ただし$h’=w_{t-n+1}…w_{t-1}$はn-gramのオーダーを一つ落とした文脈、$c(w\mid h)$はレストラン$h$でのテーブル$w$にいる客の総数、$c(h)=\sum_{w}c(w\mid h)$はその総和、$t_{hw}$は$w$を提供するテーブルの総数、$t_h=\sum_{w}t_{hw}$はその総和を表します。






たとえば単語列$\ The\ quick\ brown\ fox\ jumps\ over\ the\ lazy\ dog\ $があり、$dog$を文脈木に追加する場合を考えます。







VPYLMでは、単語列$\boldsymbol w=w_1w_2…w_T$の確率を


$\boldsymbol \Theta$は文脈木の代理客を含めたすべての客の配置を表す隠れ変数、$\boldsymbol n=n_1n_2…n_T$は$\boldsymbol w$のそれぞれの単語が生成された隠れたn-gram長を表します。

HPYLMと同様に客の配置$\boldsymbol \Theta$は推定すべきパラメータであり、VPYLMでは$\boldsymbol \Theta$と$\boldsymbol n$の両方をギブスサンプリングによって推定します。

その際、単語列$\boldsymbol w$の位置$t$の単語$w_t$の隠れたn-gramオーダー$n_t$を、

のようにギブスサンプリングする必要があるのですが、実際は$P(n_t\mid \boldsymbol w, \boldsymbol n_{-t}, \boldsymbol \Theta_{-t})$から直接サンプリングすることはできません。



$\boldsymbol n_{-t}$は、単語$w_t$を除いた単語列$\boldsymbol w_{-t} = w_1,w_2,…,w_{t-1},w_{t+1},…,w_T$に対応するn-gramオーダー$n_1,n_2,…,n_{t-1},n_{t+1},…,n_T$を表しています。


ここでの$\boldsymbol n_{-t}$はただの整数列であり、この値がそのままモデルのパラメータにあるわけではありません。

そして$n_t$をギブスサンプリングするためには$\boldsymbol n_{-t} = n_1,n_2,…,n_{t-1},n_{t+1},…,n_T$をモデルの何らかのパラメータとして持っている必要があります。




こうすることで式(7)の右辺の第二項にある$\boldsymbol n_{-t}$の条件付き確率を

のように表すことができ、$n_t$を$\boldsymbol n_{-t}$の条件付き確率からギブスサンプリングすることができるようになります。


$\boldsymbol \Theta$のサンプリング

$\boldsymbol \Theta_{-t}$は単語$w_t$をレストランから除外したあとの客の配置を表します。

これはHPYLMと同様${\rm RemoveCustomer}$をすれば得られます。

その状態で式(7)により新たなn-gramオーダー$n_t$をサンプリングし、その深さ$n_t$に単語$w_t$を追加し直すことで$\boldsymbol \Theta$が自動的にギブスサンプリングされます。

($\boldsymbol \Theta$は、代理客を含めた客の配置が決定した時点でサンプリングされたことになります。)







A Bayesian Interpretation of Interpolated Kneser-Neyの表記に従えば以下のようにも表されます。

式(10)の第二項$P(n_t=l\mid\boldsymbol w_{-t}, \boldsymbol n_{-t}, \boldsymbol \Theta_{-t})$は式(9)から求めますが、この時第二項が十分小さくなれば、それ以降の深さ$l$は見ずに計算を打ち切ります。





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