Jensenの不等式

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1. Jensen ͷෆ౳ࣜ @kogetsu0728 June 26, 2025 @kogetsu0728 Jensen ͷෆ౳ࣜ June 26, 2025 1 / 9
2. ໨࣍ 1 Ԝؔ਺ 2 Jensen ͷෆ౳ࣜ @kogetsu0728 Jensen ͷෆ౳ࣜ June 26, 2025 2 / 9
3. Ԝؔ਺ͷఆٛ ্ʹತͳؔ਺ΛԜؔ਺ (concave function) ͱ͍͏ [1]ɽ Definition ۠ؒͰఆٛ͞Εͨؔ਺ f : R ⊇ I → R ͕Ԝؔ਺Ͱ͋Δͱ͸ɼ ∀x, y ∈ I, ∀λ ∈ [0, 1], f ((1 − λ)x + λy) ≥ (1 − λ)f (x) + λf (y). Λຬͨ͢͜ͱΛ͍͏ɽ ·ͨɼ࣍ͷඞཁे෼৚͕݅஌ΒΕ͍ͯΔɽূ໌͸লུ͢Δɽ Theorem f ͕ೋ֊ඍ෼ՄೳͳΒ͹ɼ f ͕Ԝؔ਺ ⇐⇒ f ′′(x) ≤ 0. @kogetsu0728 Jensen ͷෆ౳ࣜ June 26, 2025 3 / 9
4. Ԝؔ਺ͷ൑ఆʢఆ͔ٛΒʣ Theorem log2x ͸Ԝؔ਺Ͱ͋Δɽ Proof. x, y > 0ɼλ ∈ [0, 1] ͱ͢Δɽ ॏΈ෇͖૬Ճ૬৐ฏۉͷෆ౳ࣜ (weighted AM-GM inequality) ͔Βɼ (1 − λ)x + λy ≥ x1−λyλ. ྆ลͷର਺ΛऔΔͱɼ log2 ((1 − λ)x + λy) ≥ log2 (x1−λyλ) = (1 − λ) log2 x + λ log2 y. @kogetsu0728 Jensen ͷෆ౳ࣜ June 26, 2025 4 / 9
5. Ԝؔ਺ͷ൑ఆʢೋ֊ඍ෼͔Βʣ Theorem log2x ͸Ԝؔ਺Ͱ͋Δɽ Proof. (log2 x)′ = 1 (log 2)x , (log2 x)′′ = − 1 (log 2)x2 ≤ 0. @kogetsu0728 Jensen ͷෆ౳ࣜ June 26, 2025 5 / 9
6. ໨࣍ 1 Ԝؔ਺ 2 Jensen ͷෆ౳ࣜ @kogetsu0728 Jensen ͷෆ౳ࣜ June 26, 2025 6 / 9
7. Jensen ͷෆ౳ࣜ ҎԼͷෆ౳ࣜΛ Jensen ͷෆ౳ࣜ (Jensen’s inequality) ͱ͍͏ [2]ɽ Theorem f : R ⊇ I → R ΛԜؔ਺ͱ͢Δɽ·ͨɼp1, p2, . . . , pn Λ p1 + p2 + · · · + pn = 1 Λຬͨ͢ඇෛ ࣮਺ͱ͠ɼx1, x2, . . . , xn ∈ I ͱ͢ΔͱɼҎԼͷෆ౳͕ࣜ੒Γཱͭɽ n ∑ i=1 pi f (xi ) ≤ f ( n ∑ i=1 pi xi ). ਺ֶతؼೲ๏Ͱূ໌Ͱ͖Δɽಛʹɼn = 2 ͷͱ͖͸Ԝؔ਺ͷఆٛΑΓ໌Β͔ɽ ූ߸௕Λදࣜ͢͸ log2x ͳͲԜؔ਺͕ଟ͘ɼ্քΛٻΊΔͷʹ Jensen ͷෆ౳͕ࣜ໾ཱͭɽ @kogetsu0728 Jensen ͷෆ౳ࣜ June 26, 2025 7 / 9
8. ෆ౳ࣜͷূ໌ ฏۉ৘ใྔ (entropy) ʹؔ͢ΔҎԼͷෆ౳ࣜΛ Jensen ͷෆ౳ࣜΛ༻͍ͯূ໌͢Δɽ Theorem X Λ M ݸͷࣄ৅ͷੜى֬཰ p1, p2, . . . , pM ͔ΒͳΔ֬཰ࣄ৅ܥͱ͢Δɽ͜ͷͱ͖ɼฏۉ৘ใ ྔ H(X) ʹ͍ͭͯ࣍ͷෆ౳͕ࣜ੒Γཱͭɽ H(X) ≤ log2 M. Proof. log2 x ͸Ԝؔ਺͔ͩΒɼJensen ͷෆ౳ࣜΑΓɼ H(X) = M ∑ i=1 pi log2 1 pi ≤ log2 ( M ∑ i=1 pi 1 pi ) = log2 M. @kogetsu0728 Jensen ͷෆ౳ࣜ June 26, 2025 8 / 9
9. ࢀߟจݙ [1] ߴߍ਺ֶͷඒ͍͠෺ޠ, “্ʹತɼԼʹತͳؔ਺ͱೋ֊ඍ෼”, ֶͼ Times, https://manabitimes.jp/math/927, (ࢀর: 2025-06-26). [2] ఆ݉ ๜඙, “؆ܿσʔλߏ଄”, ڞཱग़൛, 2018. @kogetsu0728 Jensen ͷෆ౳ࣜ June 26, 2025 9 / 9