribution X Y {(x,y)} / distribution Y {y}))"
873   unfolding conditional_entropy_eq[OF assms]
874   by (auto intro!: setsum_cong simp: setsum_cartesian_product')
876 subsection {* Equalities *}
878 lemma (in information_space) mutual_information_eq_entropy_conditional_entropy:
879   assumes X: "simple_function M X" and Z: "simple_function M Z"
880   shows  "\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)"
881 proof -
882   let "?XZ x z" = "joint_distribution X Z {(x, z)}"
883   let "?Z z" = "distribution Z {z}"
884   let "?X x" = "distribution X {x}"
885   note fX = X[THEN simple_function_imp_finite_random_variable]
886   note fZ = Z[THEN simple_function_imp_finite_random_variable]
887   note finite_distribution_order[OF fX fZ, simp]
888   { fix x z assume "x \<in> X`space M" "z \<in> Z`space M"
889     have "?XZ x z * log b (?XZ x z / (?X x * ?Z z)) =
890           ?XZ x z * log b (?XZ x z / ?Z z) - ?XZ x z * log b (?X x)"
891       by (auto simp: log_simps zero_le_mult_iff field_simps less_le) }
892   note * = this
893   show ?thesis
894     unfolding entropy_eq[OF X] conditional_entropy_eq[OF X Z] mutual_information_eq[OF X Z]
895     using setsum_joint_distribution_singleton[OF fZ fX, unfolded joint_distribution_commute_singleton[of Z X]]
896     by (simp add: * setsum_cartesian_product' setsum_subtractf setsum_left_distrib[symmetric]
897                      setsum_distribution)
898 qed
900 lemma (in information_space) conditional_entropy_less_eq_entropy:
901   assumes X: "simple_function M X" and Z: "simple_function M Z"
902   shows "\<H>(X | Z) \<le> \<H>(X)"
903 proof -
904   have "\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)" using mutual_information_eq_entropy_conditional_entropy[OF assms] .
905   with mutual_information_positive[OF X Z] entropy_positive[OF X]
906   show ?thesis by auto
907 qed
909 lemma (in information_space) entropy_chain_rule:
910   assumes X: "simple_function M X" and Y: "simple_function M Y"
911   shows  "\<H>(\<lambda>x. (X x, Y x)) = \<H>(X) + \<H>(Y|X)"
912 proof -
913   let "?XY x y" = "joint_distribution X Y {(x, y)}"
914   let "?Y y" = "distribution Y {y}"
915   let "?X x" = "distribution X {x}"
916   note fX = X[THEN simple_function_imp_finite_random_variable]
917   note fY = Y[THEN simple_function_imp_finite_random_variable]
918   note finite_distribution_order[OF fX fY, simp]
919   { fix x y assume "x \<in> X`space M" "y \<in> Y`space M"
920     have "?XY x y * log b (?XY x y / ?X x) =
921           ?XY x y * log b (?XY x y) - ?XY x y * log b (?X x)"
922       by (auto simp: log_simps zero_le_mult_iff field_simps less_le) }
923   note * = this
924   show ?thesis
925     using setsum_joint_distribution_singleton[OF fY fX]
926     unfolding entropy_eq[OF X] conditional_entropy_eq_cartesian_product[OF Y X] entropy_eq_cartesian_product[OF X Y]
927     unfolding joint_distribution_commute_singleton[of Y X] setsum_commute[of _ "X`space M"]
928     by (simp add: * setsum_subtractf setsum_left_distrib[symmetric])
929 qed
931 section {* Partitioning *}
933 definition "subvimage A f g \<longleftrightarrow> (\<forall>x \<in> A. f -` {f x} \<inter> A \<subseteq> g -` {g x} \<inter> A)"
935 lemma subvimageI:
936   assumes "\<And>x y. \<lbrakk> x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y"
937   shows "subvimage A f g"
938   using assms unfolding subvimage_def by blast
940 lemma subvimageE[consumes 1]:
941   assumes "subvimage A f g"
942   obtains "\<And>x y. \<lbrakk> x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y"
943   using assms unfolding subvimage_def by blast
945 lemma subvimageD:
946   "\<lbrakk> subvimage A f g ; x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y"
947   using assms unfolding subvimage_def by blast
949 lemma subvimage_subset:
950   "\<lbrakk> subvimage B f g ; A \<subseteq> B \<rbrakk> \<Longrightarrow> subvimage A f g"
951   unfolding subvimage_def by auto
953 lemma subvimage_idem[intro]: "subvimage A g g"
954   by (safe intro!: subvimageI)
956 lemma subvimage_comp_finer[intro]:
957   assumes svi: "subvimage A g h"
958   shows "subvimage A g (f \<circ> h)"
959 proof (rule subvimageI, simp)
960   fix x y assume "x \<in> A" "y \<in> A" "g x = g y"
961   from svi[THEN subvimageD, OF this]
962   show "f (h x) = f (h y)" by simp
963 qed
965 lemma subvimage_comp_gran:
966   assumes svi: "subvimage A g h"
967   assumes inj: "inj_on f (g ` A)"
968   shows "subvimage A (f \<circ> g) h"
969   by (rule subvimageI) (auto intro!: subvimageD[OF svi] simp: inj_on_iff[OF inj])
971 lemma subvimage_comp:
972   assumes svi: "subvimage (f ` A) g h"
973   shows "subvimage A (g \<circ> f) (h \<circ> f)"
974   by (rule subvimageI) (auto intro!: svi[THEN subvimageD])
976 lemma subvimage_trans:
977   assumes fg: "subvimage A f g"
978   assumes gh: "subvimage A g h"
979   shows "subvimage A f h"
980   by (rule subvimageI) (auto intro!: fg[THEN subvimageD] gh[THEN subvimageD])
982 lemma subvimage_translator:
983   assumes svi: "subvimage A f g"
984   shows "\<exists>h. \<forall>x \<in> A. h (f x)  = g x"
985 proof (safe intro!: exI[of _ "\<lambda>x. (THE z. z \<in> (g ` (f -` {x} \<inter> A)))"])
986   fix x assume "x \<in> A"
987   show "(THE x'. x' \<in> (g ` (f -` {f x} \<inter> A))) = g x"
988     by (rule theI2[of _ "g x"])
989       (insert `x \<in> A`, auto intro!: svi[THEN subvimageD])
990 qed
992 lemma subvimage_translator_image:
993   assumes svi: "subvimage A f g"
994   shows "\<exists>h. h ` f ` A = g ` A"
995 proof -
996   from subvimage_translator[OF svi]
997   obtain h where "\<And>x. x \<in> A \<Longrightarrow> h (f x) = g x" by auto
998   thus ?thesis
999     by (auto intro!: exI[of _ h]
1000       simp: image_compose[symmetric] comp_def cong: image_cong)
1001 qed
1003 lemma subvimage_finite:
1004   assumes svi: "subvimage A f g" and fin: "finite (f`A)"
1005   shows "finite (g`A)"
1006 proof -
1007   from subvimage_translator_image[OF svi]
1008   obtain h where "g`A = h`f`A" by fastsimp
1009   with fin show "finite (g`A)" by simp
1010 qed
1012 lemma subvimage_disj:
1013   assumes svi: "subvimage A f g"
1014   shows "f -` {x} \<inter> A \<subseteq> g -` {y} \<inter> A \<or>
1015       f -` {x} \<inter> g -` {y} \<inter> A = {}" (is "?sub \<or> ?dist")
1016 proof (rule disjCI)
1017   assume "\<not> ?dist"
1018   then obtain z where "z \<in> A" and "x = f z" and "y = g z" by auto
1019   thus "?sub" using svi unfolding subvimage_def by auto
1020 qed
1022 lemma setsum_image_split:
1023   assumes svi: "subvimage A f g" and fin: "finite (f ` A)"
1024   shows "(\<Sum>x\<in>f`A. h x) = (\<Sum>y\<in>g`A. \<Sum>x\<in>f`(g -` {y} \<inter> A). h x)"
1025     (is "?lhs = ?rhs")
1026 proof -
1027   have "f ` A =
1028       snd ` (SIGMA x : g ` A. f ` (g -` {x} \<inter> A))"
1029       (is "_ = snd ` ?SIGMA")
1030     unfolding image_split_eq_Sigma[symmetric]
1031     by (simp add: image_compose[symmetric] comp_def)
1032   moreover
1033   have snd_inj: "inj_on snd ?SIGMA"
1034     unfolding image_split_eq_Sigma[symmetric]
1035     by (auto intro!: inj_onI subvimageD[OF svi])
1036   ultimately
1037   have "(\<Sum>x\<in>f`A. h x) = (\<Sum>(x,y)\<in>?SIGMA. h y)"
1038     by (auto simp: setsum_reindex intro: setsum_cong)
1039   also have "... = ?rhs"
1040     using subvimage_finite[OF svi fin] fin
1041     apply (subst setsum_Sigma[symmetric])
1042     by (auto intro!: finite_subset[of _ "f`A"])
1043   finally show ?thesis .
1044 qed
1046 lemma (in information_space) entropy_partition:
1047   assumes sf: "simple_function M X" "simple_function M P"
1048   assumes svi: "subvimage (space M) X P"
1049   shows "\<H>(X) = \<H>(P) + \<H>(X|P)"
1050 proof -
1051   let "?XP x p" = "joint_distribution X P {(x, p)}"
1052   let "?X x" = "distribution X {x}"
1053   let "?P p" = "distribution P {p}"
1054   note fX = sf(1)[THEN simple_function_imp_finite_random_variable]
1055   note fP = sf(2)[THEN simple_function_imp_finite_random_variable]
1056   note finite_distribution_order[OF fX fP, simp]
1057   have "(\<Sum>x\<in>X ` space M. ?X x * log b (?X x)) =
1058     (\<Sum>y\<in>P `space M. \<Sum>x\<in>X ` space M. ?XP x y * log b (?XP x y))"
1059   proof (subst setsum_image_split[OF svi],
1060       safe intro!: setsum_mono_zero_cong_left imageI)
1061     show "finite (X ` space M)" "finite (X ` space M)" "finite (P ` space M)"
1062       using sf unfolding simple_function_def by auto
1063   next
1064     fix p x assume in_space: "p \<in> space M" "x \<in> space M"
1065     assume "?XP (X x) (P p) * log b (?XP (X x) (P p)) \<noteq> 0"
1066     hence "(\<lambda>x. (X x, P x)) -` {(X x, P p)} \<inter> space M \<noteq> {}" by (auto simp: distribution_def)
1067     with svi[unfolded subvimage_def, rule_format, OF `x \<in> space M`]
1068     show "x \<in> P -` {P p}" by auto
1069   next
1070     fix p x assume in_space: "p \<in> space M" "x \<in> space M"
1071     assume "P x = P p"
1072     from this[symmetric] svi[unfolded subvimage_def, rule_format, OF `x \<in> space M`]
1073     have "X -` {X x} \<inter> space M \<subseteq> P -` {P p} \<inter> space M"
1074       by auto
1075     hence "(\<lambda>x. (X x, P x)) -` {(X x, P p)} \<inter> space M = X -` {X x} \<inter> space M"
1076       by auto
1077     thus "?X (X x) * log b (?X (X x)) = ?XP (X x) (P p) * log b (?XP (X x) (P p))"
1078       by (auto simp: distribution_def)
1079   qed
1080   moreover have "\<And>x y. ?XP x y * log b (?XP x y / ?P y) =
1081       ?XP x y * log b (?XP x y) - ?XP x y * log b (?P y)"
1082     by (auto simp add: log_simps zero_less_mult_iff field_simps)
1083   ultimately show ?thesis
1084     unfolding sf[THEN entropy_eq] conditional_entropy_eq[OF sf]
1085     using setsum_joint_distribution_singleton[OF fX fP]
1086     by (simp add: setsum_cartesian_product' setsum_subtractf setsum_distribution
1087       setsum_left_distrib[symmetric] setsum_commute[where B="P`space M"])
1088 qed
1090 corollary (in information_space) entropy_data_processing:
1091   assumes X: "simple_function M X" shows "\<H>(f \<circ> X) \<le> \<H>(X)"
1092 proof -
1093   note X
1094   moreover have fX: "simple_function M (f \<circ> X)" using X by auto
1095   moreover have "subvimage (space M) X (f \<circ> X)" by auto
1096   ultimately have "\<H>(X) = \<H>(f\<circ>X) + \<H>(X|f\<circ>X)" by (rule entropy_partition)
1097   then show "\<H>(f \<circ> X) \<le> \<H>(X)"
1098     by (auto intro: conditional_entropy_positive[OF X fX])
1099 qed
1101 corollary (in information_space) entropy_of_inj:
1102   assumes X: "simple_function M X" and inj: "inj_on f (X`space M)"
1103   shows "\<H>(f \<circ> X) = \<H>(X)"
1104 proof (rule antisym)
1105   show "\<H>(f \<circ> X) \<le> \<H>(X)" using entropy_data_processing[OF X] .
1106 next
1107   have sf: "simple_function M (f \<circ> X)"
1108     using X by auto
1109   have "\<H>(X) = \<H>(the_inv_into (X`space M) f \<circ> (f \<circ> X))"
1110     by (auto intro!: mutual_information_cong simp: entropy_def the_inv_into_f_f[OF inj])
1111   also have "... \<le> \<H>(f \<circ> X)"
1112     using entropy_data_processing[OF sf] .
1113   finally show "\<H>(X) \<le> \<H>(f \<circ> X)" .
1114 qed
1116 end