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ディリクレ過程の定義

 ディリクレ過程の定義

seetheworld1992

December 20, 2016
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    σΟϦΫϨաఔͷఆٛ
    @seetheworld1992
    2016/12/21
    @seetheworld1992 σΟϦΫϨաఔͷఆٛ 2016/12/21 1 / 7

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  2. ࢀߟจݙ
    ຊࢿྉΛ࡞੒͢Δʹ͋ͨΓɼҎԼͷจݙΛࢀߟʹͨ͠ɽ
    Antoniak, C. (1974), Mixtures of Dirichlet processes with applications to
    Bayesian nonparametric problems, The Annals of Statistics, Vol.2, No.6,
    pp.1152-1174.
    Ferguson, T. (1973), Bayesian analysis of some nonparametric problems,
    The Annals of Statistics, Vol.1, No.2, pp.209-230.
    @seetheworld1992 σΟϦΫϨաఔͷఆٛ 2016/12/21 2 / 7

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  3. ४උ:ΨϯϚ෼෍
    Ferguron(1973) ʹΑΔΨϯϚ෼෍ͷఆٛΛࣔ͢ɽ
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    Definition 1.
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    ܗঢ়฼਺ α ≥ 0ɼई౓฼਺ β > 0 ͷΨϯϚ෼෍Λ G (α, β) ͱ͢Δɽα = 0
    ͷͱ͖ɼ͜ͷ෼෍͸ 0 ʹୀԽ͢Δ; α > 0 ͷͱ͖ɼ͜ͷ෼෍͸࣮਺্ͷϧ
    ϕʔάଌ౓ʹؔ͢ΔҎԼͷີ౓Λ΋ͭɽ
    f(z|α, β) =
    1
    Γ(α)βα
    e−z/βzα−1I(0,∞)
    (z)
    Γ(α) =


    0
    xα−1e−xdx
    IS(z) =
    {
    1 (z ∈ S)
    0 (z /
    ∈ S)
    @seetheworld1992 σΟϦΫϨաఔͷఆٛ 2016/12/21 3 / 7

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  4. ४උ:σΟϦΫϨ෼෍ (1/2)
    Ferguron(1973) ʹΑΔσΟϦΫϨ෼෍ͷఆٛΛࣔ͢ɽ
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    Definition 2.
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    αj ≥ 0 for all j, αj > 0 for some j, j = 1, · · · , k ͱ͠ɼZj ∈ G (αj, 1) Λ
    ຬͨ͢ Z1, Z2, · · · , Zk
    Λޓ͍ʹಠཱͳ֬཰ม਺ (ͷྻ) ͱ͢Δɽ
    (Y1, · · · , Yk) ͷ෼෍Λύϥϝʔλ (α1, · · · , αk) ͷσΟϦΫϨ෼෍ͱ͍͏ɽ
    ͜͜Ͱ
    Yj = Zj/
    k

    i=1
    Zi, for j = 1, · · · , k
    Ͱ͋Δɽ·ͨσΟϦΫϨ෼෍Λ D(α1, · · · , αk) ͱॻ͘ɽ
    Y1 + · · · + Yk = 1 Ͱ͋ΔͷͰɼ͜ͷ෼෍͸ k ࣍ݩϧϕʔάଌ౓ʹؔͯ͠ৗ
    ʹಛҟͰ͋Δ 1ɽ·ͨ೚ҙͷ αj = 0 ͳΒ͹ɼରԠ͢Δ Yj
    ͸ 0 ʹୀԽ͢Δɽ
    1۩ମྫͱͯ͠ k = 2 ͷͱ͖Λߟ͑Δͱɼີ౓ؔ਺͸௚ઢ Y1
    + Y2
    = 1 ্ʹ஋ΛͱΔɽ
    ೋ࣍ݩϧϕʔάଌ౓ʹؔͯ͠ಛҟͱ͸ɼ͜ͷͱ͖ʹີ౓Λੵ෼ͯ֬͠཰ΛٻΊΑ͏ͱ͠
    ͯ΋ɼີ౓ؔ਺ͷԼଆͷମੵΛٻΊΒΕͳ͍͜ͱʹରԠ͢Δɽ
    @seetheworld1992 σΟϦΫϨաఔͷఆٛ 2016/12/21 4 / 7

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  5. ४උ:σΟϦΫϨ෼෍ (2/2)
    ҰํͰɼαj > 0 for all j ͳΒ͹ɼ(Y1, · · · , Yk−1) ͷ෼෍͸ k − 1 ࣍ݩϧ
    ϕʔάଌ౓ʹରͯ͠ઈର࿈ଓͰҎԼͷີ౓Λ΋ͭ 2ɽ
    f(y1, · · · , yk−1|α1, · · · , αk)
    =
    Γ(α1 + · · · + αk)
    Γ(α1) · · · Γ(αk)
    (
    k−1

    j=1
    yαj−1
    j
    )(1 −
    k−1

    j=1
    yj)αk−1IS(y1, · · · , yk−1)
    ͜͜Ͱ S ͸ k − 1 ࣍ݩͷ୯ମ (simplex) Ͱ͋Δɽ
    S = {(y1, · · · , yk−1) : yj ≥ 0,
    k−1

    j=1
    yj ≤ 1}
    2k = 2 ͱ͢Δͱɼϕʔλ෼෍ De(α1
    , α2
    ) ͱͳΔɽ
    @seetheworld1992 σΟϦΫϨաఔͷఆٛ 2016/12/21 5 / 7

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  6. ४උ:Մଌ෼ׂ
    Ferguron(1973) ʹҎԼͷهड़͕͋Δ 3ɽͨͩ͠ɼΘ Λू߹ɼA Λ Θ ͷ෦
    ෼ू߹Λཁૉͱ͢Δ σ ू߹ମͱ͢Δɽ
    ʢ͜͜Ͱ͸ɼBi ∈ A for all i, Bi ∩ Bj = ϕ for all i ̸= j, ∪k
    j=1
    Bj = Θ ͳ
    Β͹ (B1, · · · , Bk) ΛՄଌ෼ׂͱ͍͏͜ͱʹ͢Δʣ
    3Ferguron(1973) Ͱ͸ɼΘ Ͱ͸ͳ͘ X Λ࢖༻͍ͯ͠Δɽ
    @seetheworld1992 σΟϦΫϨաఔͷఆٛ 2016/12/21 6 / 7

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  7. σΟϦΫϨաఔ
    Antoniak(1974) ʹ͸σΟϦΫϨաఔͷఆٛͱͯ͠ҎԼ͕ࣔ͞Ε͍ͯΔ 4ɽ
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    Definition 3.
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    Θ Λू߹ɼA Λ Θ ͷ෦෼ू߹Λཁૉͱ͢Δ σ ू߹ମͱ͢Δɽα Λ
    (Θ, A ) ্ͷۭͰͳ͍༗ݶଌ౓ (ඇෛͰ༗ݶՃ๏త) ͱ͢Δɽ͢΂ͯͷ
    k = 1, 2, · · · ͱ Θ ͷ͋ΒΏΔՄଌ෼ׂ (B1, · · · , Bk) ʹ͍ͭͯɼ
    (P(B1), · · · , P(Bk)) ͷಉ࣌෼෍͕ύϥϝʔλ (α(B1), · · · , α(Bk)) Λ΋ͭ
    σΟϦΫϨɼ͢ͳΘͪ (P(B1), · · · , P(Bk)) ∈ D(α(B1), · · · , α(Bk)) ͳΒ
    ͹ɼ(Θ, A ) ্ͷϥϯμϜͳ֬཰ଌ౓ P Λ (Θ, A ) ্ͷύϥϝʔλ α Λ΋
    ͭσΟϦΫϨաఔͱ͍͍ɼP ∈ D(α) ͱ͔͘ɽ(α(Bi) = 0 ͷͱ͖ɼ
    P(Bi) = 0 ͕֬཰ 1 Ͱ੒Γཱͭɽ)
    ·ͨɼAntoniak(1974) ʹ͸࣮࣭తʹҎԼΛҙຯ͢Δهड़΋͋Δɽ
    ʮ͜ͷఆٛ͸ ([0, 1]A , BFA ) ্ͷ֬཰ P ͷଘࡏΛຬ͍ͨͯ͠Δɽ͜͜
    Ͱ [0, 1]A ͸ A ͔Β [0, 1] ΁ͷશͯͷؔ਺ͷۭؒΛද͢ͱ͠ɼBFA ͸γ
    Ϧϯμʔू߹ʹΑΓੜ੒͞ΕΔ σ ू߹ମΛද͢ͱ͢Δɽ
    ʯ
    4Ferguron(1973) ʹΑΓఆٛ͞Εͨ΋ͷͰ͋Δ͕ɼ͜͜Ͱ͸͜ͷهड़Λ༻͍Δɽ
    @seetheworld1992 σΟϦΫϨաఔͷఆٛ 2016/12/21 7 / 7

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