Pitman-Yor過程に基づく可変長n-gram言語モデル(VPYLM)

2016年07月28日

概要

はじめに

VPYLMはHPYLMを拡張し、HPYLMでは固定だったn-gramオーダーを各単語ごとに推定できるようになったモデルです。

HPYLMと同様に、文脈$h=w_{t-n}…w_{t-1}$に続く単語$w$の確率は

となります。

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。

停止確率

VPYLMでは文脈木の各レストラン$u$に、木をルートからたどるときにそこで止まる確率$q_u$があると考えます。

そしてこれらの停止確率は共通のベータ事前分布から生成されていると仮定します。

また$q_u$の期待値は

となります。

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。

深さnのレストランで停止する確率を$q_{n}$とすると、文脈$h$のもとで客$dog$の停止深さが$l$となる確率は

となるので、客$dog$がルートのレストラン$\epsilon$に追加される確率は$q_{\epsilon}$、レストラン$lazy$に追加される確率は$(1-q_{\epsilon})q_{lazy}$、レストラン$the$に追加される確率は$(1-q_{\epsilon})(1-q_{lazy})q_{the}$、・・・のように計算されます。

VPYLM

ちなみに1から引いているのはそこを通過する確率を計算するためです。

通常は深いノードに行くほど停止確率が小さくなりますが、文脈に沿った経路のそれぞれの$q_n$が小さければ深いn-gramとなり、HPYLMとは違って様々な深さのノードを許すモデルになっています。

可変長ベイズn-gram言語モデル

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

と表します。

$\boldsymbol \Theta$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。

そこでこの式をベイズの定理から

と変形します。($\propto$は比例を意味します)

$\boldsymbol n_{-t}$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。

$n_t$のサンプリング

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。

そこでVPYLMでは、文脈木の各レストラン$u$で、全ての単語(客)の通過回数$a_u$と停止回数$b_u$を記録します。

そして式(3)の停止確率$q_u$の期待値をベータ事後分布の期待値として

と推定します。

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。

また式(7)の右辺の第一項はn-gramオーダーが$n_t$と決まった時の$w_t$のn-gram確率で、式(1)により計算します。

$\boldsymbol \Theta$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。)

実装

C++での実装をGitHubにあげておきました。

ハイパーパラメータのサンプリングの実装に関してはHPYLMの記事に注意点などを書いています。

$n_t$のサンプリング

$w_t$に対応する$n_t$をサンプリングするには、深さ$l=0,1,2,…$の場合それぞれについて以下の値を求めます。

第一項は式(1)から求めますが、この時深さは$l$と決まっているため、オーダー$l$の文脈$h_l$に対応するレストランを用いて以下のように計算します。

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。

また式(9)はレストランが存在しない深さ(つまり通過回数$a_u$や停止回数$b_u$が存在しない)も考慮しなければならないのですが、私はレストランが無い部分では通過回数と停止回数は$0$とし、式(3)を用いて計算しました。

実験

英語版Wikipediaからランダムに取ってきた5万文に対し単語n-gramモデルを学習させました。

学習されたVPYLMから文章を生成させると以下のようになりました。

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she has a number of notable uses . 
these were very influential constitutional conservatives , and take a liking to the first . 
otherwise it is revealed to be a part of the municipality of science fiction edited by kathryn cramer , david g . 
jackson , ms . 
the ## census it had a population of ## people . 
today , there is no longer be burned the u . 
and a cafeteria and hope that children including the major national morning of october rust to stop the german people , and the university of oxford , east of the county seat of the district of columbia . 
he served as a member of the ## census , the cdp population was ## , ## at the ## census . 
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