GM3D/須山敦史ベイズ深層学習 6.3 生成ネットワークの構造学習.ipynb

## 須山敦史ベイズ深層学習 6.3 生成ネットワークの構造学習.ipynb
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   "source": "## 参考文献\n\n[1] Thomas L. Griffiths, Zoubin Ghahramani. The Indian Buffet Process: An Introduction and Review. Journal of Machine Learning Research 12 (2011) 1185-1224\n\n[2] Ryan P. Adams, Hanna M. Wallach, Zoubin Ghahramani. LEARNING THE STRUCTURE OF DEEP SPARSE GRAPHICAL MODELS. arXiv:1001.0160v2"
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   "source": "## 6.3 生成ネットワークの構造学習\n\nこの節で扱う問題は、単純に言うと一群の確率変数 ${\\bf x} = (x_1, \\ldots x_H)$と別の確率変数${\\bf y} = (y_1, \\ldots y_N)$があったとき、$y_n$に対して$x_h$が影響を及ぼすかどうかを確率変数$m_{nh} \\in \\{0, 1\\}$で表現することである。$(m_{ij})_{i=1 \\ldots N}^{j=1 \\ldots H}$ をまとめて行列$M$で表現する。また、$y_n$への$x_h$の影響が定量的、つまり実数値で表現される場合は、その影響の強さを$F = (f_{nh})_{n=1 \\ldots N}^{h=1 \\ldots H}$としたとき、$F$を同じサイズの一般に密な実数行列$V$と$M$のアダマール積(成分ごとの積）で$F = M \\odot V$と表して、依存性の有無と強弱とを分離して扱う。このとき、$\\bf{x}$の次元は有限の場合に限定しない、$n = \\infty$の場合も考える。\n\n後に見るように事後分布の推定にこの枠組みを再帰的に用いた場合は、観測データの次元のみが固定で、${\\bf y}$が潜在変数の場合は${\\bf y}$の次元にも制限をつけずに扱うことができる。\n\n最初の問題は、このように列数の上限があらかじめ決まっていないバイナリ行列に対する適切な（性質が比較的良く分かり、推論が実行可能な）事前分布を定義することである。\n\nまず列数Hが確定している場合は、ベータ分布によるhyper priorから生成したパラメータによって列ごとにベルヌーイ分布を作成し、そこから0, 1の列を生成すればよい。"
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   },
   "cell_type": "code",
   "source": "using DataFrames\nusing SpecialFunctions\nusing Distributions\nusing LinearAlgebra\nusing Plots\ngr()",
   "execution_count": 1,
   "outputs": [
    {
     "output_type": "execute_result",
     "execution_count": 1,
     "data": {
      "text/plain": "Plots.GRBackend()"
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   "source": "lof(M) = sortslices(M, dims=2, rev=true)\n\nfunction pprint(M)\n    show(Base.stdout, \"text/plain\", M)\n    println(\"\\n\")\nend\n\nfunction tileplot(M)\n    dmr = 20 #default marker radius\n    R, C = Base.size(M)\n    if R == 0 || C == 0\n        return\n    end\n    xmarkersize = 2 * dmr * C <= 600 ? dmr : (600 / C) / 2\n    ymarkersize = 2 * dmr * R <= 400 ? dmr : (400 / R) / 2\n    mr = min(xmarkersize, ymarkersize)\n    size = (2 * mr * (C + 1), 2 * mr * (R + 1))\n    data1 = [(c-1, R - r) for r in 1:R for c in 1:C if M[r, c] == 1]\n    data0 = [(c-1, R - r) for r in 1:R for c in 1:C if M[r, c] == 0]\n    plot(xlims=(-0.5, C + 0.5), ylims=(-0.5, R - 0.5), label=\"\", size=size, showaxis=false)                              \n    scatter!(data0, xlims=(-0.5, C + 0.5), ylims=(-0.5, R - 0.5), marker=(:square, mr, 0.8, :black, stroke(0, :none)), label=\"\", size=size, showaxis=false)\n    p = scatter!(data1, xlims=(-0.5, C + 0.5), ylims=(-0.5, R - 0.5), marker=(:square, mr, 0.8, :yellow, stroke(0, :none)), label=\"\", size=size, showaxis=false)\n    return p\nend\n",
   "execution_count": 2,
   "outputs": [
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     "output_type": "execute_result",
     "execution_count": 2,
     "data": {
      "text/plain": "tileplot (generic function with 1 method)"
     },
     "metadata": {}
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   "cell_type": "code",
   "source": "function sample_M(α, β, N, H)\n    p_π = Beta(α*β/H, β)\n    π = rand(p_π, H)\n    M = hcat([rand(Bernoulli(π[h]), N, 1) for h in 1:H]...)\n    return M\nend",
   "execution_count": 3,
   "outputs": [
    {
     "output_type": "execute_result",
     "execution_count": 3,
     "data": {
      "text/plain": "sample_M (generic function with 1 method)"
     },
     "metadata": {}
    }
   ]
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   "cell_type": "code",
   "source": "α, β, N, H = 5.0, 5.0, 10, 20\nM = sample_M(α, β, N, H)\ndisplay(tileplot(M))\ndisplay(tileplot(lof(M)))",
   "execution_count": 4,
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   "source": "この分布は$π_h$が与えられれば列毎に独立、列の中では各要素独立なので容易に計算でき、特定の要素のパターンを持つ$M = (m_{nh})$が生成される確率は\n\n\\begin{align}\nP(M) &= \\Pi_{h=1}^H \\int \\{\\prod_{n=1}^N p(m_{nh}|\\pi_h)\\} p(\\pi_h) d\\pi_h\n\\end{align}\n\nであり、これに$p(\\pi_h)$の事前分布ガンマ分布と$p(m_{nh}|\\pi_h)$の具体的な形であるベルヌーイ分布を入れて$\\pi_h$についての積分を実行すると、\n\n\\begin{align}\nP(M) &= \\frac{\\Pi_{h=1}^H \\Gamma(\\alpha \\beta / H + \\beta) \\Gamma(N_h + \\alpha \\beta / H) \\Gamma(N - N_h + \\beta)}\n{\\Gamma(\\alpha \\beta / H) \\Gamma(\\beta) \\Gamma(\\alpha \\beta / H + \\beta + N)}\n\\end{align}\n\nが得られる。ただしここで$N_h$は第h行にある1の総数である。"
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   "source": "ただしこれでは$H \\rightarrow \\infty$の極限で特定の$M$の出現確率は0になるので、あまり有意義ではない。そこで個々の$M$の分布ではなく、列の入れ替えで互いに移り替わることのできる行列同士をひとまとめにした同値類[M]の分布を考えると、これは$H \\rightarrow \\infty$でも有限の値を与える。\n\nこれが意味のある設定であるためには、実際にこの行列をモデルにおいて使用する際に、列の意味付けがあらかじめ固定されていないことが必要である。たとえば後の例でみる生成ネットワークモデルではMの各列は、逐次的に生成されるネットワーク層の間の結合を表すが、これから生成する層について、たとえば1番および2番のノードを使用するネットワークも3番および4番のノードを使用するネットワークも2個のノードがある層という意味しか持たず、ノードの番号自体に固定された意味付けは存在しないので、列の入れ替えに関して同じ同値類に属する2つの行列はどちらも（その層間接続については）同じネットワークアーキテクチャを表現すると考えてよい。代表的な利用例であるPCAやCVQにおいても同様な事情になっており、この同値類に基づく定式化が意味を持つ。\n\n上の$P(M)$は、各列hにおいて決まる量の、全ての列hについての積という形をしているので、明白に列の入れ替えに対して不変である。したがって同値類の出現確率は、単に上の確率に各同値類に含まれる行列$M$の個数をかければよい。ある同値類に含まれる行列の個数は、仮にその同値類に含まれる一つの行列に注目したとき全ての列が異なるならば、同値類の他の行列は列のあらゆる可能な入れ替えで得られるので$H!$個あることになる。だが同じ列が複数あった場合は、それを入れ替えても違う行列は得られないのでその分オーバーカウントであり、あるタイプ$i$の行列が仮に$H_i$個あるのなら、$H!$を$H_i!$で割ってやらなければならない。これを列の\"タイプ\"$i$のすべてについて考慮してやる必要がある。今タイプと呼んだものは、たとえば列を一番上の要素をMSBとして二進整数とみなしてやった値で区別できる。これを[1]では列のヒストリーと呼んでおり、明らかにヒストリー$i$の取りうる値は0から$2^N - 1$である。したがって、ある特定の列入れ替えに関する同値類[M]の出現確率は、\n\n\\begin{align}\nP([M]) &= \\frac{H!}{\\Pi_{i=0}^{2^N - 1} H_i !} P(M)\n\\end{align}\n\nで与えられる。\n\n\nこの量の$H \\rightarrow \\infty$の極限は[1]において(12)-(14)式およびAppendix A. で計算されており、結果は\n\n\\begin{align}\nP([M]) &= \\frac{(\\alpha \\beta)^H_+}{\\Pi_{i=1}^{2^N - 1} H_i !} \\exp(-\\bar{H}_+) \n\\Pi_{h=1}^{H_+} \\frac{\\Gamma(N_h) \\Gamma(N - N_h + \\beta)}{\\Gamma(N + \\beta)}\n\\end{align}\n\nとなる。ここで$H_i$はヒストリー値$i$を持つ列の個数である。あるヒストリー値$i$についてそれに該当する列が存在しない($H_i = 0$)場合も積の中に含まれているが、$0! = 1$により問題は起きない。また$H_+ = \\sum_{i=1}^{2^N - 1} H_i = H - H_0$であり、$\\bar{H}_+ = \\alpha \\sum_{n=1}^N \\frac{\\beta}{n + \\beta - 1}$ である。極限をとった結果では$H_0!$が指数関数に吸収されて最初のファクターは$i$が正のもののみの積になっていることに注意。"
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   "source": "一方で行数（とハイパーパラメータ $\\alpha$、$\\beta$) のみが与えられており列数が固定でない場合は、以下のIndian buffet process によりサンプルを生成することができる。これは列数固定のときとは違い、任意のビットパターンの$M$を生成してくれるわけではないが、列の入れ替えによる同値類レベルで見ると正しく$H$有限の場合の分布の$H \\rightarrow \\infty$での極限を再現する。"
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   "cell_type": "code",
   "source": "function IBP(α, β, N)\n    M = zeros(Int64, N, 0)\n    for n in 1:N\n        for h in 1:size(M)[2]\n            Nh = sum(M[1:n-1, h])\n            p = Nh / (n + β - 1)\n            M[n, h] = rand() < p ? 1 : 0\n        end\n        new_dishes = rand(Poisson(α * β / (n + β - 1)))\n        M_expand = zeros(Int64, N, new_dishes)\n        M_expand[n, :] = ones(Int64, new_dishes)\n        M = hcat(M, M_expand)\n    end\n    return M\nend",
   "execution_count": 5,
   "outputs": [
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     "output_type": "execute_result",
     "execution_count": 5,
     "data": {
      "text/plain": "IBP (generic function with 1 method)"
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   "cell_type": "code",
   "source": "tileplot(IBP(5, 5, 10))",
   "execution_count": 6,
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   ]
  },
  {
   "metadata": {},
   "cell_type": "markdown",
   "source": "次にIBPを$N0$回行い、各行毎に1が何個あるか、その頻度分布をヒストグラムとして見る。一行目はIBP過程の定義から自明にPoisson(α)に従うが、他の行も同じ分布に従うことが見てとれる。"
  },
  {
   "metadata": {
    "trusted": true
   },
   "cell_type": "code",
   "source": "N0 = 100000\nα, β, N = 5.0, 5.0, 10\ndist = zeros(Int64, N, N0)\nfor n in 1:N0\n    M = IBP(α, β, N)\n    for i in 1:N\n        dist[i, n] = sum(M[i,:])\n    end\nend\n\nhistogram(dist[1,:], label=\"guest1\")",
   "execution_count": 7,
   "outputs": [
    {
     "output_type": "execute_result",
     "execution_count": 7,
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   "metadata": {
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   "cell_type": "code",
   "source": "println(\"α = $α, β = $β, N = $N\")\nM = IBP(α, β, N)\n#pprint(M)\ndisplay(tileplot(M))\ndisplay(tileplot(lof(M)))",
   "execution_count": 10,
   "outputs": [
    {
     "output_type": "display_data",
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     "text": "α = 5.0, β = 5.0, N = 10\n",
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   "source": "上で与えたIBPは各行の分布がそれより上の行のサンプル結果に依存するため、各行の1の個数の分布がPoisson(α)に従うことは自明ではない。その証明を以下に書いておく。\n\nまず、行列$M_1$と$M_2$が列の入れ替え、行の入れ替えで互いに移り替われることをそれぞれ$M_1 \\stackrel{c}{\\sim} M_2$、$M_1 \\stackrel{r}{\\sim} M_2$と表記する。列入れ替えに関する同値類の定義は、$[M_1] = [M_2] :\\Leftrightarrow M_1 \\stackrel{c}{\\sim} M_2$ということだった。この同値類についての分布$P([M])$がexchangeableであることは、$M_1 \\stackrel{r}{\\sim} M_2 \\Rightarrow P([M_1]) = P([M_2])$と表せる。また、\"Mの第n行の1の個数\"は$f_n(M) := \\sum_{h=1}^H m_{nh}$であるが（ここで$H$は可算無限の場合も含む)、これは明らかに列の入れ替えに対して不変である。したがって関数$f_n$は$M$の関数だが、実際には同値類$[M]$の関数とみることができる。数学っぽく言えば $f_n: \\{0, 1\\}^{N, H} \\rightarrow {\\mathbb Z}$は$f_n: \\{0, 1\\}^{N, H}/\\stackrel{c}{\\sim} \\rightarrow {\\mathbb Z}$を誘導する。$\\{0, 1\\}^{N, H}$(行列$M$の全体)を$\\stackrel{c}{\\sim}$で割った同値類の全体$\\{0, 1\\}^{N, H}/\\stackrel{c}{\\sim}$のことを以下$\\mathcal{M}$と呼ぼう。さて行の入れ替えについての$f_n$のふるまいを考えたいので、行の入れ替えを置換群$\\mathfrak{S}_N$の$M$に対する作用として表す。つまり、$\\sigma \\in \\mathfrak{S}_N$、\n\n\\begin{align}\nM &= \n\\begin{pmatrix}\n{\\bf m}_1\\\\\n{\\bf m}_2\\\\\n\\ldots\\\\\n{\\bf m}_N\n\\end{pmatrix}\n\\end{align}\n\n(${\\bf m}_n$は行ベクトル)とするとき、\n\n\\begin{align}\n\\sigma(M) &:= \n\\begin{pmatrix}\n{\\bf m}_{\\sigma(1)}\\\\\n{\\bf m}_{\\sigma(2)}\\\\\n\\ldots&\\\\\n{\\bf m}_{\\sigma(N)}\n\\end{pmatrix}\n\\end{align}\n\nとする。ちょっと考えると、$M_1 \\stackrel{c}{\\sim}M_2 \\Leftrightarrow \\sigma(M_1) \\stackrel{c}{\\sim} \\sigma(M_2)$であるので、$\\sigma: \\{0, 1\\}^{N, H} \\rightarrow \\{0, 1\\}^{N, H}$もやはり$\\sigma: \\mathcal{M} \\rightarrow \\mathcal{M}$を誘導することがわかる。しかも$\\sigma$にはかならず逆の置換が存在するので$\\sigma$は$\\mathcal{M}$上の全単射になっている。\n\n\n\n今直接興味があるのは$\\mathcal{M}$の分布であり、$f_n$も$\\mathcal{M}$上の関数とみることができるので、以後個々の$M$ではなく$\\mathcal{M}$を考察の対象とし、簡単のため$\\mathcal{M}$の要素を単に$x$とかく。つまり$x = [M]$。$\\sigma$は同値類$x$を$\\sigma(x)$に一対一に移す。$x_1 \\stackrel{r}{\\sim} x_2 \\Leftrightarrow {}^\\exists \\sigma ~x_2 = \\sigma(x_1)$ということなので、この表記のもとでは分布$P(x)$のexchangeabilityは$P(x) = P(\\sigma(x)) ~({}^\\forall \\sigma \\in \\mathfrak{S}_N)$と表せる。\n\n以上の準備のもとで、第$n$列の1の数の分布は、$P(f_n(x) = X) = \\sum_{x \\in \\mathcal{M}} P(x)\\mathbb{1}(f_n(x) = X)$と表せる($\\mathbb{1}()$はindicator function)。この和をとる変数$x$を$x = \\sigma(x^\\prime)$と書き換えれば、\n\n\\begin{align}\nP(f_n(x) = X) &= \\sum_{x \\in \\mathcal{M}} P(x)\\mathbb{1}(f_n(x) = X)\\\\\n&= \\sum_{x^\\prime \\in \\mathcal{M}} P(\\sigma(x^\\prime))\\mathbb{1}(f_n(\\sigma(x^\\prime)) = X)\n&& \\text{変数の書き換え。} \\sigma \\text{が全単射なので} x^\\prime \\text{も} \\mathcal{M} \\text{全体を動く} \\\\\n&= \\sum_{x^\\prime \\in \\mathcal{M}} P(x^\\prime) \\mathbb{1}(f_n(\\sigma(x^\\prime)) = X)\n&& P(x) \\text{のexchangeability} \\\\\n\\end{align}\n\nとなるが、ここで置換$\\sigma$を特に第1行と第$n$行を交換する互換$(1, n)$にとれば、$f_n(\\sigma(x^\\prime))$とは、「一行目と$n$行目を入れ替えた行列（の同値類）$\\sigma(x^\\prime)$における第$n$行の1の数」であり、それはあきらかに入れ替え前の行列（の同値類）$x^\\prime$における一行目の1の数と等しい。つまり$f_n(\\sigma(x^\\prime)) = f_1(x^\\prime)$。これを上の式に用いると\n\n\\begin{align}\nP(f_n(x) = X) &= \\sum_{x^\\prime \\in \\mathcal{M}} P(x^\\prime) \\mathbb{1}(f_1(x^\\prime) = X) \n&& \\text{右辺は} P(f_n(x) = X) \\text{の定義で} n = 1 \\text{としたものに一致}\\\\\n&= P(f_1(x) = X)\\\\\n&= Poisson(X; \\alpha) &&  \\text{IBPの定義から一行目の1の個数の分布は} Poisson(\\alpha) \\\\\n\\end{align}\n\nとなり、全ての行における1の数の分布は第一行のそれと等しいポアソン分布であることが分かった。"
  },
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   "metadata": {},
   "cell_type": "markdown",
   "source": "## IBPが上記P([M])を与えることの証明\n\nまずIBPでの同値類を取る前の分布$P(M)$を考える。[1]でいえば式(15)になるが、この式について詳細な導出は[1]や[2]には書かれていないのでここで与えておく。補助的に$H_i^{(n)}$という量を導入する。これは、\"Mを第一行から第n行までに制限した行列を考えたときのヒストリー値iを持つ列の数\"とする。すでに導入済みの$H_i$は$M$全体を考えてのヒストリー値に基づいていたから、$H_i = H_i^{(N)}$ということになる。実際に使用するのは$H_1^{(n)}$のみであり、これは\"n番目のお客まで考えたときにn番目のお客によってしか選択されていない料理の数\"なので、それはつまりn番目のお客が新規に選んだ料理の数を意味している。\n\n例として、具体的に\n\\begin{align}\nM &=\n\\begin{pmatrix}\n1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0\\\\\n1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\\\\n1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 1\\\\\n0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0\\\\\n\\end{pmatrix}\n\\end{align}\n\nという行列を考えてみる。これは今までに出てきたデータで記述すれば $N = 5$、$H = 9$、$(H_1^{(n)})_{n=1}^5 = (3, 3, 1, 2, 0)$、$(N_h)_{h=1}^9 = (3, 3, 2, 1, 2, 2, 1, 2, 1)$、$(H_i)_{i=0}^{31} = (0, 0, 1, 1, 1, 0, \\ldots )$となる。IBPでのこの$M$の出現確率を計算するため、$M$を以下の図のような2つの部分に分解して考える。"
  },
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   "cell_type": "code",
   "source": "display(\"image/png\", read(\"ibp1.png\"))",
   "execution_count": 11,
   "outputs": [
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   "cell_type": "markdown",
   "source": "色の明るい部分が1、暗い部分が0を表している。上の図のうち、まず青の部分がこのようになる、つまり与えられた$(H_1^{(n)})_{n=1}^N$を持つ確率$P_{blue}$を計算する。この領域において1になっている部分は全てIBPの「n番目のお客が新規に選んだ料理」に対応していることに注意すると、それらはそれぞれパラメータ$\\frac{\\alpha \\beta}{n + \\beta - 1}$をもつポアソン分布から発生しているので、\n\n\\begin{align}\nP_{blue} &= \\Pi_{n=1}^N Poisson(H_1^{(n)}; \\frac{\\alpha \\beta}{n + \\beta - 1})\\\\\n&= \\Pi_{n=1}^N \\frac{(\\frac{\\alpha \\beta}{n + \\beta - 1})^{H_1^{(n)}}}{H_1^{(n)}!} \\exp(-\\frac{\\alpha \\beta}{n + \\beta - 1})\\\\\n&= \\frac{(\\alpha \\beta)^{\\sum_{n=1}^N H_1^{(n)}} \\Pi_{n=1}^N (n + \\beta - 1)^{-H_1^{(n)}}}{\\Pi_{n=1}^N H_1^{(n)}!} \ne^{-\\alpha \\sum_{n=1}^N \\frac{\\beta}{n + \\beta - 1}}\\\\\n&= \\frac{(\\alpha \\beta)^{H_+}}{\\Pi_{n=1}^N H_1^{(n)}!} \\exp(-\\alpha \\bar{H}_+) \\Pi_{n=1}^N (n + \\beta - 1)^{-H_1^{(n)}}\n\\end{align}\n\nと求まる。ここで、$\\bar{H}_+ := \\sum_{n=1}^N \\frac{\\beta}{n + \\beta - 1}$、また上の図を見てわかるように新規に選ばれた料理の数の合計は正のヒストリーを持つ列の数と等しく、$\\sum_{n=1}^N H_1^{(n)} = H_+ := \\sum_{i=1}^{2^N - 1} H_i = H - H_0$が成り立っている。(図を家のように見立てれば、青の色が明るい部分は屋根とみなせる。この式は屋根の幅は家の幅に等しいということを言っている。）"
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   "metadata": {},
   "cell_type": "markdown",
   "source": "次に、赤い部分が与えられたパターンになる確率$P_{red}$を考えてみる。こちらは各列ごとに独立して計算できる。第n行で初めて1が出現する列を考える。その初出の1は上の図で言えば青の明るい色、屋根と呼んだ部分になる。そのような列は$H_1^{(n)}$列あるわけだが、そのうちの一つ（第h列とする）について考えると、$m_{nh} = 1$、$m_{n^\\prime h} = 0 ~(n^\\prime = 1, 2, \\ldots , n - 1)$ となっている。これらはすべて上の図で言えば青色の部分であり、上の具体例の図では、例えばn=1が屋根になっているのは左の三列である。以下考えたいのは赤色の、つまり$m_{n^\\prime h} ~(n^\\prime = n + 1, n + 2, \\ldots N)$の部分である。これらは$N - n$個あるわけだが、そのうち$N_h - 1$回は1になり、残りの$N - n - N_h + 1$回は0になっている。一見すると0と1の出る順番に依存する複雑な値になりそうだが、実はこの確率は$N_h$だけによって決まることを以下説明する。まず、$m_{n^\\prime h} = 1$となる確率を今$p_{n^\\prime h}$と書くと、IBPの定義から$n^\\prime = n + 1$のとき$p_{n^\\prime h} = \\frac{1}{(n + 1) + \\beta - 1} = \\frac{1}{n + \\beta}$となり、そこから$n^\\prime$が増えるにつれ分母が一つづつ増えていく。0が出る確率も$1-p_{n^\\prime h}$を通分して考えれば、1が出るにせよ0が出るにせよ、分母は\n\n\\begin{align}\n\\Pi_{n^\\prime = 1}^N (n^\\prime + \\beta - 1) &=\n(n + \\beta)(n + \\beta + 1) \\cdots (N + \\beta - 1)\n\\end{align}\n\nとなる。"
  },
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   "metadata": {},
   "cell_type": "markdown",
   "source": "次に分子の1が出る確率について考える。初期値は1(\"屋根\"を築いたお客によってすでに一回この料理は選ばれている)で、赤の領域には$N_h$から屋根の分を引いた$N_h - 1$個の1がある。どこで1が出たかによらず、1が出た回数に比例してこの料理の選ばれる確率は高く（分子が1ずつ増える）なっていくので、合計$(N_h - 1)!$の寄与がある。0の出る確率は少し注意が必要で、分子の初期値は、$ 1 - \\frac{1}{n^\\prime + \\beta - 1}$の分子で  $n^\\prime = n + 1$では$(n + \\beta - 1)$となり、ここから1ずつ増えていく。一般に$n^\\prime$では$n^\\prime + \\beta - 1 - N_h^{(n^\\prime)}$となるが(第$n^\\prime$行までに出た1の数を$N_h^{(n^\\prime)}$と書いた）、途中に1が出ることがあるとその分$n^\\prime$の値がジャンプし、同時に$N_h^{(n^\\prime)}$の値も増える。そのため、$n^\\prime + \\beta - 1 - N_h^{(n^\\prime)}$の値は、間に1がどのようにはさまっても初期値から必ず1ずつ順に増加していく。それが$n^\\prime = n, N_h^{(n^\\prime)} = N_h$になるまで続くので、0が出る箇所からの分子への寄与は$(n + \\beta -1)(n + \\beta) \\cdots (N + \\beta -1 - N_h)$である。分子分母をまとめると、\n\n\\begin{align}\n\\frac{(N_h - 1)! (n + \\beta -1)(n + \\beta) \\cdots (N + \\beta -1 - N_h)}{(n + \\beta)(n + \\beta + 1) \\cdots (N + \\beta - 1)}\n&= \\frac{(n + \\beta - 1)(N_h - 1)!}{(N + \\beta - N_h)\\cdots (N + \\beta - 1)}\\\\\n&= (n + \\beta - 1)(N_h - 1)! \\frac{\\Gamma(N + \\beta - N_h)}{\\Gamma(N + \\beta)}\n\\end{align}\n\nこれで列hの赤の部分の分布がわかったので、それをヒストリーが正であるようなすべての列についてかけ合わせると\n\n\\begin{align}\nP_{red} &= \\Pi_{h=1}^{H_+}  (n + \\beta - 1) (N_h - 1)! \\frac{\\Gamma(N + \\beta - N_h)}{\\Gamma(N + \\beta)}\n\\end{align}\n\nただしこの積の中でnは各列hの屋根になっている行のnの値を意味しているので、hと連動している。積を2つに分けてnを含む方の因子を、hではなくnでインデックスすれば各n毎にそのような列は$H_1^{(n)}$個あったのだから、\n\n\\begin{align}\nP_{red} &= \\Pi_{n=1}^{N}  (n + \\beta - 1)^{H_1^{(n)}} \\Pi_{h=1}^{H_+} (N_h - 1)! \\frac{\\Gamma(N + \\beta - N_h)}{\\Gamma(N + \\beta)}\n\\end{align}\n\nと書ける。よって、\n\n\\begin{align}\nP(M) &= P_{blue} P_{red}\\\\\n& = \\frac{(\\alpha \\beta)^{H_+}}{\\Pi_{n=1}^N H_1^{(n)}!} \\exp(-\\alpha \\bar{H}_+) \\Pi_{h=1}^{H_+} \\frac{(N_h - 1)! \\Gamma(N + \\beta - N_h)}{\\Gamma(N + \\beta)}\n\\end{align}\n\nと求まる。[1]の式(15)はこれの$\\beta = 1$とおいた形である。これに一つの同値類に含まれる$M$の個数$\\frac{\\Pi_{n=1}^N H_1^{(n)}!}{\\Pi_{i=1}^{2^N -1} H_i !}$をかけるとたしかに[1]における$P([M])$、テキストでの(6.51)式を得る。ただし[1]に出ている(14)式の形はやはり$\\beta = 1$とおいたものである。"
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   "execution_count": null,
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