src/HOL/Probability/Information.thy
 author huffman Sun Apr 01 16:09:58 2012 +0200 (2012-04-01) changeset 47255 30a1692557b0 parent 46905 6b1c0a80a57a child 47694 05663f75964c permissions -rw-r--r--
removed Nat_Numeral.thy, moving all theorems elsewhere
1 (*  Title:      HOL/Probability/Information.thy
2     Author:     Johannes Hölzl, TU München
3     Author:     Armin Heller, TU München
4 *)
6 header {*Information theory*}
8 theory Information
9 imports
10   Independent_Family
12   "~~/src/HOL/Library/Convex"
13 begin
15 lemma log_le: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> x \<le> y \<Longrightarrow> log a x \<le> log a y"
16   by (subst log_le_cancel_iff) auto
18 lemma log_less: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> x < y \<Longrightarrow> log a x < log a y"
19   by (subst log_less_cancel_iff) auto
21 lemma setsum_cartesian_product':
22   "(\<Sum>x\<in>A \<times> B. f x) = (\<Sum>x\<in>A. setsum (\<lambda>y. f (x, y)) B)"
23   unfolding setsum_cartesian_product by simp
25 section "Convex theory"
27 lemma log_setsum:
28   assumes "finite s" "s \<noteq> {}"
29   assumes "b > 1"
30   assumes "(\<Sum> i \<in> s. a i) = 1"
31   assumes "\<And> i. i \<in> s \<Longrightarrow> a i \<ge> 0"
32   assumes "\<And> i. i \<in> s \<Longrightarrow> y i \<in> {0 <..}"
33   shows "log b (\<Sum> i \<in> s. a i * y i) \<ge> (\<Sum> i \<in> s. a i * log b (y i))"
34 proof -
35   have "convex_on {0 <..} (\<lambda> x. - log b x)"
36     by (rule minus_log_convex[OF `b > 1`])
37   hence "- log b (\<Sum> i \<in> s. a i * y i) \<le> (\<Sum> i \<in> s. a i * - log b (y i))"
38     using convex_on_setsum[of _ _ "\<lambda> x. - log b x"] assms pos_is_convex by fastforce
39   thus ?thesis by (auto simp add:setsum_negf le_imp_neg_le)
40 qed
42 lemma log_setsum':
43   assumes "finite s" "s \<noteq> {}"
44   assumes "b > 1"
45   assumes "(\<Sum> i \<in> s. a i) = 1"
46   assumes pos: "\<And> i. i \<in> s \<Longrightarrow> 0 \<le> a i"
47           "\<And> i. \<lbrakk> i \<in> s ; 0 < a i \<rbrakk> \<Longrightarrow> 0 < y i"
48   shows "log b (\<Sum> i \<in> s. a i * y i) \<ge> (\<Sum> i \<in> s. a i * log b (y i))"
49 proof -
50   have "\<And>y. (\<Sum> i \<in> s - {i. a i = 0}. a i * y i) = (\<Sum> i \<in> s. a i * y i)"
51     using assms by (auto intro!: setsum_mono_zero_cong_left)
52   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))"
53   proof (rule log_setsum)
54     have "setsum a (s - {i. a i = 0}) = setsum a s"
55       using assms(1) by (rule setsum_mono_zero_cong_left) auto
56     thus sum_1: "setsum a (s - {i. a i = 0}) = 1"
57       "finite (s - {i. a i = 0})" using assms by simp_all
59     show "s - {i. a i = 0} \<noteq> {}"
60     proof
61       assume *: "s - {i. a i = 0} = {}"
62       hence "setsum a (s - {i. a i = 0}) = 0" by (simp add: * setsum_empty)
63       with sum_1 show False by simp
64     qed
66     fix i assume "i \<in> s - {i. a i = 0}"
67     hence "i \<in> s" "a i \<noteq> 0" by simp_all
68     thus "0 \<le> a i" "y i \<in> {0<..}" using pos[of i] by auto
69   qed fact+
70   ultimately show ?thesis by simp
71 qed
73 lemma log_setsum_divide:
74   assumes "finite S" and "S \<noteq> {}" and "1 < b"
75   assumes "(\<Sum>x\<in>S. g x) = 1"
76   assumes pos: "\<And>x. x \<in> S \<Longrightarrow> g x \<ge> 0" "\<And>x. x \<in> S \<Longrightarrow> f x \<ge> 0"
77   assumes g_pos: "\<And>x. \<lbrakk> x \<in> S ; 0 < g x \<rbrakk> \<Longrightarrow> 0 < f x"
78   shows "- (\<Sum>x\<in>S. g x * log b (g x / f x)) \<le> log b (\<Sum>x\<in>S. f x)"
79 proof -
80   have log_mono: "\<And>x y. 0 < x \<Longrightarrow> x \<le> y \<Longrightarrow> log b x \<le> log b y"
81     using `1 < b` by (subst log_le_cancel_iff) auto
83   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))"
84   proof (unfold setsum_negf[symmetric], rule setsum_cong)
85     fix x assume x: "x \<in> S"
86     show "- (g x * log b (g x / f x)) = g x * log b (f x / g x)"
87     proof (cases "g x = 0")
88       case False
89       with pos[OF x] g_pos[OF x] have "0 < f x" "0 < g x" by simp_all
90       thus ?thesis using `1 < b` by (simp add: log_divide field_simps)
91     qed simp
92   qed rule
93   also have "... \<le> log b (\<Sum>x\<in>S. g x * (f x / g x))"
94   proof (rule log_setsum')
95     fix x assume x: "x \<in> S" "0 < g x"
96     with g_pos[OF x] show "0 < f x / g x" by (safe intro!: divide_pos_pos)
97   qed fact+
98   also have "... = log b (\<Sum>x\<in>S - {x. g x = 0}. f x)" using `finite S`
99     by (auto intro!: setsum_mono_zero_cong_right arg_cong[where f="log b"]
100         split: split_if_asm)
101   also have "... \<le> log b (\<Sum>x\<in>S. f x)"
102   proof (rule log_mono)
103     have "0 = (\<Sum>x\<in>S - {x. g x = 0}. 0)" by simp
104     also have "... < (\<Sum>x\<in>S - {x. g x = 0}. f x)" (is "_ < ?sum")
105     proof (rule setsum_strict_mono)
106       show "finite (S - {x. g x = 0})" using `finite S` by simp
107       show "S - {x. g x = 0} \<noteq> {}"
108       proof
109         assume "S - {x. g x = 0} = {}"
110         hence "(\<Sum>x\<in>S. g x) = 0" by (subst setsum_0') auto
111         with `(\<Sum>x\<in>S. g x) = 1` show False by simp
112       qed
113       fix x assume "x \<in> S - {x. g x = 0}"
114       thus "0 < f x" using g_pos[of x] pos(1)[of x] by auto
115     qed
116     finally show "0 < ?sum" .
117     show "(\<Sum>x\<in>S - {x. g x = 0}. f x) \<le> (\<Sum>x\<in>S. f x)"
118       using `finite S` pos by (auto intro!: setsum_mono2)
119   qed
120   finally show ?thesis .
121 qed
123 lemma split_pairs:
124   "((A, B) = X) \<longleftrightarrow> (fst X = A \<and> snd X = B)" and
125   "(X = (A, B)) \<longleftrightarrow> (fst X = A \<and> snd X = B)" by auto
127 section "Information theory"
129 locale information_space = prob_space +
130   fixes b :: real assumes b_gt_1: "1 < b"
132 context information_space
133 begin
135 text {* Introduce some simplification rules for logarithm of base @{term b}. *}
137 lemma log_neg_const:
138   assumes "x \<le> 0"
139   shows "log b x = log b 0"
140 proof -
141   { fix u :: real
142     have "x \<le> 0" by fact
143     also have "0 < exp u"
144       using exp_gt_zero .
145     finally have "exp u \<noteq> x"
146       by auto }
147   then show "log b x = log b 0"
148     by (simp add: log_def ln_def)
149 qed
151 lemma log_mult_eq:
152   "log b (A * B) = (if 0 < A * B then log b \<bar>A\<bar> + log b \<bar>B\<bar> else log b 0)"
153   using log_mult[of b "\<bar>A\<bar>" "\<bar>B\<bar>"] b_gt_1 log_neg_const[of "A * B"]
154   by (auto simp: zero_less_mult_iff mult_le_0_iff)
156 lemma log_inverse_eq:
157   "log b (inverse B) = (if 0 < B then - log b B else log b 0)"
158   using log_inverse[of b B] log_neg_const[of "inverse B"] b_gt_1 by simp
160 lemma log_divide_eq:
161   "log b (A / B) = (if 0 < A * B then log b \<bar>A\<bar> - log b \<bar>B\<bar> else log b 0)"
162   unfolding divide_inverse log_mult_eq log_inverse_eq abs_inverse
163   by (auto simp: zero_less_mult_iff mult_le_0_iff)
165 lemmas log_simps = log_mult_eq log_inverse_eq log_divide_eq
167 end
169 subsection "Kullback\$-\$Leibler divergence"
171 text {* The Kullback\$-\$Leibler divergence is also known as relative entropy or
172 Kullback\$-\$Leibler distance. *}
174 definition
175   "entropy_density b M \<nu> = log b \<circ> real \<circ> RN_deriv M \<nu>"
177 definition
178   "KL_divergence b M \<nu> = integral\<^isup>L (M\<lparr>measure := \<nu>\<rparr>) (entropy_density b M \<nu>)"
180 lemma (in information_space) measurable_entropy_density:
181   assumes ps: "prob_space (M\<lparr>measure := \<nu>\<rparr>)"
182   assumes ac: "absolutely_continuous \<nu>"
183   shows "entropy_density b M \<nu> \<in> borel_measurable M"
184 proof -
185   interpret \<nu>: prob_space "M\<lparr>measure := \<nu>\<rparr>" by fact
186   have "measure_space (M\<lparr>measure := \<nu>\<rparr>)" by fact
187   from RN_deriv[OF this ac] b_gt_1 show ?thesis
188     unfolding entropy_density_def
189     by (intro measurable_comp) auto
190 qed
192 lemma (in information_space) KL_gt_0:
193   assumes ps: "prob_space (M\<lparr>measure := \<nu>\<rparr>)"
194   assumes ac: "absolutely_continuous \<nu>"
195   assumes int: "integrable (M\<lparr> measure := \<nu> \<rparr>) (entropy_density b M \<nu>)"
196   assumes A: "A \<in> sets M" "\<nu> A \<noteq> \<mu> A"
197   shows "0 < KL_divergence b M \<nu>"
198 proof -
199   interpret \<nu>: prob_space "M\<lparr>measure := \<nu>\<rparr>" by fact
200   have ms: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
201   have fms: "finite_measure (M\<lparr>measure := \<nu>\<rparr>)" by unfold_locales
202   note RN = RN_deriv[OF ms ac]
204   from real_RN_deriv[OF fms ac] guess D . note D = this
205   with absolutely_continuous_AE[OF ms] ac
206   have D\<nu>: "AE x in M\<lparr>measure := \<nu>\<rparr>. RN_deriv M \<nu> x = ereal (D x)"
207     by auto
209   def f \<equiv> "\<lambda>x. if D x = 0 then 1 else 1 / D x"
210   with D have f_borel: "f \<in> borel_measurable M"
211     by (auto intro!: measurable_If)
213   have "KL_divergence b M \<nu> = 1 / ln b * (\<integral> x. ln b * entropy_density b M \<nu> x \<partial>M\<lparr>measure := \<nu>\<rparr>)"
214     unfolding KL_divergence_def using int b_gt_1
215     by (simp add: integral_cmult)
217   { fix A assume "A \<in> sets M"
218     with RN D have "\<nu>.\<mu> A = (\<integral>\<^isup>+ x. ereal (D x) * indicator A x \<partial>M)"
219       by (auto intro!: positive_integral_cong_AE) }
220   note D_density = this
222   have ln_entropy: "(\<lambda>x. ln b * entropy_density b M \<nu> x) \<in> borel_measurable M"
223     using measurable_entropy_density[OF ps ac] by auto
225   have "integrable (M\<lparr>measure := \<nu>\<rparr>) (\<lambda>x. ln b * entropy_density b M \<nu> x)"
226     using int by auto
227   moreover have "integrable (M\<lparr>measure := \<nu>\<rparr>) (\<lambda>x. ln b * entropy_density b M \<nu> x) \<longleftrightarrow>
228       integrable M (\<lambda>x. D x * (ln b * entropy_density b M \<nu> x))"
229     using D D_density ln_entropy
230     by (intro integral_translated_density) auto
231   ultimately have M_int: "integrable M (\<lambda>x. D x * (ln b * entropy_density b M \<nu> x))"
232     by simp
234   have D_neg: "(\<integral>\<^isup>+ x. ereal (- D x) \<partial>M) = 0"
235     using D by (subst positive_integral_0_iff_AE) auto
237   have "(\<integral>\<^isup>+ x. ereal (D x) \<partial>M) = \<nu> (space M)"
238     using RN D by (auto intro!: positive_integral_cong_AE)
239   then have D_pos: "(\<integral>\<^isup>+ x. ereal (D x) \<partial>M) = 1"
240     using \<nu>.measure_space_1 by simp
242   have "integrable M D"
243     using D_pos D_neg D by (auto simp: integrable_def)
245   have "integral\<^isup>L M D = 1"
246     using D_pos D_neg by (auto simp: lebesgue_integral_def)
248   let ?D_set = "{x\<in>space M. D x \<noteq> 0}"
249   have [simp, intro]: "?D_set \<in> sets M"
250     using D by (auto intro: sets_Collect)
252   have "0 \<le> 1 - \<mu>' ?D_set"
253     using prob_le_1 by (auto simp: field_simps)
254   also have "\<dots> = (\<integral> x. D x - indicator ?D_set x \<partial>M)"
255     using `integrable M D` `integral\<^isup>L M D = 1`
256     by (simp add: \<mu>'_def)
257   also have "\<dots> < (\<integral> x. D x * (ln b * entropy_density b M \<nu> x) \<partial>M)"
258   proof (rule integral_less_AE)
259     show "integrable M (\<lambda>x. D x - indicator ?D_set x)"
260       using `integrable M D`
261       by (intro integral_diff integral_indicator) auto
262   next
263     show "integrable M (\<lambda>x. D x * (ln b * entropy_density b M \<nu> x))"
264       by fact
265   next
266     show "\<mu> {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<noteq> 0"
267     proof
268       assume eq_0: "\<mu> {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} = 0"
269       then have disj: "AE x. D x = 1 \<or> D x = 0"
270         using D(1) by (auto intro!: AE_I[OF subset_refl] sets_Collect)
272       have "\<mu> {x\<in>space M. D x = 1} = (\<integral>\<^isup>+ x. indicator {x\<in>space M. D x = 1} x \<partial>M)"
273         using D(1) by auto
274       also have "\<dots> = (\<integral>\<^isup>+ x. ereal (D x) * indicator {x\<in>space M. D x \<noteq> 0} x \<partial>M)"
275         using disj by (auto intro!: positive_integral_cong_AE simp: indicator_def one_ereal_def)
276       also have "\<dots> = \<nu> {x\<in>space M. D x \<noteq> 0}"
277         using D(1) D_density by auto
278       also have "\<dots> = \<nu> (space M)"
279         using D_density D(1) by (auto intro!: positive_integral_cong simp: indicator_def)
280       finally have "AE x. D x = 1"
281         using D(1) \<nu>.measure_space_1 by (intro AE_I_eq_1) auto
282       then have "(\<integral>\<^isup>+x. indicator A x\<partial>M) = (\<integral>\<^isup>+x. ereal (D x) * indicator A x\<partial>M)"
283         by (intro positive_integral_cong_AE) (auto simp: one_ereal_def[symmetric])
284       also have "\<dots> = \<nu> A"
285         using `A \<in> sets M` D_density by simp
286       finally show False using `A \<in> sets M` `\<nu> A \<noteq> \<mu> A` by simp
287     qed
288     show "{x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<in> sets M"
289       using D(1) by (auto intro: sets_Collect)
291     show "AE t. t \<in> {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<longrightarrow>
292       D t - indicator ?D_set t \<noteq> D t * (ln b * entropy_density b M \<nu> t)"
293       using D(2)
294     proof (elim AE_mp, safe intro!: AE_I2)
295       fix t assume Dt: "t \<in> space M" "D t \<noteq> 1" "D t \<noteq> 0"
296         and RN: "RN_deriv M \<nu> t = ereal (D t)"
297         and eq: "D t - indicator ?D_set t = D t * (ln b * entropy_density b M \<nu> t)"
299       have "D t - 1 = D t - indicator ?D_set t"
300         using Dt by simp
301       also note eq
302       also have "D t * (ln b * entropy_density b M \<nu> t) = - D t * ln (1 / D t)"
303         using RN b_gt_1 `D t \<noteq> 0` `0 \<le> D t`
304         by (simp add: entropy_density_def log_def ln_div less_le)
305       finally have "ln (1 / D t) = 1 / D t - 1"
306         using `D t \<noteq> 0` by (auto simp: field_simps)
307       from ln_eq_minus_one[OF _ this] `D t \<noteq> 0` `0 \<le> D t` `D t \<noteq> 1`
308       show False by auto
309     qed
311     show "AE t. D t - indicator ?D_set t \<le> D t * (ln b * entropy_density b M \<nu> t)"
312       using D(2)
313     proof (elim AE_mp, intro AE_I2 impI)
314       fix t assume "t \<in> space M" and RN: "RN_deriv M \<nu> t = ereal (D t)"
315       show "D t - indicator ?D_set t \<le> D t * (ln b * entropy_density b M \<nu> t)"
316       proof cases
317         assume asm: "D t \<noteq> 0"
318         then have "0 < D t" using `0 \<le> D t` by auto
319         then have "0 < 1 / D t" by auto
320         have "D t - indicator ?D_set t \<le> - D t * (1 / D t - 1)"
321           using asm `t \<in> space M` by (simp add: field_simps)
322         also have "- D t * (1 / D t - 1) \<le> - D t * ln (1 / D t)"
323           using ln_le_minus_one `0 < 1 / D t` by (intro mult_left_mono_neg) auto
324         also have "\<dots> = D t * (ln b * entropy_density b M \<nu> t)"
325           using `0 < D t` RN b_gt_1
326           by (simp_all add: log_def ln_div entropy_density_def)
327         finally show ?thesis by simp
328       qed simp
329     qed
330   qed
331   also have "\<dots> = (\<integral> x. ln b * entropy_density b M \<nu> x \<partial>M\<lparr>measure := \<nu>\<rparr>)"
332     using D D_density ln_entropy
333     by (intro integral_translated_density[symmetric]) auto
334   also have "\<dots> = ln b * (\<integral> x. entropy_density b M \<nu> x \<partial>M\<lparr>measure := \<nu>\<rparr>)"
335     using int by (rule \<nu>.integral_cmult)
336   finally show "0 < KL_divergence b M \<nu>"
337     using b_gt_1 by (auto simp: KL_divergence_def zero_less_mult_iff)
338 qed
340 lemma (in sigma_finite_measure) KL_eq_0:
341   assumes eq: "\<forall>A\<in>sets M. \<nu> A = measure M A"
342   shows "KL_divergence b M \<nu> = 0"
343 proof -
344   have "AE x. 1 = RN_deriv M \<nu> x"
345   proof (rule RN_deriv_unique)
346     show "measure_space (M\<lparr>measure := \<nu>\<rparr>)"
347       using eq by (intro measure_space_cong) auto
348     show "absolutely_continuous \<nu>"
349       unfolding absolutely_continuous_def using eq by auto
350     show "(\<lambda>x. 1) \<in> borel_measurable M" "AE x. 0 \<le> (1 :: ereal)" by auto
351     fix A assume "A \<in> sets M"
352     with eq show "\<nu> A = \<integral>\<^isup>+ x. 1 * indicator A x \<partial>M" by simp
353   qed
354   then have "AE x. log b (real (RN_deriv M \<nu> x)) = 0"
355     by (elim AE_mp) simp
356   from integral_cong_AE[OF this]
357   have "integral\<^isup>L M (entropy_density b M \<nu>) = 0"
358     by (simp add: entropy_density_def comp_def)
359   with eq show "KL_divergence b M \<nu> = 0"
360     unfolding KL_divergence_def
361     by (subst integral_cong_measure) auto
362 qed
364 lemma (in information_space) KL_eq_0_imp:
365   assumes ps: "prob_space (M\<lparr>measure := \<nu>\<rparr>)"
366   assumes ac: "absolutely_continuous \<nu>"
367   assumes int: "integrable (M\<lparr> measure := \<nu> \<rparr>) (entropy_density b M \<nu>)"
368   assumes KL: "KL_divergence b M \<nu> = 0"
369   shows "\<forall>A\<in>sets M. \<nu> A = \<mu> A"
370   by (metis less_imp_neq KL_gt_0 assms)
372 lemma (in information_space) KL_ge_0:
373   assumes ps: "prob_space (M\<lparr>measure := \<nu>\<rparr>)"
374   assumes ac: "absolutely_continuous \<nu>"
375   assumes int: "integrable (M\<lparr> measure := \<nu> \<rparr>) (entropy_density b M \<nu>)"
376   shows "0 \<le> KL_divergence b M \<nu>"
377   using KL_eq_0 KL_gt_0[OF ps ac int]
378   by (cases "\<forall>A\<in>sets M. \<nu> A = measure M A") (auto simp: le_less)
381 lemma (in sigma_finite_measure) KL_divergence_vimage:
382   assumes T: "T \<in> measure_preserving M M'"
383     and T': "T' \<in> measure_preserving (M'\<lparr> measure := \<nu>' \<rparr>) (M\<lparr> measure := \<nu> \<rparr>)"
384     and inv: "\<And>x. x \<in> space M \<Longrightarrow> T' (T x) = x"
385     and inv': "\<And>x. x \<in> space M' \<Longrightarrow> T (T' x) = x"
386   and \<nu>': "measure_space (M'\<lparr>measure := \<nu>'\<rparr>)" "measure_space.absolutely_continuous M' \<nu>'"
387   and "1 < b"
388   shows "KL_divergence b M' \<nu>' = KL_divergence b M \<nu>"
389 proof -
390   interpret \<nu>': measure_space "M'\<lparr>measure := \<nu>'\<rparr>" by fact
391   have M: "measure_space (M\<lparr> measure := \<nu>\<rparr>)"
392     by (rule \<nu>'.measure_space_vimage[OF _ T'], simp) default
393   have "sigma_algebra (M'\<lparr> measure := \<nu>'\<rparr>)" by default
394   then have saM': "sigma_algebra M'" by simp
395   then interpret M': measure_space M' by (rule measure_space_vimage) fact
396   have ac: "absolutely_continuous \<nu>" unfolding absolutely_continuous_def
397   proof safe
398     fix N assume N: "N \<in> sets M" and N_0: "\<mu> N = 0"
399     then have N': "T' -` N \<inter> space M' \<in> sets M'"
400       using T' by (auto simp: measurable_def measure_preserving_def)
401     have "T -` (T' -` N \<inter> space M') \<inter> space M = N"
402       using inv T N sets_into_space[OF N] by (auto simp: measurable_def measure_preserving_def)
403     then have "measure M' (T' -` N \<inter> space M') = 0"
404       using measure_preservingD[OF T N'] N_0 by auto
405     with \<nu>'(2) N' show "\<nu> N = 0" using measure_preservingD[OF T', of N] N
406       unfolding M'.absolutely_continuous_def measurable_def by auto
407   qed
409   have sa: "sigma_algebra (M\<lparr>measure := \<nu>\<rparr>)" by simp default
410   have AE: "AE x. RN_deriv M' \<nu>' (T x) = RN_deriv M \<nu> x"
411     by (rule RN_deriv_vimage[OF T T' inv \<nu>'])
412   show ?thesis
413     unfolding KL_divergence_def entropy_density_def comp_def
414   proof (subst \<nu>'.integral_vimage[OF sa T'])
415     show "(\<lambda>x. log b (real (RN_deriv M \<nu> x))) \<in> borel_measurable (M\<lparr>measure := \<nu>\<rparr>)"
416       by (auto intro!: RN_deriv[OF M ac] borel_measurable_log[OF _ `1 < b`])
417     have "(\<integral> x. log b (real (RN_deriv M' \<nu>' x)) \<partial>M'\<lparr>measure := \<nu>'\<rparr>) =
418       (\<integral> x. log b (real (RN_deriv M' \<nu>' (T (T' x)))) \<partial>M'\<lparr>measure := \<nu>'\<rparr>)" (is "?l = _")
419       using inv' by (auto intro!: \<nu>'.integral_cong)
420     also have "\<dots> = (\<integral> x. log b (real (RN_deriv M \<nu> (T' x))) \<partial>M'\<lparr>measure := \<nu>'\<rparr>)" (is "_ = ?r")
421       using M ac AE
422       by (intro \<nu>'.integral_cong_AE \<nu>'.almost_everywhere_vimage[OF sa T'] absolutely_continuous_AE[OF M])
423          (auto elim!: AE_mp)
424     finally show "?l = ?r" .
425   qed
426 qed
428 lemma (in sigma_finite_measure) KL_divergence_cong:
429   assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?\<nu>")
430   assumes [simp]: "sets N = sets M" "space N = space M"
431     "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A"
432     "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = \<nu>' A"
433   shows "KL_divergence b M \<nu> = KL_divergence b N \<nu>'"
434 proof -
435   interpret \<nu>: measure_space ?\<nu> by fact
436   have "KL_divergence b M \<nu> = \<integral>x. log b (real (RN_deriv N \<nu>' x)) \<partial>?\<nu>"
437     by (simp cong: RN_deriv_cong \<nu>.integral_cong add: KL_divergence_def entropy_density_def)
438   also have "\<dots> = KL_divergence b N \<nu>'"
439     by (auto intro!: \<nu>.integral_cong_measure[symmetric] simp: KL_divergence_def entropy_density_def comp_def)
440   finally show ?thesis .
441 qed
443 lemma (in finite_measure_space) KL_divergence_eq_finite:
444   assumes v: "finite_measure_space (M\<lparr>measure := \<nu>\<rparr>)"
445   assumes ac: "absolutely_continuous \<nu>"
446   shows "KL_divergence b M \<nu> = (\<Sum>x\<in>space M. real (\<nu> {x}) * log b (real (\<nu> {x}) / real (\<mu> {x})))" (is "_ = ?sum")
447 proof (simp add: KL_divergence_def finite_measure_space.integral_finite_singleton[OF v] entropy_density_def)
448   interpret v: finite_measure_space "M\<lparr>measure := \<nu>\<rparr>" by fact
449   have ms: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
450   show "(\<Sum>x \<in> space M. log b (real (RN_deriv M \<nu> x)) * real (\<nu> {x})) = ?sum"
451     using RN_deriv_finite_measure[OF ms ac]
452     by (auto intro!: setsum_cong simp: field_simps)
453 qed
455 lemma (in finite_prob_space) KL_divergence_positive_finite:
456   assumes v: "finite_prob_space (M\<lparr>measure := \<nu>\<rparr>)"
457   assumes ac: "absolutely_continuous \<nu>"
458   and "1 < b"
459   shows "0 \<le> KL_divergence b M \<nu>"
460 proof -
461   interpret information_space M by default fact
462   interpret v: finite_prob_space "M\<lparr>measure := \<nu>\<rparr>" by fact
463   have ps: "prob_space (M\<lparr>measure := \<nu>\<rparr>)" by unfold_locales
464   from KL_ge_0[OF this ac v.integral_finite_singleton(1)] show ?thesis .
465 qed
467 subsection {* Mutual Information *}
469 definition (in prob_space)
470   "mutual_information b S T X Y =
471     KL_divergence b (S\<lparr>measure := ereal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := ereal\<circ>distribution Y\<rparr>)
472       (ereal\<circ>joint_distribution X Y)"
474 lemma (in information_space)
475   fixes S T X Y
476   defines "P \<equiv> S\<lparr>measure := ereal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := ereal\<circ>distribution Y\<rparr>"
477   shows "indep_var S X T Y \<longleftrightarrow>
478     (random_variable S X \<and> random_variable T Y \<and>
479       measure_space.absolutely_continuous P (ereal\<circ>joint_distribution X Y) \<and>
480       integrable (P\<lparr>measure := (ereal\<circ>joint_distribution X Y)\<rparr>)
481         (entropy_density b P (ereal\<circ>joint_distribution X Y)) \<and>
482      mutual_information b S T X Y = 0)"
483 proof safe
484   assume indep: "indep_var S X T Y"
485   then have "random_variable S X" "random_variable T Y"
486     by (blast dest: indep_var_rv1 indep_var_rv2)+
487   then show "sigma_algebra S" "X \<in> measurable M S" "sigma_algebra T" "Y \<in> measurable M T"
488     by blast+
490   interpret X: prob_space "S\<lparr>measure := ereal\<circ>distribution X\<rparr>"
491     by (rule distribution_prob_space) fact
492   interpret Y: prob_space "T\<lparr>measure := ereal\<circ>distribution Y\<rparr>"
493     by (rule distribution_prob_space) fact
494   interpret XY: pair_prob_space "S\<lparr>measure := ereal\<circ>distribution X\<rparr>" "T\<lparr>measure := ereal\<circ>distribution Y\<rparr>" by default
495   interpret XY: information_space XY.P b by default (rule b_gt_1)
497   let ?J = "XY.P\<lparr> measure := (ereal\<circ>joint_distribution X Y) \<rparr>"
498   { fix A assume "A \<in> sets XY.P"
499     then have "ereal (joint_distribution X Y A) = XY.\<mu> A"
500       using indep_var_distributionD[OF indep]
501       by (simp add: XY.P.finite_measure_eq) }
502   note j_eq = this
504   interpret J: prob_space ?J
505     using j_eq by (intro XY.prob_space_cong) auto
507   have ac: "XY.absolutely_continuous (ereal\<circ>joint_distribution X Y)"
508     by (simp add: XY.absolutely_continuous_def j_eq)
509   then show "measure_space.absolutely_continuous P (ereal\<circ>joint_distribution X Y)"
510     unfolding P_def .
512   have ed: "entropy_density b XY.P (ereal\<circ>joint_distribution X Y) \<in> borel_measurable XY.P"
513     by (rule XY.measurable_entropy_density) (default | fact)+
515   have "AE x in XY.P. 1 = RN_deriv XY.P (ereal\<circ>joint_distribution X Y) x"
516   proof (rule XY.RN_deriv_unique[OF _ ac])
517     show "measure_space ?J" by default
518     fix A assume "A \<in> sets XY.P"
519     then show "(ereal\<circ>joint_distribution X Y) A = (\<integral>\<^isup>+ x. 1 * indicator A x \<partial>XY.P)"
520       by (simp add: j_eq)
521   qed (insert XY.measurable_const[of 1 borel], auto)
522   then have ae_XY: "AE x in XY.P. entropy_density b XY.P (ereal\<circ>joint_distribution X Y) x = 0"
523     by (elim XY.AE_mp) (simp add: entropy_density_def)
524   have ae_J: "AE x in ?J. entropy_density b XY.P (ereal\<circ>joint_distribution X Y) x = 0"
525   proof (rule XY.absolutely_continuous_AE)
526     show "measure_space ?J" by default
527     show "XY.absolutely_continuous (measure ?J)"
528       using ac by simp
529   qed (insert ae_XY, simp_all)
530   then show "integrable (P\<lparr>measure := (ereal\<circ>joint_distribution X Y)\<rparr>)
531         (entropy_density b P (ereal\<circ>joint_distribution X Y))"
532     unfolding P_def
533     using ed XY.measurable_const[of 0 borel]
534     by (subst J.integrable_cong_AE) auto
536   show "mutual_information b S T X Y = 0"
537     unfolding mutual_information_def KL_divergence_def P_def
538     by (subst J.integral_cong_AE[OF ae_J]) simp
539 next
540   assume "sigma_algebra S" "X \<in> measurable M S" "sigma_algebra T" "Y \<in> measurable M T"
541   then have rvs: "random_variable S X" "random_variable T Y" by blast+
543   interpret X: prob_space "S\<lparr>measure := ereal\<circ>distribution X\<rparr>"
544     by (rule distribution_prob_space) fact
545   interpret Y: prob_space "T\<lparr>measure := ereal\<circ>distribution Y\<rparr>"
546     by (rule distribution_prob_space) fact
547   interpret XY: pair_prob_space "S\<lparr>measure := ereal\<circ>distribution X\<rparr>" "T\<lparr>measure := ereal\<circ>distribution Y\<rparr>" by default
548   interpret XY: information_space XY.P b by default (rule b_gt_1)
550   let ?J = "XY.P\<lparr> measure := (ereal\<circ>joint_distribution X Y) \<rparr>"
551   interpret J: prob_space ?J
552     using rvs by (intro joint_distribution_prob_space) auto
554   assume ac: "measure_space.absolutely_continuous P (ereal\<circ>joint_distribution X Y)"
555   assume int: "integrable (P\<lparr>measure := (ereal\<circ>joint_distribution X Y)\<rparr>)
556         (entropy_density b P (ereal\<circ>joint_distribution X Y))"
557   assume I_eq_0: "mutual_information b S T X Y = 0"
559   have eq: "\<forall>A\<in>sets XY.P. (ereal \<circ> joint_distribution X Y) A = XY.\<mu> A"
560   proof (rule XY.KL_eq_0_imp)
561     show "prob_space ?J" by unfold_locales
562     show "XY.absolutely_continuous (ereal\<circ>joint_distribution X Y)"
563       using ac by (simp add: P_def)
564     show "integrable ?J (entropy_density b XY.P (ereal\<circ>joint_distribution X Y))"
565       using int by (simp add: P_def)
566     show "KL_divergence b XY.P (ereal\<circ>joint_distribution X Y) = 0"
567       using I_eq_0 unfolding mutual_information_def by (simp add: P_def)
568   qed
570   { fix S X assume "sigma_algebra S"
571     interpret S: sigma_algebra S by fact
572     have "Int_stable \<lparr>space = space M, sets = {X -` A \<inter> space M |A. A \<in> sets S}\<rparr>"
573     proof (safe intro!: Int_stableI)
574       fix A B assume "A \<in> sets S" "B \<in> sets S"
575       then show "\<exists>C. (X -` A \<inter> space M) \<inter> (X -` B \<inter> space M) = (X -` C \<inter> space M) \<and> C \<in> sets S"
576         by (intro exI[of _ "A \<inter> B"]) auto
577     qed }
578   note Int_stable = this
580   show "indep_var S X T Y" unfolding indep_var_eq
581   proof (intro conjI indep_set_sigma_sets Int_stable)
582     show "indep_set {X -` A \<inter> space M |A. A \<in> sets S} {Y -` A \<inter> space M |A. A \<in> sets T}"
583     proof (safe intro!: indep_setI)
584       { fix A assume "A \<in> sets S" then show "X -` A \<inter> space M \<in> sets M"
585         using `X \<in> measurable M S` by (auto intro: measurable_sets) }
586       { fix A assume "A \<in> sets T" then show "Y -` A \<inter> space M \<in> sets M"
587         using `Y \<in> measurable M T` by (auto intro: measurable_sets) }
588     next
589       fix A B assume ab: "A \<in> sets S" "B \<in> sets T"
590       have "ereal (prob ((X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M))) =
591         ereal (joint_distribution X Y (A \<times> B))"
592         unfolding distribution_def
593         by (intro arg_cong[where f="\<lambda>C. ereal (prob C)"]) auto
594       also have "\<dots> = XY.\<mu> (A \<times> B)"
595         using ab eq by (auto simp: XY.finite_measure_eq)
596       also have "\<dots> = ereal (distribution X A) * ereal (distribution Y B)"
597         using ab by (simp add: XY.pair_measure_times)
598       finally show "prob ((X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)) =
599         prob (X -` A \<inter> space M) * prob (Y -` B \<inter> space M)"
600         unfolding distribution_def by simp
601     qed
602   qed fact+
603 qed
605 lemma (in information_space) mutual_information_commute_generic:
606   assumes X: "random_variable S X" and Y: "random_variable T Y"
607   assumes ac: "measure_space.absolutely_continuous
608     (S\<lparr>measure := ereal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := ereal\<circ>distribution Y\<rparr>) (ereal\<circ>joint_distribution X Y)"
609   shows "mutual_information b S T X Y = mutual_information b T S Y X"
610 proof -
611   let ?S = "S\<lparr>measure := ereal\<circ>distribution X\<rparr>" and ?T = "T\<lparr>measure := ereal\<circ>distribution Y\<rparr>"
612   interpret S: prob_space ?S using X by (rule distribution_prob_space)
613   interpret T: prob_space ?T using Y by (rule distribution_prob_space)
614   interpret P: pair_prob_space ?S ?T ..
615   interpret Q: pair_prob_space ?T ?S ..
616   show ?thesis
617     unfolding mutual_information_def
618   proof (intro Q.KL_divergence_vimage[OF Q.measure_preserving_swap _ _ _ _ ac b_gt_1])
619     show "(\<lambda>(x,y). (y,x)) \<in> measure_preserving
620       (P.P \<lparr> measure := ereal\<circ>joint_distribution X Y\<rparr>) (Q.P \<lparr> measure := ereal\<circ>joint_distribution Y X\<rparr>)"
621       using X Y unfolding measurable_def
622       unfolding measure_preserving_def using P.pair_sigma_algebra_swap_measurable
623       by (auto simp add: space_pair_measure distribution_def intro!: arg_cong[where f=\<mu>'])
624     have "prob_space (P.P\<lparr> measure := ereal\<circ>joint_distribution X Y\<rparr>)"
625       using X Y by (auto intro!: distribution_prob_space random_variable_pairI)
626     then show "measure_space (P.P\<lparr> measure := ereal\<circ>joint_distribution X Y\<rparr>)"
627       unfolding prob_space_def finite_measure_def sigma_finite_measure_def by simp
628   qed auto
629 qed
631 definition (in prob_space)
632   "entropy b s X = mutual_information b s s X X"
634 abbreviation (in information_space)
635   mutual_information_Pow ("\<I>'(_ ; _')") where
636   "\<I>(X ; Y) \<equiv> mutual_information b
637     \<lparr> space = X`space M, sets = Pow (X`space M), measure = ereal\<circ>distribution X \<rparr>
638     \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = ereal\<circ>distribution Y \<rparr> X Y"
640 lemma (in prob_space) finite_variables_absolutely_continuous:
641   assumes X: "finite_random_variable S X" and Y: "finite_random_variable T Y"
642   shows "measure_space.absolutely_continuous
643     (S\<lparr>measure := ereal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := ereal\<circ>distribution Y\<rparr>)
644     (ereal\<circ>joint_distribution X Y)"
645 proof -
646   interpret X: finite_prob_space "S\<lparr>measure := ereal\<circ>distribution X\<rparr>"
647     using X by (rule distribution_finite_prob_space)
648   interpret Y: finite_prob_space "T\<lparr>measure := ereal\<circ>distribution Y\<rparr>"
649     using Y by (rule distribution_finite_prob_space)
650   interpret XY: pair_finite_prob_space
651     "S\<lparr>measure := ereal\<circ>distribution X\<rparr>" "T\<lparr> measure := ereal\<circ>distribution Y\<rparr>" by default
652   interpret P: finite_prob_space "XY.P\<lparr> measure := ereal\<circ>joint_distribution X Y\<rparr>"
653     using assms by (auto intro!: joint_distribution_finite_prob_space)
654   note rv = assms[THEN finite_random_variableD]
655   show "XY.absolutely_continuous (ereal\<circ>joint_distribution X Y)"
656   proof (rule XY.absolutely_continuousI)
657     show "finite_measure_space (XY.P\<lparr> measure := ereal\<circ>joint_distribution X Y\<rparr>)" by unfold_locales
658     fix x assume "x \<in> space XY.P" and "XY.\<mu> {x} = 0"
659     then obtain a b where "x = (a, b)"
660       and "distribution X {a} = 0 \<or> distribution Y {b} = 0"
661       by (cases x) (auto simp: space_pair_measure)
662     with finite_distribution_order(5,6)[OF X Y]
663     show "(ereal \<circ> joint_distribution X Y) {x} = 0" by auto
664   qed
665 qed
667 lemma (in information_space)
668   assumes MX: "finite_random_variable MX X"
669   assumes MY: "finite_random_variable MY Y"
670   shows mutual_information_generic_eq:
671     "mutual_information b MX MY X Y = (\<Sum> (x,y) \<in> space MX \<times> space MY.
672       joint_distribution X Y {(x,y)} *
673       log b (joint_distribution X Y {(x,y)} /
674       (distribution X {x} * distribution Y {y})))"
675     (is ?sum)
676   and mutual_information_positive_generic:
677      "0 \<le> mutual_information b MX MY X Y" (is ?positive)
678 proof -
679   interpret X: finite_prob_space "MX\<lparr>measure := ereal\<circ>distribution X\<rparr>"
680     using MX by (rule distribution_finite_prob_space)
681   interpret Y: finite_prob_space "MY\<lparr>measure := ereal\<circ>distribution Y\<rparr>"
682     using MY by (rule distribution_finite_prob_space)
683   interpret XY: pair_finite_prob_space "MX\<lparr>measure := ereal\<circ>distribution X\<rparr>" "MY\<lparr>measure := ereal\<circ>distribution Y\<rparr>" by default
684   interpret P: finite_prob_space "XY.P\<lparr>measure := ereal\<circ>joint_distribution X Y\<rparr>"
685     using assms by (auto intro!: joint_distribution_finite_prob_space)
687   have P_ms: "finite_measure_space (XY.P\<lparr>measure := ereal\<circ>joint_distribution X Y\<rparr>)" by unfold_locales
688   have P_ps: "finite_prob_space (XY.P\<lparr>measure := ereal\<circ>joint_distribution X Y\<rparr>)" by unfold_locales
690   show ?sum
691     unfolding Let_def mutual_information_def
692     by (subst XY.KL_divergence_eq_finite[OF P_ms finite_variables_absolutely_continuous[OF MX MY]])
693        (auto simp add: space_pair_measure setsum_cartesian_product')
695   show ?positive
696     using XY.KL_divergence_positive_finite[OF P_ps finite_variables_absolutely_continuous[OF MX MY] b_gt_1]
697     unfolding mutual_information_def .
698 qed
700 lemma (in information_space) mutual_information_commute:
701   assumes X: "finite_random_variable S X" and Y: "finite_random_variable T Y"
702   shows "mutual_information b S T X Y = mutual_information b T S Y X"
703   unfolding mutual_information_generic_eq[OF X Y] mutual_information_generic_eq[OF Y X]
704   unfolding joint_distribution_commute_singleton[of X Y]
705   by (auto simp add: ac_simps intro!: setsum_reindex_cong[OF swap_inj_on])
707 lemma (in information_space) mutual_information_commute_simple:
708   assumes X: "simple_function M X" and Y: "simple_function M Y"
709   shows "\<I>(X;Y) = \<I>(Y;X)"
710   by (intro mutual_information_commute X Y simple_function_imp_finite_random_variable)
712 lemma (in information_space) mutual_information_eq:
713   assumes "simple_function M X" "simple_function M Y"
714   shows "\<I>(X;Y) = (\<Sum> (x,y) \<in> X ` space M \<times> Y ` space M.
715     distribution (\<lambda>x. (X x, Y x)) {(x,y)} * log b (distribution (\<lambda>x. (X x, Y x)) {(x,y)} /
716                                                    (distribution X {x} * distribution Y {y})))"
717   using assms by (simp add: mutual_information_generic_eq)
719 lemma (in information_space) mutual_information_generic_cong:
720   assumes X: "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x"
721   assumes Y: "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x"
722   shows "mutual_information b MX MY X Y = mutual_information b MX MY X' Y'"
723   unfolding mutual_information_def using X Y
724   by (simp cong: distribution_cong)
726 lemma (in information_space) mutual_information_cong:
727   assumes X: "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x"
728   assumes Y: "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x"
729   shows "\<I>(X; Y) = \<I>(X'; Y')"
730   unfolding mutual_information_def using X Y
731   by (simp cong: distribution_cong image_cong)
733 lemma (in information_space) mutual_information_positive:
734   assumes "simple_function M X" "simple_function M Y"
735   shows "0 \<le> \<I>(X;Y)"
736   using assms by (simp add: mutual_information_positive_generic)
738 subsection {* Entropy *}
740 abbreviation (in information_space)
741   entropy_Pow ("\<H>'(_')") where
742   "\<H>(X) \<equiv> entropy b \<lparr> space = X`space M, sets = Pow (X`space M), measure = ereal\<circ>distribution X \<rparr> X"
744 lemma (in information_space) entropy_generic_eq:
745   fixes X :: "'a \<Rightarrow> 'c"
746   assumes MX: "finite_random_variable MX X"
747   shows "entropy b MX X = -(\<Sum> x \<in> space MX. distribution X {x} * log b (distribution X {x}))"
748 proof -
749   interpret MX: finite_prob_space "MX\<lparr>measure := ereal\<circ>distribution X\<rparr>"
750     using MX by (rule distribution_finite_prob_space)
751   let ?X = "\<lambda>x. distribution X {x}"
752   let ?XX = "\<lambda>x y. joint_distribution X X {(x, y)}"
754   { fix x y :: 'c
755     { assume "x \<noteq> y"
756       then have "(\<lambda>x. (X x, X x)) -` {(x,y)} \<inter> space M = {}" by auto
757       then have "joint_distribution X X {(x, y)} = 0" by (simp add: distribution_def) }
758     then have "?XX x y * log b (?XX x y / (?X x * ?X y)) =
759         (if x = y then - ?X y * log b (?X y) else 0)"
760       by (auto simp: log_simps zero_less_mult_iff) }
761   note remove_XX = this
763   show ?thesis
764     unfolding entropy_def mutual_information_generic_eq[OF MX MX]
765     unfolding setsum_cartesian_product[symmetric] setsum_negf[symmetric] remove_XX
766     using MX.finite_space by (auto simp: setsum_cases)
767 qed
769 lemma (in information_space) entropy_eq:
770   assumes "simple_function M X"
771   shows "\<H>(X) = -(\<Sum> x \<in> X ` space M. distribution X {x} * log b (distribution X {x}))"
772   using assms by (simp add: entropy_generic_eq)
774 lemma (in information_space) entropy_positive:
775   "simple_function M X \<Longrightarrow> 0 \<le> \<H>(X)"
776   unfolding entropy_def by (simp add: mutual_information_positive)
778 lemma (in information_space) entropy_certainty_eq_0:
779   assumes X: "simple_function M X" and "x \<in> X ` space M" and "distribution X {x} = 1"
780   shows "\<H>(X) = 0"
781 proof -
782   let ?X = "\<lparr> space = X ` space M, sets = Pow (X ` space M), measure = ereal\<circ>distribution X\<rparr>"
783   note simple_function_imp_finite_random_variable[OF `simple_function M X`]
784   from distribution_finite_prob_space[OF this, of "\<lparr> measure = ereal\<circ>distribution X \<rparr>"]
785   interpret X: finite_prob_space ?X by simp
786   have "distribution X (X ` space M - {x}) = distribution X (X ` space M) - distribution X {x}"
787     using X.measure_compl[of "{x}"] assms by auto
788   also have "\<dots> = 0" using X.prob_space assms by auto
789   finally have X0: "distribution X (X ` space M - {x}) = 0" by auto
790   { fix y assume *: "y \<in> X ` space M"
791     { assume asm: "y \<noteq> x"
792       with * have "{y} \<subseteq> X ` space M - {x}" by auto
793       from X.measure_mono[OF this] X0 asm *
794       have "distribution X {y} = 0"  by (auto intro: antisym) }
795     then have "distribution X {y} = (if x = y then 1 else 0)"
796       using assms by auto }
797   note fi = this
798   have y: "\<And>y. (if x = y then 1 else 0) * log b (if x = y then 1 else 0) = 0" by simp
799   show ?thesis unfolding entropy_eq[OF `simple_function M X`] by (auto simp: y fi)
800 qed
802 lemma (in information_space) entropy_le_card_not_0:
803   assumes X: "simple_function M X"
804   shows "\<H>(X) \<le> log b (card (X ` space M \<inter> {x. distribution X {x} \<noteq> 0}))"
805 proof -
806   let ?p = "\<lambda>x. distribution X {x}"
807   have "\<H>(X) = (\<Sum>x\<in>X`space M. ?p x * log b (1 / ?p x))"
808     unfolding entropy_eq[OF X] setsum_negf[symmetric]
809     by (auto intro!: setsum_cong simp: log_simps)
810   also have "\<dots> \<le> log b (\<Sum>x\<in>X`space M. ?p x * (1 / ?p x))"
811     using not_empty b_gt_1 `simple_function M X` sum_over_space_real_distribution[OF X]
812     by (intro log_setsum') (auto simp: simple_function_def)
813   also have "\<dots> = log b (\<Sum>x\<in>X`space M. if ?p x \<noteq> 0 then 1 else 0)"
814     by (intro arg_cong[where f="\<lambda>X. log b X"] setsum_cong) auto
815   finally show ?thesis
816     using `simple_function M X` by (auto simp: setsum_cases real_eq_of_nat simple_function_def)
817 qed
819 lemma (in prob_space) measure'_translate:
820   assumes X: "random_variable S X" and A: "A \<in> sets S"
821   shows "finite_measure.\<mu>' (S\<lparr> measure := ereal\<circ>distribution X \<rparr>) A = distribution X A"
822 proof -
823   interpret S: prob_space "S\<lparr> measure := ereal\<circ>distribution X \<rparr>"
824     using distribution_prob_space[OF X] .
825   from A show "S.\<mu>' A = distribution X A"
826     unfolding S.\<mu>'_def by (simp add: distribution_def [abs_def] \<mu>'_def)
827 qed
829 lemma (in information_space) entropy_uniform_max:
830   assumes X: "simple_function M X"
831   assumes "\<And>x y. \<lbrakk> x \<in> X ` space M ; y \<in> X ` space M \<rbrakk> \<Longrightarrow> distribution X {x} = distribution X {y}"
832   shows "\<H>(X) = log b (real (card (X ` space M)))"
833 proof -
834   let ?X = "\<lparr> space = X ` space M, sets = Pow (X ` space M), measure = undefined\<rparr>\<lparr> measure := ereal\<circ>distribution X\<rparr>"
835   note frv = simple_function_imp_finite_random_variable[OF X]
836   from distribution_finite_prob_space[OF this, of "\<lparr> measure = ereal\<circ>distribution X \<rparr>"]
837   interpret X: finite_prob_space ?X by simp
838   note rv = finite_random_variableD[OF frv]
839   have card_gt0: "0 < card (X ` space M)" unfolding card_gt_0_iff
840     using `simple_function M X` not_empty by (auto simp: simple_function_def)
841   { fix x assume "x \<in> space ?X"
842     moreover then have "X.\<mu>' {x} = 1 / card (space ?X)"
843     proof (rule X.uniform_prob)
844       fix x y assume "x \<in> space ?X" "y \<in> space ?X"
845       with assms(2)[of x y] show "X.\<mu>' {x} = X.\<mu>' {y}"
846         by (subst (1 2) measure'_translate[OF rv]) auto
847     qed
848     ultimately have "distribution X {x} = 1 / card (space ?X)"
849       by (subst (asm) measure'_translate[OF rv]) auto }
850   thus ?thesis
851     using not_empty X.finite_space b_gt_1 card_gt0
852     by (simp add: entropy_eq[OF `simple_function M X`] real_eq_of_nat[symmetric] log_simps)
853 qed
855 lemma (in information_space) entropy_le_card:
856   assumes "simple_function M X"
857   shows "\<H>(X) \<le> log b (real (card (X ` space M)))"
858 proof cases
859   assume "X ` space M \<inter> {x. distribution X {x} \<noteq> 0} = {}"
860   then have "\<And>x. x\<in>X`space M \<Longrightarrow> distribution X {x} = 0" by auto
861   moreover
862   have "0 < card (X`space M)"
863     using `simple_function M X` not_empty
864     by (auto simp: card_gt_0_iff simple_function_def)
865   then have "log b 1 \<le> log b (real (card (X`space M)))"
866     using b_gt_1 by (intro log_le) auto
867   ultimately show ?thesis using assms by (simp add: entropy_eq)
868 next
869   assume False: "X ` space M \<inter> {x. distribution X {x} \<noteq> 0} \<noteq> {}"
870   have "card (X ` space M \<inter> {x. distribution X {x} \<noteq> 0}) \<le> card (X ` space M)"
871     (is "?A \<le> ?B") using assms not_empty by (auto intro!: card_mono simp: simple_function_def)
872   note entropy_le_card_not_0[OF assms]
873   also have "log b (real ?A) \<le> log b (real ?B)"
874     using b_gt_1 False not_empty `?A \<le> ?B` assms
875     by (auto intro!: log_le simp: card_gt_0_iff simp: simple_function_def)
876   finally show ?thesis .
877 qed
879 lemma (in information_space) entropy_commute:
880   assumes "simple_function M X" "simple_function M Y"
881   shows "\<H>(\<lambda>x. (X x, Y x)) = \<H>(\<lambda>x. (Y x, X x))"
882 proof -
883   have sf: "simple_function M (\<lambda>x. (X x, Y x))" "simple_function M (\<lambda>x. (Y x, X x))"
884     using assms by (auto intro: simple_function_Pair)
885   have *: "(\<lambda>x. (Y x, X x))`space M = (\<lambda>(a,b). (b,a))`(\<lambda>x. (X x, Y x))`space M"
886     by auto
887   have inj: "\<And>X. inj_on (\<lambda>(a,b). (b,a)) X"
888     by (auto intro!: inj_onI)
889   show ?thesis
890     unfolding sf[THEN entropy_eq] unfolding * setsum_reindex[OF inj]
891     by (simp add: joint_distribution_commute[of Y X] split_beta)
892 qed
894 lemma (in information_space) entropy_eq_cartesian_product:
895   assumes "simple_function M X" "simple_function M Y"
896   shows "\<H>(\<lambda>x. (X x, Y x)) = -(\<Sum>x\<in>X`space M. \<Sum>y\<in>Y`space M.
897     joint_distribution X Y {(x,y)} * log b (joint_distribution X Y {(x,y)}))"
898 proof -
899   have sf: "simple_function M (\<lambda>x. (X x, Y x))"
900     using assms by (auto intro: simple_function_Pair)
901   { fix x assume "x\<notin>(\<lambda>x. (X x, Y x))`space M"
902     then have "(\<lambda>x. (X x, Y x)) -` {x} \<inter> space M = {}" by auto
903     then have "joint_distribution X Y {x} = 0"
904       unfolding distribution_def by auto }
905   then show ?thesis using sf assms
906     unfolding entropy_eq[OF sf] neg_equal_iff_equal setsum_cartesian_product
907     by (auto intro!: setsum_mono_zero_cong_left simp: simple_function_def)
908 qed
910 subsection {* Conditional Mutual Information *}
912 definition (in prob_space)
913   "conditional_mutual_information b MX MY MZ X Y Z \<equiv>
914     mutual_information b MX (MY \<Otimes>\<^isub>M MZ) X (\<lambda>x. (Y x, Z x)) -
915     mutual_information b MX MZ X Z"
917 abbreviation (in information_space)
918   conditional_mutual_information_Pow ("\<I>'( _ ; _ | _ ')") where
919   "\<I>(X ; Y | Z) \<equiv> conditional_mutual_information b
920     \<lparr> space = X`space M, sets = Pow (X`space M), measure = ereal\<circ>distribution X \<rparr>
921     \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = ereal\<circ>distribution Y \<rparr>
922     \<lparr> space = Z`space M, sets = Pow (Z`space M), measure = ereal\<circ>distribution Z \<rparr>
923     X Y Z"
925 lemma (in information_space) conditional_mutual_information_generic_eq:
926   assumes MX: "finite_random_variable MX X"
927     and MY: "finite_random_variable MY Y"
928     and MZ: "finite_random_variable MZ Z"
929   shows "conditional_mutual_information b MX MY MZ X Y Z = (\<Sum>(x, y, z) \<in> space MX \<times> space MY \<times> space MZ.
930              distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} *
931              log b (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} /
932     (joint_distribution X Z {(x, z)} * (joint_distribution Y Z {(y,z)} / distribution Z {z}))))"
933   (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))))")
934 proof -
935   let ?X = "\<lambda>x. distribution X {x}"
936   note finite_var = MX MY MZ
937   note YZ = finite_random_variable_pairI[OF finite_var(2,3)]
938   note XYZ = finite_random_variable_pairI[OF MX YZ]
939   note XZ = finite_random_variable_pairI[OF finite_var(1,3)]
940   note ZX = finite_random_variable_pairI[OF finite_var(3,1)]
941   note YZX = finite_random_variable_pairI[OF finite_var(2) ZX]
942   note order1 =
943     finite_distribution_order(5,6)[OF finite_var(1) YZ]
944     finite_distribution_order(5,6)[OF finite_var(1,3)]
946   note random_var = finite_var[THEN finite_random_variableD]
947   note finite = finite_var(1) YZ finite_var(3) XZ YZX
949   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>
950           \<Longrightarrow> joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} = 0"
951     unfolding joint_distribution_commute_singleton[of X]
952     unfolding joint_distribution_assoc_singleton[symmetric]
953     using finite_distribution_order(6)[OF finite_var(2) ZX]
954     by auto
956   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)))) =
957     (\<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))))"
958     (is "(\<Sum>(x, y, z)\<in>?S. ?L x y z) = (\<Sum>(x, y, z)\<in>?S. ?R x y z)")
959   proof (safe intro!: setsum_cong)
960     fix x y z assume space: "x \<in> space MX" "y \<in> space MY" "z \<in> space MZ"
961     show "?L x y z = ?R x y z"
962     proof cases
963       assume "?XYZ x y z \<noteq> 0"
964       with space have "0 < ?X x" "0 < ?Z z" "0 < ?XZ x z" "0 < ?YZ y z" "0 < ?XYZ x y z"
965         using order1 order2 by (auto simp: less_le)
966       with b_gt_1 show ?thesis
967         by (simp add: log_mult log_divide zero_less_mult_iff zero_less_divide_iff)
968     qed simp
969   qed
970   also have "\<dots> = (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?X x * ?YZ y z))) -
971                   (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XZ x z / (?X x * ?Z z)))"
972     by (auto simp add: setsum_subtractf[symmetric] field_simps intro!: setsum_cong)
973   also have "(\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XZ x z / (?X x * ?Z z))) =
974              (\<Sum>(x, z)\<in>space MX \<times> space MZ. ?XZ x z * log b (?XZ x z / (?X x * ?Z z)))"
975     unfolding setsum_cartesian_product[symmetric] setsum_commute[of _ _ "space MY"]
976               setsum_left_distrib[symmetric]
977     unfolding joint_distribution_commute_singleton[of X]
978     unfolding joint_distribution_assoc_singleton[symmetric]
979     using setsum_joint_distribution_singleton[OF finite_var(2) ZX]
980     by (intro setsum_cong refl) (simp add: space_pair_measure)
981   also have "(\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?X x * ?YZ y z))) -
982              (\<Sum>(x, z)\<in>space MX \<times> space MZ. ?XZ x z * log b (?XZ x z / (?X x * ?Z z))) =
983              conditional_mutual_information b MX MY MZ X Y Z"
984     unfolding conditional_mutual_information_def
985     unfolding mutual_information_generic_eq[OF finite_var(1,3)]
986     unfolding mutual_information_generic_eq[OF finite_var(1) YZ]
987     by (simp add: space_sigma space_pair_measure setsum_cartesian_product')
988   finally show ?thesis by simp
989 qed
991 lemma (in information_space) conditional_mutual_information_eq:
992   assumes "simple_function M X" "simple_function M Y" "simple_function M Z"
993   shows "\<I>(X;Y|Z) = (\<Sum>(x, y, z) \<in> X`space M \<times> Y`space M \<times> Z`space M.
994              distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} *
995              log b (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} /
996     (joint_distribution X Z {(x, z)} * joint_distribution Y Z {(y,z)} / distribution Z {z})))"
997   by (subst conditional_mutual_information_generic_eq[OF assms[THEN simple_function_imp_finite_random_variable]])
998      simp
1000 lemma (in information_space) conditional_mutual_information_eq_mutual_information:
1001   assumes X: "simple_function M X" and Y: "simple_function M Y"
1002   shows "\<I>(X ; Y) = \<I>(X ; Y | (\<lambda>x. ()))"
1003 proof -
1004   have [simp]: "(\<lambda>x. ()) ` space M = {()}" using not_empty by auto
1005   have C: "simple_function M (\<lambda>x. ())" by auto
1006   show ?thesis
1007     unfolding conditional_mutual_information_eq[OF X Y C]
1008     unfolding mutual_information_eq[OF X Y]
1009     by (simp add: setsum_cartesian_product' distribution_remove_const)
1010 qed
1012 lemma (in information_space) conditional_mutual_information_generic_positive:
1013   assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y" and Z: "finite_random_variable MZ Z"
1014   shows "0 \<le> conditional_mutual_information b MX MY MZ X Y Z"
1015 proof (cases "space MX \<times> space MY \<times> space MZ = {}")
1016   case True show ?thesis
1017     unfolding conditional_mutual_information_generic_eq[OF assms] True
1018     by simp
1019 next
1020   case False
1021   let ?dXYZ = "distribution (\<lambda>x. (X x, Y x, Z x))"
1022   let ?dXZ = "joint_distribution X Z"
1023   let ?dYZ = "joint_distribution Y Z"
1024   let ?dX = "distribution X"
1025   let ?dZ = "distribution Z"
1026   let ?M = "space MX \<times> space MY \<times> space MZ"
1028   note YZ = finite_random_variable_pairI[OF Y Z]
1029   note XZ = finite_random_variable_pairI[OF X Z]
1030   note ZX = finite_random_variable_pairI[OF Z X]
1031   note YZ = finite_random_variable_pairI[OF Y Z]
1032   note XYZ = finite_random_variable_pairI[OF X YZ]
1033   note finite = Z YZ XZ XYZ
1034   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>
1035           \<Longrightarrow> joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} = 0"
1036     unfolding joint_distribution_commute_singleton[of X]
1037     unfolding joint_distribution_assoc_singleton[symmetric]
1038     using finite_distribution_order(6)[OF Y ZX]
1039     by auto
1041   note order = order
1042     finite_distribution_order(5,6)[OF X YZ]
1043     finite_distribution_order(5,6)[OF Y Z]
1045   have "- conditional_mutual_information b MX MY MZ X Y Z = - (\<Sum>(x, y, z) \<in> ?M. ?dXYZ {(x, y, z)} *
1046     log b (?dXYZ {(x, y, z)} / (?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z})))"
1047     unfolding conditional_mutual_information_generic_eq[OF assms] neg_equal_iff_equal by auto
1048   also have "\<dots> \<le> log b (\<Sum>(x, y, z) \<in> ?M. ?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z})"
1049     unfolding split_beta'
1050   proof (rule log_setsum_divide)
1051     show "?M \<noteq> {}" using False by simp
1052     show "1 < b" using b_gt_1 .
1054     show "finite ?M" using assms
1055       unfolding finite_sigma_algebra_def finite_sigma_algebra_axioms_def by auto
1057     show "(\<Sum>x\<in>?M. ?dXYZ {(fst x, fst (snd x), snd (snd x))}) = 1"
1058       unfolding setsum_cartesian_product'
1059       unfolding setsum_commute[of _ "space MY"]
1060       unfolding setsum_commute[of _ "space MZ"]
1061       by (simp_all add: space_pair_measure
1062                         setsum_joint_distribution_singleton[OF X YZ]
1063                         setsum_joint_distribution_singleton[OF Y Z]
1064                         setsum_distribution[OF Z])
1066     fix x assume "x \<in> ?M"
1067     let ?x = "(fst x, fst (snd x), snd (snd x))"
1069     show "0 \<le> ?dXYZ {?x}"
1070       "0 \<le> ?dXZ {(fst x, snd (snd x))} * ?dYZ {(fst (snd x), snd (snd x))} / ?dZ {snd (snd x)}"
1071      by (simp_all add: mult_nonneg_nonneg divide_nonneg_nonneg)
1073     assume *: "0 < ?dXYZ {?x}"
1074     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)}"
1075       by (cases x) (auto simp add: zero_le_mult_iff zero_le_divide_iff less_le)
1076   qed
1077   also have "(\<Sum>(x, y, z) \<in> ?M. ?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z}) = (\<Sum>z\<in>space MZ. ?dZ {z})"
1078     apply (simp add: setsum_cartesian_product')
1079     apply (subst setsum_commute)
1080     apply (subst (2) setsum_commute)
1081     by (auto simp: setsum_divide_distrib[symmetric] setsum_product[symmetric]
1082                    setsum_joint_distribution_singleton[OF X Z]
1083                    setsum_joint_distribution_singleton[OF Y Z]
1084           intro!: setsum_cong)
1085   also have "log b (\<Sum>z\<in>space MZ. ?dZ {z}) = 0"
1086     unfolding setsum_distribution[OF Z] by simp
1087   finally show ?thesis by simp
1088 qed
1090 lemma (in information_space) conditional_mutual_information_positive:
1091   assumes "simple_function M X" and "simple_function M Y" and "simple_function M Z"
1092   shows "0 \<le> \<I>(X;Y|Z)"
1093   by (rule conditional_mutual_information_generic_positive[OF assms[THEN simple_function_imp_finite_random_variable]])
1095 subsection {* Conditional Entropy *}
1097 definition (in prob_space)
1098   "conditional_entropy b S T X Y = conditional_mutual_information b S S T X X Y"
1100 abbreviation (in information_space)
1101   conditional_entropy_Pow ("\<H>'(_ | _')") where
1102   "\<H>(X | Y) \<equiv> conditional_entropy b
1103     \<lparr> space = X`space M, sets = Pow (X`space M), measure = ereal\<circ>distribution X \<rparr>
1104     \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = ereal\<circ>distribution Y \<rparr> X Y"
1106 lemma (in information_space) conditional_entropy_positive:
1107   "simple_function M X \<Longrightarrow> simple_function M Y \<Longrightarrow> 0 \<le> \<H>(X | Y)"
1108   unfolding conditional_entropy_def by (auto intro!: conditional_mutual_information_positive)
1110 lemma (in information_space) conditional_entropy_generic_eq:
1111   fixes MX :: "('c, 'd) measure_space_scheme" and MY :: "('e, 'f) measure_space_scheme"
1112   assumes MX: "finite_random_variable MX X"
1113   assumes MZ: "finite_random_variable MZ Z"
1114   shows "conditional_entropy b MX MZ X Z =
1115      - (\<Sum>(x, z)\<in>space MX \<times> space MZ.
1116          joint_distribution X Z {(x, z)} * log b (joint_distribution X Z {(x, z)} / distribution Z {z}))"
1117 proof -
1118   interpret MX: finite_sigma_algebra MX using MX by simp
1119   interpret MZ: finite_sigma_algebra MZ using MZ by simp
1120   let ?XXZ = "\<lambda>x y z. joint_distribution X (\<lambda>x. (X x, Z x)) {(x, y, z)}"
1121   let ?XZ = "\<lambda>x z. joint_distribution X Z {(x, z)}"
1122   let ?Z = "\<lambda>z. distribution Z {z}"
1123   let ?f = "\<lambda>x y z. log b (?XXZ x y z * ?Z z / (?XZ x z * ?XZ y z))"
1124   { fix x z have "?XXZ x x z = ?XZ x z"
1125       unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) }
1126   note this[simp]
1127   { fix x x' :: 'c and z assume "x' \<noteq> x"
1128     then have "?XXZ x x' z = 0"
1129       by (auto simp: distribution_def empty_measure'[symmetric]
1130                simp del: empty_measure' intro!: arg_cong[where f=\<mu>']) }
1131   note this[simp]
1132   { fix x x' z assume *: "x \<in> space MX" "z \<in> space MZ"
1133     then have "(\<Sum>x'\<in>space MX. ?XXZ x x' z * ?f x x' z)
1134       = (\<Sum>x'\<in>space MX. if x = x' then ?XZ x z * ?f x x z else 0)"
1135       by (auto intro!: setsum_cong)
1136     also have "\<dots> = ?XZ x z * ?f x x z"
1137       using `x \<in> space MX` by (simp add: setsum_cases[OF MX.finite_space])
1138     also have "\<dots> = ?XZ x z * log b (?Z z / ?XZ x z)" by auto
1139     also have "\<dots> = - ?XZ x z * log b (?XZ x z / ?Z z)"
1140       using finite_distribution_order(6)[OF MX MZ]
1141       by (auto simp: log_simps field_simps zero_less_mult_iff)
1142     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)" . }
1143   note * = this
1144   show ?thesis
1145     unfolding conditional_entropy_def
1146     unfolding conditional_mutual_information_generic_eq[OF MX MX MZ]
1147     by (auto simp: setsum_cartesian_product' setsum_negf[symmetric]
1148                    setsum_commute[of _ "space MZ"] *
1149              intro!: setsum_cong)
1150 qed
1152 lemma (in information_space) conditional_entropy_eq:
1153   assumes "simple_function M X" "simple_function M Z"
1154   shows "\<H>(X | Z) =
1155      - (\<Sum>(x, z)\<in>X ` space M \<times> Z ` space M.
1156          joint_distribution X Z {(x, z)} *
1157          log b (joint_distribution X Z {(x, z)} / distribution Z {z}))"
1158   by (subst conditional_entropy_generic_eq[OF assms[THEN simple_function_imp_finite_random_variable]])
1159      simp
1161 lemma (in information_space) conditional_entropy_eq_ce_with_hypothesis:
1162   assumes X: "simple_function M X" and Y: "simple_function M Y"
1163   shows "\<H>(X | Y) =
1164     -(\<Sum>y\<in>Y`space M. distribution Y {y} *
1165       (\<Sum>x\<in>X`space M. joint_distribution X Y {(x,y)} / distribution Y {(y)} *
1166               log b (joint_distribution X Y {(x,y)} / distribution Y {(y)})))"
1167   unfolding conditional_entropy_eq[OF assms]
1168   using finite_distribution_order(5,6)[OF assms[THEN simple_function_imp_finite_random_variable]]
1169   by (auto simp: setsum_cartesian_product'  setsum_commute[of _ "Y`space M"] setsum_right_distrib
1170            intro!: setsum_cong)
1172 lemma (in information_space) conditional_entropy_eq_cartesian_product:
1173   assumes "simple_function M X" "simple_function M Y"
1174   shows "\<H>(X | Y) = -(\<Sum>x\<in>X`space M. \<Sum>y\<in>Y`space M.
1175     joint_distribution X Y {(x,y)} *
1176     log b (joint_distribution X Y {(x,y)} / distribution Y {y}))"
1177   unfolding conditional_entropy_eq[OF assms]
1178   by (auto intro!: setsum_cong simp: setsum_cartesian_product')
1180 subsection {* Equalities *}
1182 lemma (in information_space) mutual_information_eq_entropy_conditional_entropy:
1183   assumes X: "simple_function M X" and Z: "simple_function M Z"
1184   shows  "\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)"
1185 proof -
1186   let ?XZ = "\<lambda>x z. joint_distribution X Z {(x, z)}"
1187   let ?Z = "\<lambda>z. distribution Z {z}"
1188   let ?X = "\<lambda>x. distribution X {x}"
1189   note fX = X[THEN simple_function_imp_finite_random_variable]
1190   note fZ = Z[THEN simple_function_imp_finite_random_variable]
1191   note finite_distribution_order[OF fX fZ, simp]
1192   { fix x z assume "x \<in> X`space M" "z \<in> Z`space M"
1193     have "?XZ x z * log b (?XZ x z / (?X x * ?Z z)) =
1194           ?XZ x z * log b (?XZ x z / ?Z z) - ?XZ x z * log b (?X x)"
1195       by (auto simp: log_simps zero_le_mult_iff field_simps less_le) }
1196   note * = this
1197   show ?thesis
1198     unfolding entropy_eq[OF X] conditional_entropy_eq[OF X Z] mutual_information_eq[OF X Z]
1199     using setsum_joint_distribution_singleton[OF fZ fX, unfolded joint_distribution_commute_singleton[of Z X]]
1200     by (simp add: * setsum_cartesian_product' setsum_subtractf setsum_left_distrib[symmetric]
1201                      setsum_distribution)
1202 qed
1204 lemma (in information_space) conditional_entropy_less_eq_entropy:
1205   assumes X: "simple_function M X" and Z: "simple_function M Z"
1206   shows "\<H>(X | Z) \<le> \<H>(X)"
1207 proof -
1208   have "\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)" using mutual_information_eq_entropy_conditional_entropy[OF assms] .
1209   with mutual_information_positive[OF X Z] entropy_positive[OF X]
1210   show ?thesis by auto
1211 qed
1213 lemma (in information_space) entropy_chain_rule:
1214   assumes X: "simple_function M X" and Y: "simple_function M Y"
1215   shows  "\<H>(\<lambda>x. (X x, Y x)) = \<H>(X) + \<H>(Y|X)"
1216 proof -
1217   let ?XY = "\<lambda>x y. joint_distribution X Y {(x, y)}"
1218   let ?Y = "\<lambda>y. distribution Y {y}"
1219   let ?X = "\<lambda>x. distribution X {x}"
1220   note fX = X[THEN simple_function_imp_finite_random_variable]
1221   note fY = Y[THEN simple_function_imp_finite_random_variable]
1222   note finite_distribution_order[OF fX fY, simp]
1223   { fix x y assume "x \<in> X`space M" "y \<in> Y`space M"
1224     have "?XY x y * log b (?XY x y / ?X x) =
1225           ?XY x y * log b (?XY x y) - ?XY x y * log b (?X x)"
1226       by (auto simp: log_simps zero_le_mult_iff field_simps less_le) }
1227   note * = this
1228   show ?thesis
1229     using setsum_joint_distribution_singleton[OF fY fX]
1230     unfolding entropy_eq[OF X] conditional_entropy_eq_cartesian_product[OF Y X] entropy_eq_cartesian_product[OF X Y]
1231     unfolding joint_distribution_commute_singleton[of Y X] setsum_commute[of _ "X`space M"]
1232     by (simp add: * setsum_subtractf setsum_left_distrib[symmetric])
1233 qed
1235 section {* Partitioning *}
1237 definition "subvimage A f g \<longleftrightarrow> (\<forall>x \<in> A. f -` {f x} \<inter> A \<subseteq> g -` {g x} \<inter> A)"
1239 lemma subvimageI:
1240   assumes "\<And>x y. \<lbrakk> x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y"
1241   shows "subvimage A f g"
1242   using assms unfolding subvimage_def by blast
1244 lemma subvimageE[consumes 1]:
1245   assumes "subvimage A f g"
1246   obtains "\<And>x y. \<lbrakk> x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y"
1247   using assms unfolding subvimage_def by blast
1249 lemma subvimageD:
1250   "\<lbrakk> subvimage A f g ; x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y"
1251   using assms unfolding subvimage_def by blast
1253 lemma subvimage_subset:
1254   "\<lbrakk> subvimage B f g ; A \<subseteq> B \<rbrakk> \<Longrightarrow> subvimage A f g"
1255   unfolding subvimage_def by auto
1257 lemma subvimage_idem[intro]: "subvimage A g g"
1258   by (safe intro!: subvimageI)
1260 lemma subvimage_comp_finer[intro]:
1261   assumes svi: "subvimage A g h"
1262   shows "subvimage A g (f \<circ> h)"
1263 proof (rule subvimageI, simp)
1264   fix x y assume "x \<in> A" "y \<in> A" "g x = g y"
1265   from svi[THEN subvimageD, OF this]
1266   show "f (h x) = f (h y)" by simp
1267 qed
1269 lemma subvimage_comp_gran:
1270   assumes svi: "subvimage A g h"
1271   assumes inj: "inj_on f (g ` A)"
1272   shows "subvimage A (f \<circ> g) h"
1273   by (rule subvimageI) (auto intro!: subvimageD[OF svi] simp: inj_on_iff[OF inj])
1275 lemma subvimage_comp:
1276   assumes svi: "subvimage (f ` A) g h"
1277   shows "subvimage A (g \<circ> f) (h \<circ> f)"
1278   by (rule subvimageI) (auto intro!: svi[THEN subvimageD])
1280 lemma subvimage_trans:
1281   assumes fg: "subvimage A f g"
1282   assumes gh: "subvimage A g h"
1283   shows "subvimage A f h"
1284   by (rule subvimageI) (auto intro!: fg[THEN subvimageD] gh[THEN subvimageD])
1286 lemma subvimage_translator:
1287   assumes svi: "subvimage A f g"
1288   shows "\<exists>h. \<forall>x \<in> A. h (f x)  = g x"
1289 proof (safe intro!: exI[of _ "\<lambda>x. (THE z. z \<in> (g ` (f -` {x} \<inter> A)))"])
1290   fix x assume "x \<in> A"
1291   show "(THE x'. x' \<in> (g ` (f -` {f x} \<inter> A))) = g x"
1292     by (rule theI2[of _ "g x"])
1293       (insert `x \<in> A`, auto intro!: svi[THEN subvimageD])
1294 qed
1296 lemma subvimage_translator_image:
1297   assumes svi: "subvimage A f g"
1298   shows "\<exists>h. h ` f ` A = g ` A"
1299 proof -
1300   from subvimage_translator[OF svi]
1301   obtain h where "\<And>x. x \<in> A \<Longrightarrow> h (f x) = g x" by auto
1302   thus ?thesis
1303     by (auto intro!: exI[of _ h]
1304       simp: image_compose[symmetric] comp_def cong: image_cong)
1305 qed
1307 lemma subvimage_finite:
1308   assumes svi: "subvimage A f g" and fin: "finite (f`A)"
1309   shows "finite (g`A)"
1310 proof -
1311   from subvimage_translator_image[OF svi]
1312   obtain h where "g`A = h`f`A" by fastforce
1313   with fin show "finite (g`A)" by simp
1314 qed
1316 lemma subvimage_disj:
1317   assumes svi: "subvimage A f g"
1318   shows "f -` {x} \<inter> A \<subseteq> g -` {y} \<inter> A \<or>
1319       f -` {x} \<inter> g -` {y} \<inter> A = {}" (is "?sub \<or> ?dist")
1320 proof (rule disjCI)
1321   assume "\<not> ?dist"
1322   then obtain z where "z \<in> A" and "x = f z" and "y = g z" by auto
1323   thus "?sub" using svi unfolding subvimage_def by auto
1324 qed
1326 lemma setsum_image_split:
1327   assumes svi: "subvimage A f g" and fin: "finite (f ` A)"
1328   shows "(\<Sum>x\<in>f`A. h x) = (\<Sum>y\<in>g`A. \<Sum>x\<in>f`(g -` {y} \<inter> A). h x)"
1329     (is "?lhs = ?rhs")
1330 proof -
1331   have "f ` A =
1332       snd ` (SIGMA x : g ` A. f ` (g -` {x} \<inter> A))"
1333       (is "_ = snd ` ?SIGMA")
1334     unfolding image_split_eq_Sigma[symmetric]
1335     by (simp add: image_compose[symmetric] comp_def)
1336   moreover
1337   have snd_inj: "inj_on snd ?SIGMA"
1338     unfolding image_split_eq_Sigma[symmetric]
1339     by (auto intro!: inj_onI subvimageD[OF svi])
1340   ultimately
1341   have "(\<Sum>x\<in>f`A. h x) = (\<Sum>(x,y)\<in>?SIGMA. h y)"
1342     by (auto simp: setsum_reindex intro: setsum_cong)
1343   also have "... = ?rhs"
1344     using subvimage_finite[OF svi fin] fin
1345     apply (subst setsum_Sigma[symmetric])
1346     by (auto intro!: finite_subset[of _ "f`A"])
1347   finally show ?thesis .
1348 qed
1350 lemma (in information_space) entropy_partition:
1351   assumes sf: "simple_function M X" "simple_function M P"
1352   assumes svi: "subvimage (space M) X P"
1353   shows "\<H>(X) = \<H>(P) + \<H>(X|P)"
1354 proof -
1355   let ?XP = "\<lambda>x p. joint_distribution X P {(x, p)}"
1356   let ?X = "\<lambda>x. distribution X {x}"
1357   let ?P = "\<lambda>p. distribution P {p}"
1358   note fX = sf(1)[THEN simple_function_imp_finite_random_variable]
1359   note fP = sf(2)[THEN simple_function_imp_finite_random_variable]
1360   note finite_distribution_order[OF fX fP, simp]
1361   have "(\<Sum>x\<in>X ` space M. ?X x * log b (?X x)) =
1362     (\<Sum>y\<in>P `space M. \<Sum>x\<in>X ` space M. ?XP x y * log b (?XP x y))"
1363   proof (subst setsum_image_split[OF svi],
1364       safe intro!: setsum_mono_zero_cong_left imageI)
1365     show "finite (X ` space M)" "finite (X ` space M)" "finite (P ` space M)"
1366       using sf unfolding simple_function_def by auto
1367   next
1368     fix p x assume in_space: "p \<in> space M" "x \<in> space M"
1369     assume "?XP (X x) (P p) * log b (?XP (X x) (P p)) \<noteq> 0"
1370     hence "(\<lambda>x. (X x, P x)) -` {(X x, P p)} \<inter> space M \<noteq> {}" by (auto simp: distribution_def)
1371     with svi[unfolded subvimage_def, rule_format, OF `x \<in> space M`]
1372     show "x \<in> P -` {P p}" by auto
1373   next
1374     fix p x assume in_space: "p \<in> space M" "x \<in> space M"
1375     assume "P x = P p"
1376     from this[symmetric] svi[unfolded subvimage_def, rule_format, OF `x \<in> space M`]
1377     have "X -` {X x} \<inter> space M \<subseteq> P -` {P p} \<inter> space M"
1378       by auto
1379     hence "(\<lambda>x. (X x, P x)) -` {(X x, P p)} \<inter> space M = X -` {X x} \<inter> space M"
1380       by auto
1381     thus "?X (X x) * log b (?X (X x)) = ?XP (X x) (P p) * log b (?XP (X x) (P p))"
1382       by (auto simp: distribution_def)
1383   qed
1384   moreover have "\<And>x y. ?XP x y * log b (?XP x y / ?P y) =
1385       ?XP x y * log b (?XP x y) - ?XP x y * log b (?P y)"
1386     by (auto simp add: log_simps zero_less_mult_iff field_simps)
1387   ultimately show ?thesis
1388     unfolding sf[THEN entropy_eq] conditional_entropy_eq[OF sf]
1389     using setsum_joint_distribution_singleton[OF fX fP]
1390     by (simp add: setsum_cartesian_product' setsum_subtractf setsum_distribution
1391       setsum_left_distrib[symmetric] setsum_commute[where B="P`space M"])
1392 qed
1394 corollary (in information_space) entropy_data_processing:
1395   assumes X: "simple_function M X" shows "\<H>(f \<circ> X) \<le> \<H>(X)"
1396 proof -
1397   note X
1398   moreover have fX: "simple_function M (f \<circ> X)" using X by auto
1399   moreover have "subvimage (space M) X (f \<circ> X)" by auto
1400   ultimately have "\<H>(X) = \<H>(f\<circ>X) + \<H>(X|f\<circ>X)" by (rule entropy_partition)
1401   then show "\<H>(f \<circ> X) \<le> \<H>(X)"
1402     by (auto intro: conditional_entropy_positive[OF X fX])
1403 qed
1405 corollary (in information_space) entropy_of_inj:
1406   assumes X: "simple_function M X" and inj: "inj_on f (X`space M)"
1407   shows "\<H>(f \<circ> X) = \<H>(X)"
1408 proof (rule antisym)
1409   show "\<H>(f \<circ> X) \<le> \<H>(X)" using entropy_data_processing[OF X] .
1410 next
1411   have sf: "simple_function M (f \<circ> X)"
1412     using X by auto
1413   have "\<H>(X) = \<H>(the_inv_into (X`space M) f \<circ> (f \<circ> X))"
1414     by (auto intro!: mutual_information_cong simp: entropy_def the_inv_into_f_f[OF inj])
1415   also have "... \<le> \<H>(f \<circ> X)"
1416     using entropy_data_processing[OF sf] .
1417   finally show "\<H>(X) \<le> \<H>(f \<circ> X)" .
1418 qed
1420 end