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ɹ135*.&4ࣾ಺ษڧձ 8PSEMFΛ༡ΜͰ ৘ใՊֶΛษڧ͢Δ

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ΞδΣϯμ ࣗݾ঺հ 8PSEMFͱ͸ ස౓෼ੳ Τϯτϩϐʔ ·ͱΊ

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ࣗݾ঺հ ໊લɿߐؒ༸ฏ ৬ۀɿόοΫΤϯυΤϯδχΞ (JU)VCNFJIFJ 5XJUUFSBQQF@T

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8PSEMFͱ͸

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8PSEMFͰ༡Μͩ͜ͱ͕͋Δਓ🙋

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8PSEMFͱ͸ ̑จࣈͷ୯ޠΛ౰ͯΔήʔϜ จࣈͱҐஔ͕౰͍ͨͬͯͨΒ🟩 จࣈ͕౰͍ͨͬͯͨΒ🟨 จࣈ͕౴͑ʹؚ·Εͳ͍৔߹͸⬜ ௅ઓ͸̒ճ·Ͱ

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ྫ͑͹ɾɾɾ ਖ਼ղ͕803-%ͷ৔߹ 41&",ͱೖྗ͢Δͱ⬜⬜⬜⬜⬜͕ฦ͞ΕΔ 108&3ͱೖྗ͢Δͱ⬜🟩🟨⬜🟨͕ฦ͞ΕΔ 80345ͱೖྗ͢Δͱ🟩🟩🟩⬜⬜͕ฦ͞ΕΔ 803-%ͱೖྗ͢Δͱ🟩🟩🟩🟩🟩͕ฦ͞ΕͯɺήʔϜʹউར

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ࠓ೔͸͜ͷήʔϜͰ༡ͼͳ͕Β ৘ใՊֶ $4 ͷษڧΛ͠·͢

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ΞδΣϯμ ࣗݾ঺հ 8PSEMFͱ͸ ස౓෼ੳ Τϯτϩϐʔ ·ͱΊ ࠓ೔ֶͿࣄ

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ͬͦ͘͞༡ΜͰߦ͖·͠ΐ͏

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ॳखɺԿʹ͠·͔͢ʁ

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🟩🟩🟩🟩🟩ˡউͪ ⬜🟩🟩🟩🟩ˡ̐จࣈ౰͍ͨͬͯΔɺઌ಄̍จࣈ͕ҧ͏ ⬜🟨⬜🟨⬜ˡ̎จࣈ౰͍ͨͬͯΔ ⬜⬜⬜⬜⬜ˡશવΘ͔Βͳ͍ʁ

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🟩ͱ🟨͕དྷΔͱخ͍͠

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ͳΔ΂͘ਖ਼ղͷ̑จࣈʹ ࢖ΘΕ͍ͯΔจࣈΛೖྗ͍ͨ͠

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ස౓෼ੳ

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୯Ұஔ׵҉߸ͷ࿩Λ͠·͢ ͍͖ͳΓͰ͕͢ʜ

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୯Ұஔ׵҉߸ ͭͷจࣈʹରͯ͠ɺৗʹಉ͡จࣈʹ ม׵͞ΕΔΑ͏ͳ҉߸ γʔβʔ҉߸ɺ305ͳͲ͕༗໊ γʔβʔ҉߸

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ྫʢ305ʣ ΞϧϑΝϕοτΛจࣈγϑτͤͨ͞୯Ұஔ׵҉߸ ฏจ ҉߸จ

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ྫʢ305ʣ ΞϧϑΝϕοτΛจࣈγϑτͤͨ͞୯Ұஔ׵҉߸ ҉߸จ 🤔

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͜Εɺ࣮͸؆୯ʹղ͚·͢

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ม׵લ͸zFz͕ଟ͘ɺม׵ޙ͸zSz͕ଟ͍ ฏจ ҉߸จ

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ม׵લ͸zWz͕গͳ͘ɺม׵ޙ͸zJz͕গͳ͍ ฏจ ҉߸จ

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จࣈͷग़ݱස౓͸ภΓ͕͋Γͦ͏

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จষதͷจࣈͷग़ݱස౓ ฏจ ҉߸จ

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୯Ұஔ׵҉߸͸͜͏΍ͬͯղ͘ ฏจ Ұൠతͳจষͷจࣈͷग़ݱස౓ͱൺֱ ҉߸จ ҉߸จͷจࣈͷग़ݱස౓Λ෼ੳ

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8PSEMF΋ಉ͡ΞϓϩʔνͰ ղ͚ͳ͍Ζ͏͔ʁ

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ස౓෼ੳGPS8PSEMF ొ࿥͞Ε͍ͯΔ̑จࣈͷӳ୯ޠ͸ ݸ͋Δ ͔͠͠ɺ࣮ࡍʹXPSEMFͰ౴͑ͱͯ͠࢖ΘΕΔ୯ޠ͸ ݸ ϫʔυϦετ͸(JU)VC͔ΒऔಘՄೳ ˠΑ͠ɺ෼ੳ͠Α͏ ˞͜Ε͸ެࣜͷ΋ͷͰ͸ͳ͍Ͱ͢

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୯ޠʹจࣈ͕̍จࣈҎ্ग़ݱ͢Δස౓ ΍͸ΓF͕ڧ͍ʢ໿ͷ ୯ޠʹؚ·ΕΔʣ ࣍఺ͰB΍Sʢ໿ʣ J M O P T UͳͲ͸Α͘ग़ ݱ͢Δʢ໿ʣ K R Y [͸΄ͱΜͲग़ݱ͠ ͳ͍ʢ໿ʣ

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ͳΔ΂͘ਖ਼ղͷ̑จࣈʹ ࢖ΘΕ͍ͯΔจࣈΛೖྗ͍ͨ͠ ໋୊

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ग़ݱස౓ͷߴ͍F B SΛඞͣ࢖͏ Ճ͑ͯJ M O P T UΛ࢖ͬͯ୯ޠ Λ࡞Δ ΍Δ͜ͱ

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୯ޠͷީิ w FBSO w FBSMT w FBSOT w FBSOU w FBSTU w MFBST w OFBST w TUFBS w UFBST

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୯ޠͷީิ w FBSO w FBSMT w FBSOT w FBSOU w FBSTU w MFBST w OFBST w TUFBS w UFBST ˢࠓճ͸͜ΕΛ࠾༻

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ॳख5&"34ͰຒΊͯߦ͘

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ྑ͍ॳख͕ଧͯͨ🙌

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࣍ͷखɺԿʹ͠·͔͢ʁ

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·ͣ͸۪௚ʹߟ͑ͯΈΔ

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ࠓ෼͔͍ͬͯΔࣄ w "͸จࣈͱҐஔ͕߹͍ͬͯΔ w & 4͸Ͳ͔͜Ͱ࢖ΘΕ͍ͯΔ w 5 3͸࢖ΘΕͳ͍ ɹɹˠ͜ͷ৘ใ͔Β৚݅Λ࡞Δ

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ਖ਼ղͰ͋Δ৚݅ w ̏จࣈ໨͸" w ̎จࣈ໨Ҏ֎ʹ&ؚ͕·ΕΔ w ̑จࣈ໨Ҏ֎ʹ4ؚ͕·ΕΔ w 5 3͸จࣈʹؚ·Εͳ͍

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୯ޠͷީิʢ̍̕௨Γʣ w BCBTF w DFBTF w DIBTF w MFBTF w MFBTI w QIBTF w TDBMF w TIBEF w TIBLF w TIBMF w TIBNF w TIBQF w TIBWF w TMBWF w TOBLF w TQBDF w TQBEF w TVBWF w VTBHF

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ස౓෼ੳΑΓJ M O Pؚ͕·ΕΔจࣈ w BCBTF w DFBTF w DIBTF w MFBTF w MFBTI w QIBTF w TDBMF w TIBEF w TIBLF w TIBMF w TIBNF w TIBQF w TIBWF w TMBWF w TOBLF w TQBDF w TQBEF w TVBWF w VTBHF

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ස౓෼ੳΑΓJ M O Pؚ͕·ΕΔจࣈ w BCBTF w DFBTF w DIBTF w MFBTF w MFBTI w QIBTF w TDBMF w TIBEF w TIBLF w TIBMF w TIBNF w TIBQF w TIBWF w TMBWF w TOBLF w TQBDF w TQBEF w TVBWF w VTBHF ˣࠓճ͸͜ΕΛ࠾༻

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̎ख໨4$"-&ͰຒΊͯߦ͘

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🟩͕૿͑ͨ🙌

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ߋʹ୯ޠͷީิΛߜΔʢ̕௨Γʣ w TIBEF w TIBLF w TIBNF w TIBQF w TIBWF w TOBLF w TQBDF w TQBEF w TVBWF

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ߋʹ୯ޠͷީิΛߜΔʢ̕௨Γʣ w TIBEF w TIBLF w TIBNF w TIBQF w TIBWF w TOBLF w TQBDF w TQBEF w TVBWF ˣࠓճ͸͜ΕΛ࠾༻

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̏ख໨4)"%&ͰຒΊͯߦ͘

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🟩͕૿͑ͨ🙌

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ߋʹ୯ޠͷީิΛߜΔʢ̐௨Γʣ w TIBLF w TIBNF w TIBQF w TIBWF ˡ࣍͸͜ΕΛ࠾༻

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̐ख໨4)".&ͰຒΊͯߦ͘

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͸ͣΕͨ😥

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ߋʹ୯ޠͷީิΛߜΔʢ̏௨Γʣ w TIBLF w TIBQF w TIBWF ˡ࣍͸͜ΕΛ࠾༻

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̑ख໨4)"7&ͰຒΊͯߦ͘

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͸ͣΕͨ😰

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ߋʹ୯ޠͷީิΛߜΔʢ̎௨Γʣ w TIBLF w TIBQF ˡ࠷ޙ͸͜ΕΛ࠾༻

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ϥετ4)"1&ͰຒΊͯߦ͘

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͸ͣΕͨ😱

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ͳͥͳͷ͔

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ৼΓฦͬͯΈΔ ̍ख໨͸ ௨Γ ̎ख໨͸௨Γ ̏ख໨͸௨Γ ̐ख໨͸௨Γ ̑ख໨͸௨Γ ̒ख໨͸௨Γ

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ৼΓฦͬͯΈΔ ̍ख໨͸ ௨Γ ̎ख໨͸௨Γ ̏ख໨͸௨Γ ̐ख໨͸௨Γ ̑ख໨͸௨Γ ̒ख໨͸௨Γ ·Ͱ࡟ݮɺྑ͍ख

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ৼΓฦͬͯΈΔ ̍ख໨͸ ௨Γ ̎ख໨͸௨Γ ̏ख໨͸௨Γ ̐ख໨͸௨Γ ̑ख໨͸௨Γ ̒ख໨͸௨Γ ·Ͱ࡟ݮɺඍົ

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ৼΓฦͬͯΈΔ ̍ख໨͸ ௨Γ ̎ख໨͸௨Γ ̏ख໨͸௨Γ ̐ख໨͸௨Γ ̑ख໨͸௨Γ ̒ख໨͸௨Γ ·Ͱ࡟ݮɺඍົ

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ৼΓฦͬͯΈΔ ̍ख໨͸ ௨Γ ̎ख໨͸௨Γ ̏ख໨͸௨Γ ̐ख໨͸௨Γ ̑ख໨͸௨Γ ̒ख໨͸௨Γ ͻͲ͍ ͻͲ͍

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ͳͥͳͷ͔

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ॳखͰ෼͔ͬͨࣄ w "͸จࣈͱҐஔ͕߹͍ͬͯΔ w & 4͸Ͳ͔͜Ͱ࢖ΘΕ͍ͯΔ w 5 3͸࢖ΘΕͳ͍

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ॳखͰ͸ଟ͘ͷ৘ใ͕ಘΒΕͨ

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௨Γ͔Β௨Γ·Ͱ ࡟ݮͨ͠

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̐ख໨Ͱ෼͔ͬͨࣄ w .͸࢖ΘΕͳ͍

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̐ख໨͸গ͠ͷࣄ͔͠ ෼͔Βͳ͔ͬͨ

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௨Γ͔Β௨Γʹͳ͚ͬͨͩ

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ͭ·ΓɺಘΒΕΔ৘ใ͕ ଟ͍΄͏͕ྑ͍ख

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ಘΒΕΔ৘ใͷྔΛ͸͔Δࣄ͸ ग़དྷͳ͍ͩΖ͏͔ʁ

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Τϯτϩϐʔ

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৘ใྔ ͦͷࣄ৅͕ͲΕ͚ͩ৘ใΛग़͔ͨ͠Λ͸͔Δई౓ ࣄ৅͕ى͜Δ֬཰ ͱ͢Δͱ ৘ใྔ p I(p) = − log p

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؆୯ʹݴ͏ͱɺ௝͍͠ࣄ͸ ಘΒΕΔ৘ใ͕ଟ͍

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৘ใྔͷྫ छྨͷΞϧϑΝϕοτ͔Β̍ͭΛ౰ͯΔ * −log2 1 26

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৘ใྔͷྫ छྨͷΞϧϑΝϕοτ͔Β̍ͭΛ౰ͯΔ * ɹˠ෼͔Γ͖͍ͬͯΔࣄ৅ ɹɹˠಘΒΕΔ৘ใྔ͸̌ −log2 1 1

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৘ใྔΛ࢖͑͹8PSEMF ΋ղ͚ͦ͏ͩʂ

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Ͳ͏΍ͬͯ8PSEMFʹԠ༻͢Δ͔ʁ

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ΤϯτϩϐʔΛ࠷େԽ͢Δ

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Τϯτϩϐʔʢฏۉ৘ใྔʣ ͷ͢΂ͯͷࣄ৅ʹରͯ͠৘ใྔͷฏۉΛͱͬͨ΋ͷ ฏۉ৘ใྔ ɹɹˠ͍ͭ͜Λ࠷େԽ͢Ε͹Αͦ͞͏ X H(X) = − ∑ i pi log pi

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࣮ࡍʹ8PSEMFͰΤϯτϩϐʔ Λ࢖ͬͯΈΔ

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̐ख໨ͷީิ w TIBLF w TIBNF w TIBQF w TIBWF

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4)".&ΛબΜͩ৔߹ w TIBLFɹˠɹ🟩🟩🟩⬜🟩 w TIBNFɹˠɹ🟩🟩🟩🟩🟩 w TIBQFɹˠɹ🟩🟩🟩⬜🟩 w TIBWFɹˠɹ🟩🟩🟩⬜🟩

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4)".&ΛબΜͩ৔߹ w TIBLFɹˠɹ🟩🟩🟩⬜🟩 w TIBNFɹˠɹ🟩🟩🟩🟩🟩 w TIBQFɹˠɹ🟩🟩🟩⬜🟩 w TIBWFɹˠɹ🟩🟩🟩⬜🟩 🟩🟩🟩⬜🟩 🟩🟩🟩🟩🟩 p 3 4 p 1 4

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4)".&ΛબΜͩ৔߹ w TIBLFɹˠɹ🟩🟩🟩⬜🟩 w TIBNFɹˠɹ🟩🟩🟩🟩🟩 w TIBQFɹˠɹ🟩🟩🟩⬜🟩 w TIBWFɹˠɹ🟩🟩🟩⬜🟩 🟩🟩🟩⬜🟩 🟩🟩🟩🟩🟩 p 3 4 p 1 4 4)".& H − 3 4 log2 3 4 − 1 4 log2 1 4 ≒ 0.81

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Τϯτϩϐʔ͕େ͖͍୯ޠ Ͱ͸Ͳ͏ͳΔͷ͔

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4,*.1ΛબΜͩ৔߹ w TIBLFɹˠɹ🟩🟨⬜⬜⬜ w TIBNFɹˠɹ🟩⬜⬜⬜⬜ w TIBQFɹˠɹ🟩⬜⬜⬜🟨 w TIBWFɹˠɹ🟩⬜⬜⬜⬜ ˞༧Ίܭࢉͯ͠ɺ͜ͷ୯ޠΛબΜͰ͍·͢

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4,*.1ΛબΜͩ৔߹ w TIBLFɹˠɹ🟩🟨⬜⬜⬜ w TIBNFɹˠɹ🟩⬜⬜⬜⬜ w TIBQFɹˠɹ🟩⬜⬜⬜🟨 w TIBWFɹˠɹ🟩⬜⬜⬜⬜ 🟩⬜⬜⬜⬜ 🟩🟨⬜⬜⬜ 🟩⬜⬜⬜🟨 p 2 4 p 1 4 p 1 4

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4,*.1ΛબΜͩ৔߹ w TIBLFɹˠɹ🟩🟨⬜⬜⬜ w TIBNFɹˠɹ🟩⬜⬜⬜⬜ w TIBQFɹˠɹ🟩⬜⬜⬜🟨 w TIBWFɹˠɹ🟩⬜⬜⬜⬜ 🟩⬜⬜⬜⬜ 🟩🟨⬜⬜⬜ 🟩⬜⬜⬜🟨 p 2 4 p 1 4 p 1 4 4,*.1 H − 2 4 log2 2 4 − 1 4 log2 1 4 − 1 4 log2 1 4 ≒ 1.5

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4)".&ΑΓ΋4,*.1ͷํ͕ ಘΒΕΔ৘ใ͕ଟ͍

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4,*.1ͰຒΊͯߦ͘

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͸ͣΕͨ😥

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Ͱ΋৘ใྔ͸େ͖͍😄

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̑ख໨ͷ୯ޠͷީิ͸ʜ w TIBLF ɹˠɹ̍௨ΓʹߜΒΕͨʂ

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উͬͨʂʂ

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Τϯτϩϐʔ͛͢ʔ

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ͦ͏͍͑͹

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🟩ͱ🟨͸ݮͬͯྑ͍ͷ͔ʁ 🤔

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🟩ͱ🟨͕དྷΔͱخ͍͠

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🟩ͱ🟨͕དྷΔͱخ͍͠ ͜Ε͸ؒҧ͑

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ਖ਼͘͠͸ Τϯτϩϐʔ͕࠷େԽ͞ΕΔ ͱخ͍͠

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ͪͳΈʹ Τϯτϩϐʔͷ࠷େԽ͸ॳखͰ΋దԠग़དྷΔ ɹˠɹιϧόʔ࡞ΔͳΒස౓෼ੳΑΓΤϯτϩϐʔ 4"-&5͕࠷ڧͱ͍͏ͷ͕௨આ

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·ͱΊ

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ࠓ೔ֶΜͩ͜ͱ ୯Ұஔ׵҉߸͸ස౓෼ੳͰղ͚Δ ࣗવݴޠͷจࣈʹ͸ग़ݱස౓ʹภΓ͕͋Δ ৘ใྔͱ͸ͦͷࣄ৅͕ͲΕ͚ͩ৘ใΛग़͔ͨ͠Λ͸͔Δई౓ ৘ใྔ Τϯτϩϐʔͱ͸͢΂ͯͷࣄ৅ͷ৘ใྔͷฏۉ ฏۉ৘ใྔ I(p) = − log p H(X) = − ∑ i pi log pi

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ࢀߟ w IUUQTYDMPDIFIBUFCMPKQFOUSZ w IUUQTTPOPSPVTDIPDPMBUFDPNOPUFTJOEFYQIQ UJUMF5IF@CFTU@TUSBUFHJFT@GPS@8PSEMF w IUUQTHJUIVCDPNBMFYXPSEMF w IUUQTRJJUBDPNNBTBLB@QSPHSBNNJOHJUFNT BGFBBFECG w IUUQTGSFTINBOEFWXPSEMFMFBEFSCPBSE