ノンパラメトリックベイズ統計と自然言語処理持橋大地
統計数理研究所
daichi@ism.ac.jp
Summer School 数理物理
2020-8-27 (金)自己紹介
l 統計数理研究所 数理・推論研究系 准教授／

総合研究大学院大学 統計科学専攻
l 電子情報通信学会 IBISML (情報論的学習理論と機械学習) 研究会 専門委員 (2016-2021)l 専門: 自然言語処理 、機械学習 (特に教師なし学習)

立川・統数研1 / 124

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自己紹介 (2)l 文科三類 12組フランス語

à 教養学部基礎科学科第二 (文科全体から2名の定数外)– 現在の広域科学科
l NAIST博士課程àATR音声研àNTT CS基礎研
à統数研

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「ガウス過程と機械学習」
l 講談社機械学習プロフェッショナルシリーズ(MLP),
2019年3月発売– 持橋大地・大羽成征著– 現在、レビュー49件
l 線形回帰モデルの非常にやさしい導入から入っています
l 確率過程としての話ではなく、統計の道具としての意味と使い方の話
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岩波データサイエンスl 岩波データサイエンス2巻 :「統計的自然言語処理

ーことばを扱う機械」
編集：持橋 (統数研)・
中谷 (サイボウズラボ)
数解研のご出身
p 特集統括・記事の一部を執筆
p 「統計数理」2016年12月

号も自然言語処理の特集です

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機械学習界の変化と歴史
l 「機械学習」が盛んになったのは、Webが普及した

2000年代〜
l 2000年代：統計的機械学習のブーム– SVM, カーネル法, LDA, … ← 2004年 ニューラル言語モデル(Bengio)l 2010年代〜：深層学習へと移行– 2012年 AlexNet (画像処理)– 2013年 Word2Vec (自然言語処理)
l 2020年代：両者が融合？ (研究レベルでは多数例あり)

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ノンパラメトリックベイズ統計
l 「パラメトリック」な統計モデル…

少数(＜データ数)のパラメータで確率分布を表現– ガウス分布、多項分布、ガンマ分布やその混合分布

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ノンパラメトリックベイズ統計
l 「ノンパラメトリック」な統計モデル…

データを直接使った、柔軟な確率分布を実現
0.10

1

5

0.05

1

0.5

0.00
!5
0

0

0.5

0

-0.5
0

0.2

5

0.4

0.6

0.8

1

!5

0

– これをベイズ的に考えたものàノンパラメトリック
ベイズ法
l 離散的な場合と連続的な場合がある
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持橋担当分のスケジュール
l 1日目：概要と目次、ノンパラメトリックベイズ法(離散的な場合; 無限モデル)
l 2日目：ガウス過程とその適用(連続的な場合; ベイズ的関数回帰)
l 3日目：ノンパラメトリックベイズ法と自然言語処理への応用(研究紹介)

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深層学習について
l 自然言語処理での深層学習は、中で何が起きているか

ほとんど分かっていない ( $ 画像処理)– 上手く動くこともあるが、変な動きをしないかは保証できない– 多数のハイパーパラメータを手作業で調整する必要
l 深層学習の幾つかは、数学的に等価な表現がある– Word2Vecは、特別な行列のSVDと等価– (多層)ニューラルネットは、(多層)ガウス過程と等価à数学的な性質が明らか、計算が簡単、モデルとして拡張しやすい<latexit sha1_base64="slqMW9ZQna37kMCePjiTUI0+n18=">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</latexit>

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言語学と統計的自然言語処理
l 言語の研究à言語学科に行けばよいか?
l いわゆる「言語学」＝手で書いたルールの固まり！
l 例：構文解析

S→NP VP
VP→V PP
VP→V NP
PP→P NP
NP→DET N
NP→N
文法ルール
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S (文)VP (動詞句)

NP
N

VP

He V

NP P

PP (前置詞句)
NP (名詞句)

saw N with a telescopeher言語学と統計的自然言語処理 (2)He saw her with a telescope

He saw her with a hat

l 解釈が名詞によってなぜ違うのか?
l 古典的な言語学では答えが出せない・そもそも主観的à 確率モデル・統計学として数学的に考え直す必要– cf. 中世の天動説から地動説の物理理論へ

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統計的自然言語処理
l 構文解析, 文書モデル, 評判分析, 古文書解読, ...

彼女 は 花 を 買 った 。
0.92

0.85
0.37

文書2
文書1

0.61

1.0

観測値:
単語列

ある単語xの品詞が形容詞である確率12 / 124 Summer School 数理物理2021 Lecture1

統計的自然言語処理の特徴
l 観測値が離散・超高次元の時系列“国連 安保理 は 経済制裁 を 実行 し た”↓
“45701 14332 46 9734 7 2077 672 55”
l データ量が膨大– 数万〜数百万〜数億文の学習データ– 計算はR/MATLAB等ではほぼ不可能
l C++の最適化されたコードでも数時間〜数日の計算
l 億単位の学習テキストの場合、数週間計算する場合も

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統計的自然言語処理の特徴 (2)l 観測値が離散・超高次元の時系列↓
本当は、無限次元– 可能な単語の種類は無限にある– “キュラソ星人” “時雨P” “升” “水素水” “今津線”…
l 可能なカテゴリの数も無限– 動詞、名詞、名詞-鉄道-阪急、動詞-他動詞-抽象、…

l 無限次元離散分布を統計的にどうやって扱うか?

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言語の巾乗則l 自然言語の単語の出現は、巾乗則 (Power law)に従っているといわれている (Zipf則; Zipf 1935)
l 単語wについて、その頻度n(w)はn(w) / r ↵( r：頻度順の順位)

l よって

log n(w) /↵ log r– log n(w)とlog rが右下がりの直線関係 (↵ ⇡ 1)<latexit sha1_base64="IV8QKLmY7DQstTLfi0PT9N4C+eM=">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</latexit>

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京大コーパスの統計
l 毎日新聞1995年度のテキスト約4万記事に、

形態素解析(単語分割・品詞付与・その他)を行ったテキストデータ
後 に 昭和 に 入って 、 反軍 演説 で 名 を 馳せた 憲政 会 の 斎藤隆夫 が 、 日記 に こう 記した の は 、 いわゆる 男子 普通 選挙 法が 成立 した 一九二五 年 三 月 二十九 日 の こと であった 。
それ まで 総 人口 の わずか 二 ％ 程度 だった 有権者 数 が 、 一挙に 二〇 ％ に 拡大 さ れ 、 確かに 、 選挙 制度 と して 画期的な もの で は あった 。
「 我國 の 政界 に 新 時代 を 畫す べき 當日 の 兩院 の 傍聴 席 には 流石 に 熱心な 聴衆 の 緊張 した 顔 が 幾重にも ぎっしり と 重なり合ひ …… 」
毎日新聞 の 前身 、 「 東京 日日新聞 」 は 同年 三 月 三十 日 付 朝
刊 で 成立 の 様子 を こう 伝えて いる 。
：
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単語の確率l 観測した全単語数をNとすると、単語wの確率はn(w)p(w) =
N
↑
確率
p(w)

→単語番号
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単語の確率l 単語確率 p(w) の順(=頻度順)にソートしてみる

↑
確率
p(w)

ソート済
→単語番号

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京大コーパスの統計
l 京大コーパスでの確率を両対数プロットにしたもの

単語頻度n(w)順位r

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言語の巾乗則 (2)l どうして、こんな巾乗分布が生まれるのか?
l ‘Rich gets richer’の現象
l 回答(の一つ)：Dirichlet過程、Pitman-Yor過程– 数学(確率論)では、Gibbs partition, Ewens公式などとして詳しく研究されている

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準備: 多項分布 (離散分布)

1

2

3

K

l K種類のアイテムのどれかが出る確率分布– 離散データの統計モデルの基本中の基本(0,0,1)lは(K-1)次元の単体(Simplex)の内部に存在(1,0,0)21 / 124 Summer School 数理物理2021 Lecture1(0,1,0)ディリクレ分布(
Dir(p|↵) = QPk<latexit sha1_base64="BAVAfpSRC0O7ZoDgD5D/TwN+neo=">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</latexit>

k

K
Y
↵k )

(↵k )

パラメータ:l ランダムな多項分布を生成する確率分布のとき、単体上で Uniform な分布
l 「期待値」:「分散」 :l22 / 124 Summer School 数理物理2021 Lecture1

k=1

k 1
p↵
k

ディリクレ分布 (2)のとき、上に凸lのとき、下に凸– 統計的自然言語処理等では、多くの場合
(
くらい)
l

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ディリクレ分布 (3)l ディリクレ分布からのサンプル

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ディリクレ分布に基づく予測
l ゆがんだ三面サイコロを振ったら、結果は(1=1回,2=3回,3=2回) だった。
次の目は?
l ベイズの定理:→

lの期待値は、

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ディリクレ過程
0.06

0.08

0.04

0.06
2

0.02

0.04
2

0.02

0.00
0
!2

0.00
0
!2

0

!2

0

2

!2
2

l 基底測度measure)に似た、無限次元の離散測度 (atomicを生成する確率過程

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ディリクレ過程 (2)l Dirichlet processとは要するに何?

à 無限次元ディリクレ分布.
l DPの定義 (Ferguson 1973):
A stochastic process P is said to be a Dirichlet processonwith parameter if for any measurable partitionof , the random vector
has a Dirichlet distribution with parameter.
l つまり..

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ディリクレ過程 (3)A5<latexit sha1_base64="uhUdAJjmoo95DsRomcx+rwiBT3w=">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</latexit>l ディリクレ過程: 任意に細かいPartitionに対して、常にその上で離散分布がディリクレ分布に従う.
l 有限次元に周辺化すれば、普通のディリクレ分布

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ディリクレ過程 (4)

予測確率:29 / 124 Summer School 数理物理2021 Lecture1

ディリクレ過程による予測
l 無限次元多項分布 G を積分消去することで、予測ルールが得られる

p(x|x1 , · · · , xn ) =Z

p(x|G)p(G|x1 , · · · , xn )dG↵nk=(x = Xk ) +G0 (x)↵+n↵+n
前に出たものl つまり、

G | x1 , · · · , xn ⇠ DP(↵ +<latexit sha1_base64="DOkJ21dNCNGLP6q9Ba0iE7mbrkQ=">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</latexit>

30 / 124 Summer School 数理物理2021 Lecture1

Pn

i=1

新しいもの

(xi ))

ディリクレ過程による予測 (証明)G | x1 , · · · , xn ⇠ DP(↵ +<latexit sha1_base64="DOkJ21dNCNGLP6q9Ba0iE7mbrkQ=">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</latexit>

l n=1のとき示せば充分なので、

Pn

i=1

(xi ))G ⇠ DP(↵), X ⇠ G のとき、
G | X ⇠ DP(↵ + (X))
を示す.
<latexit sha1_base64="WGnV6H/yuGpBlMtqIX+juFrncFo=">AAAHy3ichZW/b9NAFMdfW8Cl/GgLCxKLRRRUJFRdAhKIqYVWbdVSpQlpUzVVZZtLY8WOLdtJmpiwMcA/wMAEEgNiYIWZhX+AoX8CYiwSCwPPZ0dO4pfWVuK79z7f5/fuzneqbeiux9jx2PjEufMXpMmLU5cuX7k6PTN7bdu1Go7Gi5plWE5JVVxu6HVe9HTP4CXb4YqpGnxHrT0J/DtN7ri6VX/mtW2+byqHdb2ia4qHpoOZ1IpcdnVT9suOKS/lunNlxbCryp278ku5FLpWkGLzTFxyspGJGimIrpw1O7EKZXgOFmjQABM41MHDtgEKuHjvQQYY2GjbBx9tDrZ04efQhSnUNpDiSChoreH/Ifb2Imsd+0FMV6g1fIuBPweVMqTZT/aJnbAf7DP7xf6NjOWLGEEubXyqoZbbB9NvbhT+nqky8elBNVadolCRPr0mDyrwUNSiY222sARVamH8ZuftSeFRPu3fZh/Yb6zvPTtm37HCevOP9nGL599h9LQYb46RytDEpwYpHGMfe71cVfT5wtqN8on5QYUl+g7eOuZYRbUieha0RkY4It51RHBtgmsTXIfgOgTXIrgWwVUIrkJwNYKrEdwiwS0S3BrBrRHcOsGtE1yJ4EoEt0twuwTHCI4lODWa32AtU/OqRvMa+pPzqYovoefnhN/EVR8TZdFPUjbmqA9woYUi3QTpkmSQW7DC+9meLUkHo9Tj+xWxvUuoDLFvqGgZ1vV7KGUuUXFuRMUF8bWaQ5XE1qRiYyCrWLMxlFO/ajPKX0OfgbvTZoJYHlpRKvaXE1SeoPIJaknsNinICrqCe1Cwd4az2Ts1wjzCXYn2ZIO4eIZlhk+sZGM7O5+5N5/dup9aeBydZpNwE27BHMZ/AAuwiuNfxBxewxf4Ct+kp5IrdaQXITo+Fmmuw8AlvfoPbO+bDA==</latexit><latexit sha1_base64="XtZ4sQGjAEDXDBfAf01OkM1/Hes=">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</latexit>l 考えている空間 X の可測な分割 A1 , · · · , Ak について
<latexit sha1_base64="t01nwMZq9n4ZNTiCMcEvSmMVck4=">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</latexit><latexit sha1_base64="Bf9qKgLyvfAxLuzlhszlIg+KIlE=">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</latexit>(G(A1 ), · · · , G(Ak )) | X<latexit sha1_base64="Za2v8UfOC/uqRX01u79JWfwKiTg=">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</latexit>⇠ Dir(↵(A1 ) + X (A1 ), · · · , ↵(Ak ) + X (Ak ))を示せばよい.
31 / 124 Summer School 数理物理2021 Lecture1

ディリクレ過程による予測 (2)l 考えている X の含まれる集合を B とおくと<latexit sha1_base64="dEGTQjnN9FQEM2/rdYpKLBkCIOo=">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</latexit><latexit sha1_base64="rvvtQEkVGjqbL4/iwoQfe8dVd70=">AAAHp3ichZXNbtNAEMenLeBSPtrCBYlLRRQEl2gTkECc2kKlVq1KPkiaqqkqr7tprTi2ZTtJkygvAFcQB04gcUA8BhdegEMfAXEsEhcOjNeOnGSnxVbi3Znffzyzu97lrmX6AWOnU9Mzly5f0Wavzl27fuPm/MLirYrvtDxDlA3Hcrwq131hmbYoB2ZgiarrCb3JLbHDG89D/05beL7p2K+Criv2m/qRbdZNQw/QVFg9WEixDJPXktrIxo0UxFfeWZxZhxocggMGtKAJAmwIsG2BDj7ee5AFBi7a9qGPNg9bpvQLGMAcaltICSR0tDbw/wh7e7HVxn4Y05dqA99i4c9D5RKk2Q/2hZ2x7+wr+8n+nhurL2OEuXTxySOtcA/mX98p/fmvqonPAI4T1QUKjvTFNQVQh6eyFhNrc6UlrNKI4rd7789Kz4rp/n32if3C+j6yU/YNK7Tbv43PBVH8gNHTcrwFRqpBG58GpHCM+9gb5srR15fWQZxPwo8rHNn38DYxx2NU67LnQOfcCCfEu04IrktwXYLrEVyP4DoE1yG4OsHVCa5BcA2CWyG4FYLbILgNgtskuE2CqxJcleB2CW6X4BjBMYXj8fyGa5maVx7Pa+RX55PLL2HoF4S/ias+IWqyr1Iu5miOcZGFIn2F9EkyzC1c4aPs0KbS4SgN+VFFYh8QKkvuGxwtk7pRD6XMKxXnz6m4JL/W5kQliVVVbI1llWi2JnIaVW3H+Rvos3B32laItYkVxbG/plBFgioq1Au526QgJ+k67kHh3hnN5vDUiPKIdiXakwvj4hmWnTyx1EYll8k+yuQKj1PLq/FpNgt34R48wPhPYBnWcfzLmIOAN/AW3mkPtZdaRatG6PRUrLkNY5em/wNPHI6V</latexit>p(X 2 B, G(A1 )  y1 , · · · , G(Ak )  yk )= EX⇠G [D(y1 , · · · , yk |↵(A1 ) + X (A1 ), · · · , ↵(Ak ) + X (Ak )),{X 2 B}]<latexit sha1_base64="4F/rRKXAk0UyCrJPA8nyRsna/mc=">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</latexit>

を示せばよい.
l ここでDはディリクレ分布の分布関数でD(y1 , · · · , yk |↵1 , · · · , ↵k )= p(Y1  y1 , · · · , Yk  yk ),Y ⇠ Dir(↵1 , · · · , ↵k )A1<latexit sha1_base64="oDd5w/c6CzhSzQqRoyn9BID8CD4=">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</latexit><latexit sha1_base64="rvvtQEkVGjqbL4/iwoQfe8dVd70=">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</latexit><latexit sha1_base64="uW08uxxStnsarJGo+lKn1R+QgHo=">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</latexit><latexit sha1_base64="Bg/uP//TPIiIqbjAQ6hxC0amtSQ=">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</latexit>

A2<latexit sha1_base64="2njEDqOxg9Ne/7XWpG3V763GliQ=">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</latexit>

32 / 124 Summer School 数理物理2021 Lecture1

B A
3

Ak
<latexit sha1_base64="hU7rJqO9Og16+UzpU5/zKL3G2CU=">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</latexit>

X<latexit sha1_base64="dEGTQjnN9FQEM2/rdYpKLBkCIOo=">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</latexit>

ディリクレ過程による予測 (3)A1

p(X 2 B, G(A1 )  y1 , · · · , G(Ak )  yk )A2<latexit sha1_base64="oDd5w/c6CzhSzQqRoyn9BID8CD4=">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</latexit>

B A
3<latexit sha1_base64="rvvtQEkVGjqbL4/iwoQfe8dVd70=">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</latexit><latexit sha1_base64="uW08uxxStnsarJGo+lKn1R+QgHo=">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</latexit>

Ak

X<latexit sha1_base64="dEGTQjnN9FQEM2/rdYpKLBkCIOo=">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</latexit><latexit sha1_base64="hU7rJqO9Og16+UzpU5/zKL3G2CU=">AAAHqXichZXNbtNAEMenLeBSPtrCBYlLRRSEhBRtAhKIUwNUatWqygdpU5oqss0mteIv2U7SJMoLIHGlB04gcUA8BhdegEMfAXEsEhcOjNeOnGQnxVbi3Znffzyzu97VXNPwA8bO5uYXLl2+oixeXbp2/cbN5ZXVW3u+0/Z0XtEd0/Gqmupz07B5JTACk1ddj6uWZvJ9rfUi9O93uOcbjv0q6Ln8yFKbttEwdDVAUzlfb9VXUizDxLUmN7JxIwXxVXBWFzahBm/AAR3aYAEHGwJsm6CCj/chZIGBi7YjGKDNw5Yh/ByGsITaNlIcCRWtLfxvYu8wttrYD2P6Qq3jW0z8eahcgzT7wb6wc/adfWU/2d+ZsQYiRphLD59apOVuffntnfKf/6osfAZwnKguUGhIX1xTAA14KmoxsDZXWMIq9Sh+p396Xn5WSg/us0/sF9b3kZ2xb1ih3fmtfy7y0geMnhbjzTFSDTr41CGFYzzA3ihXDX0DYR3G+ST8pMIRfQ9vA3M8RrUqeg50Z0Y4Id51QnA9gusRXJ/g+gTXJbguwTUIrkFwLYJrEVye4PIEt0VwWwS3TXDbBFcluCrBHRDcAcExgmMSp8XzG65lal61eF4jvzyfmvgSRn5O+C1c9QlRE32ZcjFHY4KLLBTpS6RPkmFu4QofZ0c2mQ5HacSPKxL7kFCZYt/Q0DKtG/dQyoJUcWFGxWXxtVpTlSRWWbEzkVWi2ZnKaVy1G+evo8/E3WlXIjamVpSG/Q2JKhFUSaJeit0mBTlBN3APCvfOaDZHp0aUR7Qr0Z5cGBfPsOz0iSU39nKZ7KNMrvg4tf48Ps0W4S7cgwcY/wmswyaOfwVzaMI7eA+nykOlqFSV1xE6PxdrbsPEpej/AFk6j3I=</latexit><latexit sha1_base64="2njEDqOxg9Ne/7XWpG3V763GliQ=">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</latexit>= EX⇠G [D(y1 , · · · , yk |↵(A1 ) + X (A1 ), · · · , ↵(Ak ) + X (Ak )),{X 2 B}]<latexit sha1_base64="4F/rRKXAk0UyCrJPA8nyRsna/mc=">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</latexit>

(j)
↵i =

l の右辺を計算すると、<latexit sha1_base64="GcP6rVh57dcOglYepjJ9bhFDtrw=">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</latexit>

Z

⇢

↵i
(i 6= j)↵i + 1 (i = j)↵(x)
D(y1 , · · · , yk |↵(A1 ) + X (A1 ), · · · , ↵(Ak ) + X (Ak )) d
↵(X )
B
k Z
X
↵(x)
(j)
(j)
=
D(y1 , · · · , yk |↵1 , · · · , ↵k ) d↵(X )j=1 Aj \B=<latexit sha1_base64="+rgqbQHiiGjEBCwFuKLQkZQOO9E=">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</latexit>k
X↵(Aj \B)
j=1

↵(X )

(j)

(j)

D(y1 , · · · , yk |↵1 , · · · , ↵k ) = 左辺.

33 / 124 Summer School 数理物理2021 Lecture1

補題1
A<latexit sha1_base64="QJfs+k4Xg8AGAa5Wv9A5DZMSB4o=">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</latexit>l G ⇠ DP(↵) のとき、8A 2 A について
<latexit sha1_base64="Di8KeQ0wzRhn6g1lE09sr5XUmOM=">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</latexit><latexit sha1_base64="Rg229dwN3GqyFVtVRvJDYHSdHoQ=">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</latexit><latexit sha1_base64="A7sdKZQrnj25FBhgPsDozFYuYik=">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</latexit>l Proof:E[G(A)] = ↵(A)/↵(X ).

X \A<latexit sha1_base64="O1J91jD/B2NyEkcDHGRvqss8Epg=">AAAHvnichZXNbtNAEMenLZBSPpqChJC4pERBnKJNQAJxSoBKrVpVaULaVHUV2WbTWvGXbCdpYvICvAAHTlTigMpbcOEFOPQREMciceHAeO3ISTwpthLvzvz+45nd9a5i65rrMXY+N79w5eq11OL1pRs3b91eTq/c2XWtjqPyumrpltNQZJfrmsnrnubpvGE7XDYUne8p7VeBf6/LHVezzDde3+aHhnxkai1NlT00NdP3fEmV9UxjKK1KLvcMzey40mq5mc6yPBNXJtkoRI0sRFfFWllYBwneggUqdMAADiZ42NZBBhfvAygAAxtth+CjzcGWJvwchrCE2g5SHAkZrW38P8LeQWQ1sR/EdIVaxbfo+HNQmYEc+8G+sAv2nZ2xn+zvzFi+iBHk0senEmq53Vx+f7/2578qA58eHMeqSxQK0pfX5EELnotaNKzNFpagSjWM3x18uKi9qOb8R+yU/cL6PrFz9g0rNLu/1c87vPoRo+fEeHOMJEEXnypkcYx97I1yVdDnC+swyifmJxWW6Dt4a5jjMapl0bOgNzPCCfGuE4LrE1yf4AYENyC4HsH1CK5FcC2CaxNcm+DKBFcmuA2C2yC4TYLbJLgGwTUIbp/g9gmOERxLcEo0v8FapuZVieY19CfnUxFfwsjPCb+Bqz4mJNFPUjbmqE1woYUi3QTpkmSQW7DCx9mRLUkHozTixxWxfUiodLFvKGiZ1o17KGUlUXFlRsU18bUaU5XE1qRiayKrWLM1ldO4ajvKX0WfjrvTdoJYm1pRCvbXElSVoKoJ6rXYbbJQFHQL96Bg7wxnc3RqhHmEuxLtKQZx8QwrTJ9YycZuMV94ki/uPM2WXkan2SI8gIfwGOM/gxKs4/jXMYd3cApn8DVVSrVSRsoK0fm5SHMXJq7UyT/dDJdh</latexit>

X<latexit sha1_base64="t01nwMZq9n4ZNTiCMcEvSmMVck4=">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</latexit>

ディリクレ過程の定義より、A1 = A, A2 = X \Aととれば、ディリクレ分布の性質から<latexit sha1_base64="WzyF5AlAl6H/npfYbUEaujkxpww=">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</latexit>(G(A1 ), G(A2 )) = (G(A), G(X \A)) ⇠ Be(↵(A), ↵(X \A))<latexit sha1_base64="WQX3/HFb01Wc4bPV/B9clgNgZ+c=">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</latexit>

よって、
↵(A)
↵(A)
E[G(A)] ==.
↵(A) + ↵(X \A)↵(X )<latexit sha1_base64="DCYEQSTUlub0JzdzGx+GOUR69cg=">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</latexit>

34 / 124 Summer School 数理物理2021 Lecture1

補題2
l G ⇠ DP(↵), X ⇠ G とすると、<latexit sha1_base64="WGnV6H/yuGpBlMtqIX+juFrncFo=">AAAHy3ichZW/b9NAFMdfW8Cl/GgLCxKLRRRUJFRdAhKIqYVWbdVSpQlpUzVVZZtLY8WOLdtJmpiwMcA/wMAEEgNiYIWZhX+AoX8CYiwSCwPPZ0dO4pfWVuK79z7f5/fuzneqbeiux9jx2PjEufMXpMmLU5cuX7k6PTN7bdu1Go7Gi5plWE5JVVxu6HVe9HTP4CXb4YqpGnxHrT0J/DtN7ri6VX/mtW2+byqHdb2ia4qHpoOZ1IpcdnVT9suOKS/lunNlxbCryp278ku5FLpWkGLzTFxyspGJGimIrpw1O7EKZXgOFmjQABM41MHDtgEKuHjvQQYY2GjbBx9tDrZ04efQhSnUNpDiSChoreH/Ifb2Imsd+0FMV6g1fIuBPweVMqTZT/aJnbAf7DP7xf6NjOWLGEEubXyqoZbbB9NvbhT+nqky8elBNVadolCRPr0mDyrwUNSiY222sARVamH8ZuftSeFRPu3fZh/Yb6zvPTtm37HCevOP9nGL599h9LQYb46RytDEpwYpHGMfe71cVfT5wtqN8on5QYUl+g7eOuZYRbUieha0RkY4It51RHBtgmsTXIfgOgTXIrgWwVUIrkJwNYKrEdwiwS0S3BrBrRHcOsGtE1yJ4EoEt0twuwTHCI4lODWa32AtU/OqRvMa+pPzqYovoefnhN/EVR8TZdFPUjbmqA9woYUi3QTpkmSQW7DC+9meLUkHo9Tj+xWxvUuoDLFvqGgZ1vV7KGUuUXFuRMUF8bWaQ5XE1qRiYyCrWLMxlFO/ajPKX0OfgbvTZoJYHlpRKvaXE1SeoPIJaknsNinICrqCe1Cwd4az2Ts1wjzCXYn2ZIO4eIZlhk+sZGM7O5+5N5/dup9aeBydZpNwE27BHMZ/AAuwiuNfxBxewxf4Ct+kp5IrdaQXITo+Fmmuw8AlvfoPbO+bDA==</latexit><latexit sha1_base64="75IZ0Bmqwtm9+PEkoskv78hoedI=">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</latexit>l Proof:

8A 2 A, p(X 2 A) = ↵(A)/↵(X ).

定義より p(X 2 A | G) = G(A) なので、
<latexit sha1_base64="TN+KNGTg/kP2KnbiKrHtO66nSTI=">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</latexit><latexit sha1_base64="ciisLvZzCzbn/tvQdrLqRCrd1u8=">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</latexit>↵(A)p(X 2 A) = EG [p(X 2 A | G)] = EG [G(A)] =.
↵(X )35 / 124 Summer School 数理物理2021 Lecture1

補題3
l G ⇠ DP(↵), X ⇠ G とする. このとき、<latexit sha1_base64="v5Mvgh0W0tZvDlz1zP/8SPtF86I=">AAAHynichZXNbtNAEMenLZBSPprCBYlL1CioSKjaBCQQpxZatVWjKk1Im6qpIttsGiv+ku0kTaxw4oB4AQ6cQOKAkLjCnQsvwKGPgDgWiQsHxmtHTuJJsZV4d+b3H8/srndlS1Mdl7HTqemZCxcvJWYvz125eu36fHLhxp5jtmyFlxVTM+2KLDlcUw1edlVX4xXL5pIua3xfbj71/fttbjuqaTxzuxY/0qVjQ62riuSiqZZc3EhVHVVPeVVbT60V+ktVSbMa0t17L1KVwLNRS6bZMhNXKt7Iho00hFfBXJjZhCo8BxMUaIEOHAxwsa2BBA7eh5AFBhbajsBDm40tVfg59GEOtS2kOBISWpv4f4y9w9BqYN+P6Qi1gm/R8GejMgUZ9oN9ZGfsO/vEfrK/E2N5IoafSxefcqDlVm3+9a3Sn/+qdHy60IhU5yhkpM+vyYU6PBK1qFibJSx+lUoQv917c1Z6XMx4d9h79gvre8dO2Tes0Gj/Vj7s8uJbjJ4R480xUhXa+FQgjWPsYW+Qq4w+T1j7YT4RP6owRd/GW8UcG6iWRM+EzsQIJ8S7TgiuS3BdgusRXI/gOgTXIbg6wdUJrklwTYJbJbhVgtsiuC2C2ya4bYKrEFyF4A4I7oDgGMGxGCeH8+uvZWpe5XBeA398PmXxJQz8nPDruOojoir6ccrCHNURLrBQpBMjHZL0c/NX+DA7sMVpf5QG/LAisvcJlSb2DRkt47phD6UsxCouTKi4JL5WfaySyBpX5EeyijT5sZyGVTth/gr6NNyddmLE+tiKkrG/HqOKBFWMUWtit0lDTtB13IP8vTOYzcGpEeQR7Eq0J+fHxTMsO35ixRt7ueXs/eXc7oP0ypPwNJuF27AISxj/IazAJo5/GXN4BZ/hC3xN5BN2opvwAnR6KtTchJEr8fIfUEqa4g==</latexit><latexit sha1_base64="UWV8kcLQEHwxTjNq2GCP32cQwm8=">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</latexit>

8 B 2 A, 8 {A1 , · · · , Ak } について

p(X 2 B, G(A1 )  y1 , · · · , G(Ak )  yk )kX↵(Aj \B)(j)(j)=
D(y1 , · · · , yk | ↵1 , · · · , ↵k ) .
↵(X )
j=1
Aj,1
l Proof:<latexit sha1_base64="6GL60RpNxyQpW5BlwmCvOqIpyaM=">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</latexit><latexit sha1_base64="bxKNbJeyT4T345FOs8RC2iYeTv0=">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</latexit>

Aj,1 = Aj \ B, Aj,0 = Aj \ B̄<latexit sha1_base64="fqc50jVSVu94SlP0FdYF/8rXbT4=">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</latexit>

とおく.

Yj,v = G(Aj,v )

Aj,0

A1
<latexit sha1_base64="oDd5w/c6CzhSzQqRoyn9BID8CD4=">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</latexit>

B A
3<latexit sha1_base64="rvvtQEkVGjqbL4/iwoQfe8dVd70=">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</latexit><latexit sha1_base64="uW08uxxStnsarJGo+lKn1R+QgHo=">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</latexit><latexit sha1_base64="F98/PMBlVrECf+9haBf9WEOEi24=">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</latexit><latexit sha1_base64="6PrCuAX3HlN+c+puCosDmWAbJhQ=">AAAHunichZXdahNBFMdPWzW1fjStN4I3xRCpIGESBUUUGrXY0lLSxLQJTQk766Rds1/sbr7JC/gCXgiCgqj4GN74Al70EcTLCt544dnZDfnYk7pLsjPn/P5nz5mZneG2rrkeYyczs3Pnzl+IzV9cuHT5ytXF+NLynms1HFUUVUu3nBJXXKFrpih6mqeLku0IxeC62Of1p75/vykcV7PMF17HFoeGcmRqNU1VPDRV48vlau/VnWb/8fPVbNC6XY0nWIrJayXaSIeNBIRXzlqa24AKvAQLVGiAAQJM8LCtgwIu3geQBgY22g6hhzYHW5r0C+jDAmobSAkkFLTW8f8Ieweh1cS+H9OVahXfouPPQeUKJNkP9oWdsu/sK/vJ/k6N1ZMx/Fw6+OSBVtjVxdfXC3/+qzLw6cHxUHWGgiN9dk0e1OCBrEXD2mxp8atUg/jN7pvTwsN8sneLfWC/sL737IR9wwrN5m/1467Iv8XoSTneAiNVoIlPFRI4xj3sDXLl6OtJaz/MZ8iPKyzZd/DWMMdjVCuyZ0FraoQ28a42wXUIrkNwXYLrElyL4FoEVyO4GsHVCa5OcFmCyxLcJsFtEtwWwW0RXIngSgRXJrgywTGCYxGOh/Prr2VqXnk4r4E/Op9cfgkDvyD8Bq76IVGR/ShlY47aGBdYKNKNkC5J+rn5K3yUHdiitD9KA35UMbT3CZUu9w2OlkndqIdS5iIV56ZUXJBfqzFRydAaVWyPZTXUbE/kNKraCfNX0afj7rQTIdYnVhTH/nqEyhNUPkI9k7tNAjKSruEe5O+dwWwOTo0gj2BXoj0ZPy6eYenJEyva2Muk0ndTmd17ibUn4Wk2DzfgJqxi/PuwBhs4/kXMoQ3v4BN8jj2K8ZgWqwfo7EyouQZjV8z7B7unlWM=</latexit>

と表せば、

A2
<latexit sha1_base64="2njEDqOxg9Ne/7XWpG3V763GliQ=">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</latexit>

36 / 124 Summer School 数理物理2021 Lecture1

Ak<latexit sha1_base64="hU7rJqO9Og16+UzpU5/zKL3G2CU=">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</latexit>

X<latexit sha1_base64="dEGTQjnN9FQEM2/rdYpKLBkCIOo=">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</latexit>

Aj,1

補題3 (2)<latexit sha1_base64="bxKNbJeyT4T345FOs8RC2iYeTv0=">AAAHrXichZXNbtNAEMenLeBSPtrCBYlLRRTEAZVNQAJxaoBKrVpVaUKaVE2JbLNJTRzbsp2kiZUX4MYJiZ5A4oB4DC68AIc+AuJYJC4cGK8dOclOiq3EuzO//3hmd72rOabh+YydzszOXbh4SZm/vHDl6rXri0vLN/Y8u+3qvKTbpu1WNNXjpmHxkm/4Jq84LldbmsnLWvN56C93uOsZtvXS7zn8sKU2LKNu6KqPpnKuFry5nxnUllJslYlrRW5k4kYK4itvL89tQBVegw06tKEFHCzwsW2CCh7eB5ABBg7aDiFAm4stQ/g5DGABtW2kOBIqWpv438DeQWy1sB/G9IRax7eY+HNRuQJp9oN9YWfsO/vKfrK/U2MFIkaYSw+fWqTlTm3x7a3in/+qWvj04ShRnaPQkD6/Jh/q8ETUYmBtjrCEVepR/E7//VnxaSEd3GWf2C+s7yM7Zd+wQqvzW/+8ywsnGD0txptjpCp08KlDCsc4wN4wVw19gbAO4nwSflxhi76Lt4E5HqFaFT0bulMjHBPvOia4HsH1CK5PcH2C6xJcl+DqBFcnuCbBNQkuR3A5gtskuE2C2yK4LYKrEFyF4PYJbp/gGMExidPi+Q3XMjWvWjyvkV+eT018CUM/J/wtXPUJURV9mXIwR2OMiywU6UmkR5JhbuEKH2WHNpkOR2nIjyoS+4BQmWLf0NAyqRv1UMq8VHF+SsVF8bW2JipJrLJieyyrRLM9kdOoaifOX0efibvTjkSsT6woDfvrElUgqIJEvRC7TQqygq7jHhTundFsDk+NKI9oV6I92TAunmGZyRNLbuxlVzMPV7O7j1Jrz+LTbB5uwx24h/Efwxps4PiXxP79Dj7AifJAKSlV5VWEzs7EmpswdimNf6+nkO4=</latexit>

B A
3

A1

<latexit sha1_base64="rvvtQEkVGjqbL4/iwoQfe8dVd70=">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</latexit><latexit sha1_base64="oDd5w/c6CzhSzQqRoyn9BID8CD4=">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</latexit>

Aj,0<latexit sha1_base64="uW08uxxStnsarJGo+lKn1R+QgHo=">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</latexit><latexit sha1_base64="F98/PMBlVrECf+9haBf9WEOEi24=">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</latexit>

l 定義からp(X 2 B | Yj,v ; j = 1, · · · , k, j = 0, 1)
k
k
X
X
=
G(X 2 Bj,1 |Yj,v ) =
Yj,1
j=1

A2
<latexit sha1_base64="2njEDqOxg9Ne/7XWpG3V763GliQ=">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</latexit>

j=1

l よって
p(X 2 B, Yj,v  yj,v ; j = 1, · · · , k, v = 0, 1)
" k
#
X
=E
Yj,1 , {Yj,v  yj,v ; j = 1, · · · , k, v = 0, 1}<latexit sha1_base64="0050eMkXdgQIAoSm6xBotIRgFl8=">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</latexit>

j=1

=

k
X
↵(Aj,1 )D(y1,0 , · · · , yk,0 , y1,1 , · · · , yk,1 ))↵(X )D(y1 , · · · , yk | ↵1 , · · · , ↵k ) .

j=1=<latexit sha1_base64="NyZq5OefHW7uCHJMn2QVJM0HQ+s=">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</latexit>

↵(X )

k
X
↵(Aj \B)j=1

37 / 124 Summer School 数理物理2021 Lecture1

(j)

(j)

⇤

Ak
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X<latexit sha1_base64="dEGTQjnN9FQEM2/rdYpKLBkCIOo=">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</latexit>

Chinese Restaurant Process (CRP)l 予測確率(DP)(Dirichlet),– ディリクレ分布/過程に従うと、頻度 の高いものはさらに現れやすくなる (rich-gets-richer) à CRP

1

2

3

4

2

3

1

0

確率:
38 / 124 Summer School 数理物理2021 Lecture1?

ディリクレ過程と言語モデル
l ディリクレ過程は、語彙が無限の場合の単語の確率分布ともみることができる
n(w)↵p(w|X) =+G0 (w)↵+n ↵+n<latexit sha1_base64="4TmNS1UAJGOFgr8KfPCn0B/AaXQ=">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</latexit>– カウントn(w)が0のどんな未知の単語 w でも、の確率を持つ39 / 124 Summer School 数理物理2021 Lecture1

ディリクレ過程混合モデル
l 混合モデルのパラメータがディリクレ過程に従うとすると、クラスタ数を決めない無限混合モデルが可能になる40 / 124 Summer School 数理物理2021 Lecture1

ディリクレ過程混合モデル (2)l MCMC法による学習: 各データ

クラスタ番号– ベイズの定理:l よって、

41 / 124 Summer School 数理物理2021 Lecture1に、それを生成したを割り当てるディリクレ過程混合モデル (3)l Gibbs samplerによる学習 (物理では熱浴法)

電子情1: while not converged dofor n in randperm (1, · · · , N ) do
3:
xn をクラスタ zn から削除してパラメータを更新
4:zn ∼ p(zn |X, Z−i ) をサンプル
5:
xn をクラスタ zn に追加してパラメータを更新
6:
end for
7: end while
8: z1 , · · · , zN を出力
2:K

6 ディリクレ過程混合モデルのギブスサンプリング.
–図
randperm():
引数のランダムな並び替えを返す関数
randperm (X) は X のランダムな並び換えを表す.

6

6

4

4

2

2

0

0

42 / 124 Summer School 数理物理2021 Lecture1
−2

−2

3. ノ

ノンパラ

ンフォマテ

8: z1 , · · · , zN を出力1

図 6 ディリクレ過程混合モデルのギブスサンプリング.
randperm (X) は X のランダムな並び換えを表す.

ディリクレ過程混合モデル (4)
6

6

4

4

2

2

0

0

−2

−2

−4

−4

−6

−6
−6

−4

−2

0

2

4

6

(a) データ

4

4

2

2

0

0

−2

−2

−4

−4

−6

−6
−6

−4

−2

0

2

4

(c) 繰り返し数=100

図7

−4

−2

0

2

4

6

(b) 繰り返し数=106620 40 60 80
Gib

図8

混合数 K の

3. ノンパラメトリ
l データから自動

−6

6

0

的にクラスタ数
ノンパラメトリックベイを決定できる, データ
ンフォマティクス
[8], そl に適用されているが対数尤度が収束
トデータに対する自然言
すれば、学習はい. ここではその中から,終了

する基礎的なモデルとして(Infinite HMM, 以下 IHMHMM は図 9 に表したよ
裏に隠れた状態系列 s1 s2

値が生成されたと考えるモ
−6

−4

−2

0

2

4

6

(d) 繰り返し数=200

ディリクレ過程混合モデルの MCMC による学習.

43 / 124 Summer School 数理物理2021 Lecture1

総数は事前に決めておく必の隠れ状態の総数も自動的
期的な) モデルである [9].

まず, 通常の HMM は本電子情報通信学会論文誌 xxxx/xx Vol. Jxx–X No. xx

ディリクレ過程混合モデル (5)1098
クラスタ数 7
6
K
5
4
3
2
1
0

20 40 60 80 100 120 140 160 180 200
Gibbs iterationl Gibbsの繰り返し毎に、クラスタ数がだんだん収束
図 8 混合数 K の学習過程での変化.

– データがもっと多いと、収束はかなり顕著
3.– クラスタ数の推定には、それまで多数のヒューリス
ノンパラメトリックベイズ法の応用
ティックが提案されていた, バイオイ
44 /ノンパラメトリックベイズ法は画像処理
124 Summer School 数理物理2021 Lecture1

Pitman-Yor過程 (Pitman and Yor 1997)l Dirichlet過程で、新しいカテゴリが作られる確率↵G0 (w)
はn+↵→ nが大きくなると、新しいカテゴリを作ることはどんどん難しくなる
l 現実の単語確率分布をより忠実に反映するにはどうすればよいか?

45 / 124 Summer School 数理物理2021 Lecture1

Pitman-Yor過程 (2)l 新しいカテゴリの出る確率を、これまでに現れたカテゴリ数に依存させる
8>>< p(k  K)

nk d
/
↵+n
↵ + dK>>: p(k = K + 1) /↵+n<latexit sha1_base64="TS/grF+B5WDHymN5viNWnPGXeVY=">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</latexit>l CRP形式で書くと、

p(x|x, PD(d, ↵)) =<latexit sha1_base64="VeuuOw9InReSdi9ndNMnzCgDN2c=">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</latexit>

K
X
nk

k=1

d
n+↵

↵ + dK
G0 (x)Xk +n+↵( K：これまでに出たテーブルの数)

46 / 124 Summer School 数理物理2021 Lecture1

Pitman-Yor過程 (3)l Pitman-Yor過程 (Pitman and Yor 1997): PY(α,d)– ディリクレ過程の拡張, Poisson-Dirichlet過程とも– 新たにディスカウント係数dを持つ
l CRPの違い
新しいテーブルが作られやすい
1

2

3

4

2

3

1

0

?

確率

差が小さい
47 / 124 Summer School 数理物理2021 Lecture1

これまでのテーブルの数ディリクレ過程から生成される離散分布
l SBP(1)からのサンプル (最初の1000次元)

確率分布p48 / 124 Summer School 数理物理2021 Lecture1

対数 log(p)Pitman-Yor過程から生成される離散分布
l PY(1,0.5)からのサンプル (最初の1000次元)

確率分布p49 / 124 Summer School 数理物理2021 Lecture1

対数 log(p)ディリクレ過程とPitman-Yor過程の比較
l 『数学セミナー』2007年11月号「生きたことばをモデル化する」に使った図– Pitman-Yor過程の方が、言語の冪乗則をよく再現

Frequency

1000
DP(500)PD(100,0.5)
100
10
1
1

10

50 / 124 Summer School 数理物理2021 Lecture1

100
1000
Rank

10000

無限木構造隠れMarkovモデル(情報処理学会自然言語処理研究会 NL-226, 2016)

51 / 80

本研究の概要l HMMを、無限の木構造上に状態を持つように拡張– Infinite HMM (Beal+ 2001; Teh+ 2006)の拡張– 「品詞体系」の教師なし導出

木のノードからノードへの状態遷移
普通のHMM

52 / 124 Summer School 数理物理2021 Lecture1

説明の流れl 隠れMarkovモデル(HMM)とは
l 品詞の教師なし・半教師あり学習l 無限隠れMarkovモデル
l 木構造Stick-breaking過程 (Adams+ 2010)l 階層的木構造Stick-breaking過程
l iTHMMと特別なMCMC法による学習
l 実験 (日本語・英語・クリンゴン語)
l まとめと展望

53 / 124 Summer School 数理物理2021 Lecture1単語の時系列データ
l 本当は、言語の入力は時系列

When he was a young boy, a book …
l これをどのようにモデル化するか？

– 面白い複雑なモデルは色々考えられるが、– 最も簡単な隠れマルコフモデル (HMM)について
l HMMは、言語に限らずコンピュータサイエンス全般の基礎 (ロボティクス、バイオ、経済学、…)

54 / 124 Summer School 数理物理2021 Lecture1

HMMの基礎l 各時刻tの観測値 yt に、隠れ状態 st が存在<latexit sha1_base64="Xiao0VD4L2ygXb+SNJGkIY+OctE=">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</latexit><latexit sha1_base64="Q/vSLu0ZX7tVAOZkuMtXyGc8kaU=">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</latexit>

d
K
y
2
R
,
s
2
R
t
– 一般には t<latexit sha1_base64="J9X4t5aed65dWKMA4lct7LRHSjM=">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</latexit>– 自然言語処理での最も簡単な場合は、
yt = wt 2 {1, · · · , V } : 単語、st 2 {1, · · · , K} : 隠れ状態
<latexit sha1_base64="95j1Zwp9ihBpwRxmEJu8sRljHRE=">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</latexit><latexit sha1_base64="C6b3Ny8MuTyHF1RsByExBGBU9/8=">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</latexit>

55 / 124 Summer School 数理物理2021 Lecture1

統計的自然言語処理での離散HMM
l 最もわかりやすい例→品詞の学習 (形態素解析)

名詞

助詞

副詞

動詞

首相

が

そう

言う

– 日本語形態素解析器・茶筌はHMMの教師あり学習としてモデル化 (竹内1997)– 半教師あり学習にも不可欠 (Suzuki+2008)
56 / 124 Summer School 数理物理2021 Lecture1

HMMの定式化
隠れ状態

観測値

l 観測系列の背後に、隠れ状態の列が存在
l 観測系列の確率を最大化 :57 / 124 Summer School 数理物理2021 Lecture1

HMMの学習法: 最尤推定

K個

8
>
>
>
>
>
>
<
>
>
>
>
>
>
:

s1
<latexit sha1_base64="6NkJmQgw7NY9u+MHZQkgD3KGrTw=">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</latexit>

s2<latexit sha1_base64="hK2CRmzuRwqrEmdtCA+LA721gFc=">AAAHqXichZXNbtNAEMenLeBSPtrCBYlLRRSEhBRtAhKIUwtUatWqygdpU5oq8ppNasWOLdtJmkR5ASSu9MAJJA6Ix+DCC3DoIyCOReLCgfHakZPstNhKvDvz+49ndte73LVMP2DsdGZ27tLlK9r81YVr12/cXFxavrXrO23PEGXDsRyvwnVfWGZLlAMzsETF9YRuc0vs8eaL0L/XEZ5vOq1XQc8Vh7beaJl109ADNJX8Wq62lGIZJq8VtZGNGymIr7yzPLcBVXgDDhjQBhsEtCDAtgU6+HgfQBYYuGg7hAHaPGyZ0i9gCAuobSMlkNDR2sT/BvYOYmsL+2FMX6oNfIuFPw+VK5BmP9gXdsa+s6/sJ/t7bqyBjBHm0sMnj7TCrS2+vVP681+Vjc8AjhLVBQqO9MU1BVCHp7IWE2tzpSWs0ojid/onZ6VnxfTgPvvEfmF9H9kp+4YVtjq/jc8FUfyA0dNyvAVGqkIHnwakcIwH2BvlytE3kNZhnE/CTyoc2ffwNjHHI1TrsudA99wIx8S7jgmuR3A9gusTXJ/gugTXJbg6wdUJrklwTYJbI7g1gtskuE2C2yK4LYKrEFyF4PYJbp/gGMExhePx/IZrmZpXHs9r5Ffnk8svYeQXhN/GVZ8QVdlXKRdzNCe4yEKRvkL6JBnmFq7wcXZkU+lwlEb8uCKxDwmVJfcNjpZp3biHUuaVivPnVFySX6s9VUliVRXbE1klmu2pnMZVO3H+Bvos3J12FGJ9akVx7K8rVJGgigr1Uu42KchJuo57ULh3RrM5OjWiPKJdifbkwrh4hmWnTyy1sZvLZB9lcoXHqdXn8Wk2D3fhHjzA+E9gFTZw/MuYQwPewXs40R5qBa2ivY7Q2ZlYcxsmLs34BybJj2s=</latexit>

s3

s4<latexit sha1_base64="zaOnXORu7aiIjbqGEEA5fXKOGRk=">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</latexit><latexit sha1_base64="eh41yoXv1JvD/mwGfzwyjq7gIs4=">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</latexit>sT<latexit sha1_base64="yvKYdkAfp9uEUTWovBwvXGJhCT8=">AAAHqXichZXNbtNAEMenLeBSPtrCBYlLRRSEhBRtAhKIUwtUatWqykfTpjRVZLub1Iq/ZDtJkygvgMSVHjiBxAHxGFx4AQ59BMSxSFw4MF47cpKdFFuJd2d+//HM7npXc03DDxg7n5mdu3L1mjJ/feHGzVu3F5eW7+z5TsvTeVl3TMeraKrPTcPm5cAITF5xPa5amsn3tear0L/f5p5vOPZu0HX5kaU2bKNu6GqAppJf260tpViGiWtFbmTjRgriK+8sz21AFY7BAR1aYAEHGwJsm6CCj/chZIGBi7Yj6KPNw5Yh/BwGsIDaFlIcCRWtTfxvYO8wttrYD2P6Qq3jW0z8eahcgTT7wb6wC/adfWU/2d+psfoiRphLF59apOVubfHtvdKf/6osfAZwkqguUWhIX15TAHV4LmoxsDZXWMIq9Sh+u3d2UXpRTPcfsk/sF9b3kZ2zb1ih3f6tfy7w4geMnhbjzTFSFdr41CGFY9zH3jBXDX19YR3E+ST8uMIRfQ9vA3M8QbUqeg50pkY4Jd51SnBdgusSXI/gegTXIbgOwdUJrk5wTYJrEtwawa0R3CbBbRLcFsFtEVyF4CoEd0BwBwTHCI5JnBbPb7iWqXnV4nmN/PJ8auJLGPo54bdw1SdEVfRlysUcjTEuslCkL5E+SYa5hSt8lB3aZDocpSE/qkjsA0Jlin1DQ8ukbtRDKfNSxfkpFZfE12pNVJJYZcX2WFaJZnsip1HVTpy/jj4Td6cdiVifWFEa9tclqkhQRYl6LXabFOQEXcc9KNw7o9kcnhpRHtGuRHtyYVw8w7KTJ5bc2Mtlsk8yucLT1OrL+DSbh/vwAB5h/GewChs4/mXMoQHv4D2cKY+VglJR3kTo7EysuQtjl6L/Ax2+j40=</latexit><latexit sha1_base64="ENRwBZnjFwwyARXl2peR1TGkjbs=">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</latexit>

l 可能なパスは指数的( KT個)に存在…動的計画法(内側確率)t (s) = p(yt = s, x1 · · · xt )X=p(xt |yt = s)p(yt = s|yt 1 = k) t 1 (k)kl デコード時には、確率最大のパスを1つだけ、動的計画法で求める (Viterbiパス)
58 / 124 Summer School 数理物理2021 Lecture1

HMMの学習法: ベイズ推定l MCMC法: 各データの持つ状態系列を実際に

サンプリング à 局所解に陥らない
l Forward Filtering-Backward Sampling (Scott 2002)
文末
確率的に選択– 内側確率を計算しておいて、文末から確率的に選択(確率的 Viterbi)
59 / 124 Summer School 数理物理2021 Lecture1

教師なし品詞解析

ベイズ推定

状態遷移行列最尤推定

l 1990年代: Merialdo+(1994), Kupiec(1992)→失敗
l 2000年代: ベイズ学習で成功 (Goldwater+07, van Gael+09)– Baum-Welchは最尤推定なので局所解にはまる– MCMC法による学習
60 / 124 Summer School 数理物理2021 Lecture1

無限隠れMarkovモデル
l ノンパラメトリックベイズ法により、隠れ状態数

Kすら推定できるl Forward-backwardも可能 (van Gael+ 07)
尤度

0 100 200 300 400 500 600 700 800 900 1000
Gibbs iteration

Log Likelihood

10
9
8
7
K 6
5
4
3
2

-120000-122000-124000-126000-128000-130000-132000-134000-136000
0 100 200 300 400 500 600 700 800 9001000
Gibbs iteration

61 / 124 Summer School 数理物理2021 Lecture1

“Alice in Wonderland”の解析
State 1
she
432
to
387
i
324
it
265
you
218
alice
166
and
147
they
76
there 61
he
55
that
39
who
37
what 27i'll
26

State 2
the
1026
a
473
her
116
very
84
its
50
my
46
no
44
his
44
this
39
$
39
an
37
your
36
as
31
that
27

62 / 124 Summer School 数理物理2021 Lecture1

State 3
was
277
had
126
said
113
$
87
be
77
is
73
went
58
were
56
see
52
could 52know 50thought 44
herself 42
began 40

状態遷移行列State 5way
45
mouse 41thing
39queen 37
head
36
cat
35
hatter34
duchess 34
well
31
time
31
tone
28
rabbit28
door
28
march 26

階層ディリクレ過程 (HDP)(Teh+ 2006)l ディリクレ過程から、さらにディリクレ過程を生成する

63 / 124 Summer School 数理物理2021 Lecture1

HDP-HMM (無限HMM)l なぜHDPが必要?à

例えば、HMMではHDPを使わず別々に状態遷移分布を生成すると、遷移先がバラバラになってしまう

64 / 124 Summer School 数理物理2021 Lecture1

これで充分か…?
l 京大コーパスや国立国語研究所コーパス等の実際の言語の品詞は、階層化されている– 名詞→一般名詞→地名– 動詞→他動詞→サ変l 構文解析でのシンボル細分化 (松崎05, 進藤12など):– VP−1, ADVP-5 のように文法的カテゴリを細分化– ただし、一段階のみしか不可能
l 「品詞体系」を統計的に導出できないか?

65 / 124 Summer School 数理物理2021 Lecture1

階層的な隠れ状態の学習
?
?

?

l 問題:

?

– 各分岐の数を何個にすればよいのか? (無限の選択)– どの深さまで階層を考えればよいのか? (指数爆発)– ノード間の遷移確率をどう考えればよいのか?
ナイーブな方法では不可能！
66 / 124 Summer School 数理物理2021 Lecture1

無限木構造を生成するモデル
l 木構造Stick-breaking過程 (Tree-structured stick-breaking process, TSSB)(Adams+ NIPS 2010):– 無限の深さと分岐を持つ木構造上の離散分布を生成する確率過程– Stick-breaking過程 (=Dirichlet process)の拡張

67 / 124 Summer School 数理物理2021 Lecture1

Stick-breaking process (SBP)l 無限次元の多項分布

を生成する

確率過程
– ディリクレ過程と等価 (Sethuraman 1994)

68 / 124 Summer School 数理物理2021 Lecture1

ディリクレ過程とStick-breaking表現
0.08
0.06

0.06

0.04

0.04
2

0.02

2

0.02
0.00

0.00

0

0

!2

!2
0

0

!2
2

!2
2

l ディリクレ過程からのサンプル

は、次のようにして構成できる (Sethuraman 1994)
kY1
1
X(1 ✓j ), ✓k ⇠ Be(1, ↵)G=⇡k (Xk ) ⇡k = ✓k<latexit sha1_base64="rmMBQtmZKVrdoOb23XI9h4USDn0=">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</latexit>

k=1

j=1
X k ⇠ G0<latexit sha1_base64="uJ3NRsrZxlUXeuCag68Zx/B41TA=">AAAINXichZXPT9RAFMffgrq4/gD0ovHSsIFAApvpitGYkABKhEAILAJLKG7aMsuWbbdNWxaWZv0DTDx78KSJB+Mf4B/gxX/AAwdvxsR48ICJFw++znazP/rANrudee/zfX1vZjqjOabh+YydJHp6L1y8lOy7nLpy9dr1/oHBGxuefeDqfF23TdvNa6rHTaPC133DN3necblqaSbf1MqPQv9mlbueYVee+jWH71jqXsUoGrrqo6kw8FJxjEJZGpmSFL/EfRXbiuPau4Vgf0quPwvKE3JdGpUnIu/+2PjzVIv0DEsKFNeSZnl9VB5XVNMpqWOSomxPyJl7/GgnlQ+DC+5JgaWUkueoOg8muYVRy1Mo0Xdt3xtXjErRr40VBtIsw8QlxRty1EhDdK3Yg73zoMAu2KDDAVjAoQI+tk1QwcN7G2Rg4KBtBwK0udgyhJ9DHVKoPUCKI6GitYz/e9jbjqwV7IcxPaHW8S0m/lxUSjDMvrD37JR9Zh/YD/b3zFiBiBHmUsOn1tByp9D/4tban/+qLHz6UGqpzlFoSJ9fkw9FeCBqMbA2R1jCKvVG/Orxq9O1h7nhYIS9ZT+xvjfshH3CCivV3/q7VZ57jdGHxXhzjKRAFZ86pHGMA+w1c9XQFwhrPcqnxXcqbNF38TYwxxKqVdGz4fDMCEfEu44IrkZwNYI7JrhjgjskuEOCKxJckeDKBFcmuBmCmyG4BYJbILhFglskuDzB5Qlui+C2CI4RHItxWjS/4Vqm5lWL5rXhj8+nJr6Epp8TfgtXfYtQRD9OOZij0cE1LBTpxUiPJMPcwhXezjZtcTocpSbfrmjZ64TKFPuGhpZuXbuHUq7EKl45o+I18bVaXZW0rHHFUkdWLc1SV07tquUofx19Ju5OyzFirmtFadifi1E5gsrFqMdit0lDVtBF3IPCvbMxm81To5FHY1eiPdkwLp5hcveJFW9sZDPy3Ux2dTI9PRudZn1wB4ZgFOPfh2mYx/FfxxxOE7cTQ4l08mPya/Jb8nsD7UlEmpvQcSV//QPqd7/5</latexit>

69 / 124 Summer School 数理物理2021 Lecture1(k = 1, · · · , 1)

Stick-breaking process (3)l ディリクレ過程⇡k = ✓ k

kY1

は、SBP(α)で表現できる
0.06

(1

✓j ),

0.04

0.00
0
!2

j=1

✓k ⇠ Be (1, ↵),X k ⇠ G0 ,1X
G=⇡k X k<latexit sha1_base64="/vX9NbHRvgN3PFsYAHLN3iHHhy0=">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</latexit>

k=1

2

0.02

0

!2
2

0.08
0.06
0.04
2

0.02
0.00
0
!2
0

!2
2

– G0からランダムに選んだ場所 Xk に、高さ ⇡k の棒を立てていったものがG<latexit sha1_base64="JIvdKCSpPatHnXj1m5sUkZ9u9tk=">AAAIz3ichZXNaxNREMCnrR9r/WirF8FLMUREpLxEQREKrVqwtJQ2sW1KW8Lu9qVdsl/ubpOmIeJVwas9eFLwIP4ZXvwHPBT/AvFYwYsH581uyMdO7IbNvjfzm9mZN2/nGb5thZEQx0PDI2fOnjuvXRi9eOnylbHxiatrobcfmHLV9GwvKBl6KG3LlauRFdmy5AdSdwxbrhvVJ0q/XpNBaHnu86jhy21H33WtimXqEYqKpXK1PJ4RU4KuyfQglwwykFzL3sTID9iCHfDAhH1wQIILEY5t0CHE3ybkQICPsm1ooizAkUV6CS0YRdt9pCQSOkqr+L+Ls81E6uJc+QzJ2sS32HgHaDkJWfFdfBYn4pv4In6KvwN9NcmHiqWBTyO2lX557M314p9TrRx8RrDXsfqPhYH04JyaeDso28Pni1Nyj6ACDylnC9fAJ4laDTOOo3Z4dFJ8VMg2b4mP4heuwwdxLL7iSri13+anFVl4j96zVBeJnraghk8TMliLJs7aORmoa5K0lcTT4XstPJoH+LMwxj201mnmQX2ghwPmXQcM12C4BsMdMtwhw9UZrs5wFYarMFyV4aoMN8twsww3z3DzDLfAcAsMV2K4EsNtMNwGwwmGEynOSOqr9jxXVyOpa6xP19OgL6Ktl4zewV3fIbZonqbU92T1cLGEI8MUGbKkik3t8G62LUvTEX3RMd9t0ZG3GCub+ouBkn67bg1nuZzKeHlAxkX6Wp2+TDrStMViT1Qdm8W+mLqtlpL4TdTZ2J2WUsRc344ycD6XogoMVUhRT6nbZCBPdAV7kOqdcTXbp0scR9yVeE0+5XeHurKi7+IdUt1UTw5QbqN8h3xN478iWtRfJ7H/xVUOcBx7Uv3aond5OArpHS5q6tS/aygPSNM+KVVvUjn46DOLtIqg+5zYpJOyguR0SmfjGsX5yySq+Iy1qcIu3KEsYyuX4us6f/Ckz/Wf6+nBWn4qd28qv3I/M/M4OfM1uAE34Tau7gOYgWe4+1Yxgl14C+/gSFvR6tpL7VWMDg8lNteg59Je/wP+qsoz</latexit>

70 / 124 Summer School 数理物理2021 Lecture1<latexit sha1_base64="RS949+Krewc+pqhN7MQs3EGVTpM=">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</latexit>DPのstick-breaking表現の証明
l まず、

G=

1
X

k=1
<latexit sha1_base64="WNd5huVRf33fRspfkUplFEpQU6o=">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</latexit>

から

✓k

kY1

(1

j=1

✓j ) · Xk= ✓1 X1 + (1✓1 )✓2 X2 + · · ·d✓1 )G0G = ✓1 X1 + (1<latexit sha1_base64="y7J6fXNHLTHq1iljpLIfWIhB7yU=">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</latexit>が成り立つことに注意する. ここでG’はGと同じ分布からの独立なサンプル.

71 / 124 Summer School 数理物理2021 Lecture1

DPのstick-breaking表現の証明 (2)l よって、X の可測な分割 A1 , · · · , Ak に対して、<latexit sha1_base64="Y0EM13qMiMCy4/N6GJJ7vEk4ekE=">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</latexit><latexit sha1_base64="7yuU+E31i44KF9aHAiYW2twkmEU=">AAAIoXichZXNbtNAEMenlI9QPtrCBYlLRRSEUFVtAhIICakBKlG1qtKUtqmSKordTWvFX9hu0jTqC/ACHHoC0QPiGThx4QU49BEQxyJx4cDfa0f58ITaSrw785vd2Z3dGc01DT8Q4nTswvjFS5evpK5OXLt+4+bk1PStDd/Z93S5rjum45W0mi9Nw5brgRGYsuR6smZpptzUGi9D/WZTer7h2G+Ctiu3rdqubdQNvRZAVJ2azFezsxV9xwn82Xy1MVGdSos5oZ6ZZCMbN9IUPwVnevwrVWiHHNJpnyySZFOAtkk18vGWKUuCXMi2qQOZh5ah9JKOaAK2+6AkiBqkDfzvoleOpTb64Zi+stYxi4mfB8sZyogf4rM4E9/FF/FT/B05VkeNEfrSxleLbKVbnXx3Z+3PuVYWvgHt9az+Y6GBHr2mDn4WZHv4vj1n7QHV6alas4E9cJUk3A098qN5+P5s7Vkx07kvPopf2IcP4lR8w07Yzd/6yaosHmP0jIqLxEgVauKrUxqx6KDXXZMGXUdJj2J/evyghaP6Hl4DPu7BuqZ6DrVGjnDAzHXAcG2GazPcIcMdMlyL4VoMV2e4OsM1GK7BcHmGyzPcIsMtMtwSwy0xXInhSgy3xXBbDCcYTiQ4LY5veOa5uGpxXCN9Mp6auhFdvWT0Fk59j6iofpIK75MxwEUSjvQTpM+SoW/hCe9nu7IkHagbHfH9Fj35EWNlqvyiQTJs16/hLAuJFRdGrHhN3VZraCU9adJiecCrns3ykE/9Viux/zp0JrLTSoJYGDpRGvoLCarIUMUE9UplmzTlFF1HDgpzZxTNbnWJ/IiyEq/JxeNm0HPiSHloR7OEOddQvIOWr+a3oWmpHNyE3FOabrUL80voh4sxM6DDOftzfVlVuzrI5wmdiXVGa5DQzGKEqE6aKko2PVSeRla28q+vhqBaZ4drc7KxkZvLPprLrT5Oz7+I63aK7tI9eoAdekLz9BonaF3V72P6RCepdGoxVUgVI/TCWGxzmwaeVPkfGLTBrg==</latexit>

この測度Gはd(G(A1 ), · · · , G(Ak )) = ✓1 ( X1 (A1 ), · · · , X1 (Ak ))+ (1<latexit sha1_base64="sMaZlNtmO/Rb0cnQrkEdasoSI80=">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</latexit>✓1 )(G0 (A1 ), · · · , G0 (Ak ))を満たす.
l W = ✓1 , U = ( X1 (A1 ), · · · , X1 (Ak )),<latexit sha1_base64="ypPIxXNR9N0+rLlbzb8CnjnarBM=">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</latexit>

V = (G(A1 ), · · · , G(Ak )) とおくと、<latexit sha1_base64="gda1W4RrzUFtWo5PebhMArTJMIQ=">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</latexit>d

V = W U + (1W )V<latexit sha1_base64="+l8ryflTQPjsK9w88rQsaAjpPm8=">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</latexit>となり、これを満たすVは一意に定まる.

72 / 124 Summer School 数理物理2021 Lecture1

補題4
l 補題1: W 2 ( 1, 1) , U を確率ベクトルとする. U と同じ<latexit sha1_base64="qYWT4fxafx/YGTf3lZsiCFcyL9Y=">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</latexit><latexit sha1_base64="qYWT4fxafx/YGTf3lZsiCFcyL9Y=">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</latexit><latexit sha1_base64="sOpvbkFYBgr2EIfXg8FZSW8KjGs=">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</latexit>

次元の確率ベクトル V が W, U とは独立で<latexit sha1_base64="dmpUrhmRl+GATf2msj3tV/VlTL4=">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</latexit><latexit sha1_base64="Q3D4vkb5U9NV0YYOxUT5PMjGMZc=">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</latexit>d

V = U + WV(*)

<latexit sha1_base64="yVj7LEoX1ayxaEXFU4xPm5XyPnE=">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</latexit>のとき、V の分布は一意に定まる.
Proof.
V, V’ がともに(*)を満たすとする. (Wn,Un)を(W,U)の
独立なコピーとし, V1 = V, V10 = V 0とすると、(*)から⇢⇢
V2 = U 1 + W1 V1
Vn = U n + W n Vn
同様にして
0
0
V2 = U 1 + W 1 V 1
Vn0 = Un + Wn Vn0<latexit sha1_base64="dmpUrhmRl+GATf2msj3tV/VlTL4=">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</latexit><latexit sha1_base64="MNldfA+xwFJ1wUd+r4XA4CvU4cM=">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</latexit><latexit sha1_base64="BgIRc4RAHMns1WQLWut3KG4Mqzw=">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</latexit><latexit sha1_base64="lWLiG20fB3Y1gVYn9NT97IL7vOs=">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</latexit>

73 / 124 Summer School 数理物理2021 Lecture1

補題4 (2)⇢<latexit sha1_base64="lWLiG20fB3Y1gVYn9NT97IL7vOs=">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</latexit>

Vn = U n + W n Vn
Vn0 = Un + Wn Vn0

l を順番に作ったとき、|Vn+1<latexit sha1_base64="/h5CcpL1DB1z1NWonk9xLm4ctSQ=">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</latexit>0
Vn+1| = |Wn ||Vn Vn0 | = · · ·nY=|Wi | · |V V 0 | ! 0 (* |Wi | < 1)i=1d

よって V = V 0 . ⇤l これから、<latexit sha1_base64="hcJEXZ73u2oMK+kVjDhNj5aBeQc=">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</latexit>

V = W U + (1
= (1

W )V

W 0 )V + W 0 V<latexit sha1_base64="bf4NLZ2jiRBC9Xb9/zt1oYm7T6s=">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</latexit>

を満たすVは一意に定まる. □74 / 124 Summer School 数理物理2021 Lecture1(W 0 = 1W)DPのstick-breaking表現の証明 (3)l 元に戻って、d

V = W U + (1W )V<latexit sha1_base64="+l8ryflTQPjsK9w88rQsaAjpPm8=">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</latexit>

ゆえ、V ⇠ Dir(↵(A1 ), · · · , ↵(Ak )) のとき<latexit sha1_base64="bCdYD7IuX9vGR3ZhOZftkkXLaHg=">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</latexit>

W U + (1<latexit sha1_base64="YQqpvAYNxmVjFthL6PFrQXRIwp8=">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</latexit>

W )V ⇠ Dir(↵(A1 ), · · · , ↵(Ak ))を示せばよい. ただし W ⇠ Be(1, ↵) .
l Uは特定のjについて U = ej となるから、左辺のUの条件付き分布は<latexit sha1_base64="BQv8WP1rIpQEgm/BpP6WZc/V94o=">AAAI53ichZXNTxNREMAH8WPFD0AvGi+NTQ0aQl6ricaEBFASCYRAESihhOwur3TT/XJ3aSlN492LNzXxhAkHY6J/hBf/AQ/Ev8B4xMSLB+e93aYfbyrbbPe9md/Mzrx5O8/wbSuMGDseODN49tz5C9rFoUuXr1wdHhm9thZ6e4HJV03P9oKCoYfctly+GlmRzQt+wHXHsPm6UXki9OtVHoSW5z6P6j7fcvRd1ypZph6haHvk5nqqGFpOqlEMnNQMb45lx4u67Zf1u9sjaTbB5JVSB9lkkIbkWvJGB39AEXbAAxP2wAEOLkQ4tkGHEH+bkAUGPsq2oIGyAEeW1HNowhDa7iHFkdBRWsH/XZxtJlIX58JnKK1NfIuNd4CWKciw7+wjO2Hf2Cf2k/3t66shfYhY6vg0Ylvubw+/urHy51QrB58RlNtW/7EwkO6fUwNvB2VlfL44JfcISvBI5mzhGvhSIlbDjOOoHrw9WXmczzTusA/sF67DITtmX3El3Opv82iZ59+j94ysC0dPRaji04Q01qKBs1ZOBuoaUtpM4mnz3RaenAf4szDGMlrrcuZBra+HfeJd+wRXJ7g6wR0Q3AHB1QiuRnAlgisRXIXgKgQ3TXDTBDdHcHMEN09w8wRXILgCwW0Q3AbBMYJjCmck9RV7nqqrkdQ11qv1NOQX0dJzQu/grm8TRTlXKfE9WV1cLKHIUCFDkhSxiR3eybZkKh3JLzrmOy3a8iZhZcv+YqCk165TQ1kuKRkv9cl4RX6tTk8mbalqsdAVVdtmoSemTqvFJH4TdTZ2p0WFmO3ZUQbOZxUqT1B5hXoqu00acpIuYQ8SvTOuZut0ieOIuxKtySl+d2RXFvQ43qGsm+jJAcptlO9IX5P4L4im7K8p7H9xlQMcx55Ev7bkuzwchfIdLmpqsn9XUR5ITeukFL1J5OCjzwzSIoLOc2JTnpQlJCcVnY1rFOfPk6jiM9aWFXbhnswytnJlfB3nD5702d5zXR2s5Say9ydyyw/SUzPJma/BLbgNY7i6D2EKnuHuW8UIXsIRfIYvmqW91t5o72L0zEBicx26Lu3wH4cb0mQ=</latexit><latexit sha1_base64="cAaifvqW/4ML9r8z5mdjQ/uKQGA=">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</latexit>

W U + (1W )U | U = ej(j)<latexit sha1_base64="dFd3ZAZ2DIc6TwcDf40DniPcr3o=">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</latexit>

– ここで

(j)

⇠ Dir(↵1 , · · · , ↵k )
(j)

(j)

↵i = ↵i (i 6= j), ↵i = ↵i + 1 (i = j)<latexit sha1_base64="ITataSOIVrMhZx7sFtFvpwm8g5Q=">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</latexit>

75 / 124 Summer School 数理物理2021 Lecture1

補題5
l U ⇠ Dir(↵1 , · · · , ↵k ), V ⇠ Dir( 1 , · · · , k ),PP<latexit sha1_base64="/2HNxTlSHKmpZh/GJDtr0PxM8dE=">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</latexit>

W ⇠ Be (<latexit sha1_base64="OURehLPTAIi05gKN2f7nrXIrM+I=">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</latexit>のとき、k ↵k ,

W U + (1<latexit sha1_base64="Ah2olpzkskuqICKvsimKg4kasvo=">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</latexit>kk)W )V ⇠ Dir(↵1 +l Proof: Z1 , · · · , Z2k について

1 , · · · , ↵k +k)<latexit sha1_base64="d39LMO20ApwjVEBI2o2j5CgQTYU=">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</latexit>

Zj ⇠ Ga(↵j , 1), Zk+j ⇠ Ga( j , 1)<latexit sha1_base64="CPu1TykFeNL9QLiUdE8G08Mw9/4=">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</latexit>(j = 1, · · · , k)<latexit sha1_base64="4LFE0EKsXJJahUxZ2Mnsgd+5DbI=">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</latexit>とする. ディリクレ分布は正規化ガンマ分布として書けるので、d(U, V, W ) =<latexit sha1_base64="kqtlIrI0KcvyfNY45/3xxi+nR2g=">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</latexit>

Pk

Z1
Zk
Zk+1
Z2k
j=1 ZjP,··· , P,P,··· , P
, P2k
Z
Z
Z
Z
j j
j j
j k+j
j k+j
j=1 Zj76 / 124 Summer School 数理物理2021 Lecture1!

補題5 (2)l このとき、U W + (1d

W )V ==<latexit sha1_base64="EckIxeAJFyndER5X8SWPPhHXGag=">AAAKUXichZVLb9RIEMcrA+wSw24IXEBcLEaJks0QtYeVFiFF4imBQCgk5AHp7Mj29GSc8WNieyYPyxeO+wX2wImV9oD4GHvZO+KA+ASII0hoJQ5b7vYwD9eARx53V/3+1dXddrXVdp0oZuztROnY8RM//HhyUjt1+qefp85Mn12Pgk5oizU7cINw0zIj4Tq+WIud2BWb7VCYnuWKDat1K/NvdEUYOYH/OD5si23P3PGdhmObMZpq0xPP1jb0BX3OuLwxv67N8rrY07krGvGcxhuhaSdPa0aa8Kjj1Xb1p7XdNPcq54BdtZPdJSP9Pam2UmnkobPTjOe1heGYX2VJa2E3paVaT9tLA1kjTUe1FY3b9SCOerjOuTa7NG4KI2PgxKnoI1RFLwyxZYiDbVytyPH0hIeeftsJU16Z46bbbpo1Y4FbIsZnRSkrub2V21vzvLKo8b2OWec3g4PamTJbZPLSiw0jb5Qhv5aD6WPvgEMdArChAx4I8CHGtgsmRPjbAgMYtNG2DQnaQmw50i8gBQ21HaQEEiZaW/i/g72t3OpjP4sZSbWNo7h4h6jUYYa9YS/ZR/Yve8Xesy9jYyUyRpbLIT4tpRXt2tQf51c/f1fl4TOGZl/1DYWF9Pg5JXh7aGvic+87c4+hAVflnB1cg7a0ZKthqzy6R39+XL22MpPMsr/YB1yHF+wt+wdXwu9+sv9+JFaeY/QZuS8CI3Ho4tOGMu5Fgr3enCz0JdKa5vn0+WFFIPsh/hzMsYlqU/YC2B8b4YAY64DgDgnukOCOCO6I4PYJbp/gGgTXILgWwbUI7gbB3SC4ewR3j+DuE9x9gtskuE2Ce0JwTwiOERwrcFa+v9k7T+2rle+r8hf305JfRM8vCL+Hb32f4LJfpLLvyRnilIUiowIZkWSWW/aGD7I9W5GO5Ret+EFF354SKlfWFwsto7pBD6VcLsx4ecyMV+XX6o3MpG8tKh4MZdXXPBjJaVD1MM/fRp+L1elhgbgz8kZZ2L9ToFYIaqVA3ZbVpgxVSTewBmW1U+1m73RReaiqRHuqhbh1WZUzuoJ3JPctq8kh2l2012WsJfzPiFTWVx3rn9rlENsqUlavHTlWgK1IjuGjZ1/W7y7aQ+npnZRZbcrm0MaYM0hnGQyeE1vypGwguVTwubhGav4iz0qdsa7cYR9+kbNUKl/mN3D+4ElvjJ7rxcZ6ddG4slh99Gv5+s38zD8JF+ESzOHq/gbX4S6+fWtgT7wvTZXOly5Mvp78TwOtpNDSRK45B0OXdup/vtVQVg==</latexit>

P

j Zj
P2k
j=1 Zj

Z
P 1
j Zj
Z1

!

+

Zk+1
+
,···
P2k
P2k
j=1 Zjj=1 Zj⇠ Dir (↵1 +

1 , · · · , ↵k +77 / 124 Summer School 数理物理2021 Lecture1

P

j Zk+jP2k
j=1 Zj

!

k) .

⇤

!

Z
P k+1 , · · ·j Zk+j!

DPのstick-breaking表現の証明 (4)l Uについて期待値をとれば、

W U + (1 W )V⇥
= EU W U + (1=kXW )V | U = ejp(U ) p(W U + (1W )V )⇤

j=1=<latexit sha1_base64="myKPFG5svvg7pX9TZVE0ANkWez8=">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</latexit>

k
X
↵j
j=1

↵

(j)

(j)

Dir(↵1 , · · · , ↵k ) ⇠ Dir(↵1 , · · · , ↵k ) .
ここも補題が必要だが省略(単純計算)

78 / 124 Summer School 数理物理2021 Lecture1⇤

Stick-breaking process (SBP)l 無限次元の多項分布

を生成する

確率過程
– ディリクレ過程と等価 (Sethuraman 1994)

79 / 124 Summer School 数理物理2021 Lecture1

Stick-breaking process (4)l SBPの要素

はベータ分布に従う→ の事後分布が求まる
l 無限次元多項分布 からk番目の値が選ばれる

までは棒を折り続けるで止まる

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Chinese District Process (CDP)l SBPのCRP表現 (Paisley+ (2008))
l k番目の領域(番人)で止まった回数を通過した回数を

81 / 124 Summer School 数理物理2021 Lecture1

とすると、

、

Chinese District Process (2)l よって、CDP表現を使えば、の事後期待値は– 基本的に、止まった数と通過した数を数えているだけ!
l DP(α)・CRP(α)・CDP(α)はすべて等価な表現

82 / 124 Summer School 数理物理2021 Lecture1

階層的離散分布l 最も簡単な階層的離散分布:

SBPの各stick(Polya trees)

を、再帰的にさらにSBPで分割

l これだと、データは常に最も細かいカテゴリにしか存在しない– 「よくわからないが、動詞なことは確か」な言葉?
– “thing”, “way” など、抽象的な名詞?

83 / 124 Summer School 数理物理2021 Lecture1

Tree-structured stick-breaking processl TSSB: 先にまず、“そのカテゴリで止まる確率”

を生成(1)
で棒を分割して、 を生成(2) 残ったをSBP(γ)で分割して、各stickに同じ操作を適用.

SBP(γ)

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TSSBの構成lから実際に生成したサンプル

TSSB(1,1,1)

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TSSB(1,5,0.5)

TSSBのCRP(CDP)表現l 客を木の根から辿って追加– 確率νでそのノードに残る– 確率(1-ν)で子供に降り、CRPで子供を選択
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TSSBの確率モデル
l TSSBは無限木構造に対応し、そのノードは整数列で番号づけられる (例:
l ノード

の確率

)

は、縦方向と横方向のSBPの積

SBP(γ)

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Posterior of TSSBl Probability of descent at

node s is: letbe thenumber of times customers
stop at s andbethe times customers havedescent,

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Posterior of TSSB (2)l In a descent from node s,

probability to choose achild k is given by SBP:l Let

be the number ofhorizontal customers stayed at node t andbe the numbers left,

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TSSB as a diffusion priorl [0,1) contains everything in the world!

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Hierarchical clustering with TSSBl Adams+(NIPS 2010)
l CIFAR-10
image data,
infinitehierarchy
modeling

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TSSB上の状態遷移モデル

木のノードからノードへの状態遷移

l HMMでは、無限木構造のノード間に状態遷移
l 木構造の各ノードが、次の時刻の木構造への確率分布を持っている– 普通のHMMのときは単純なKxKの遷移行列
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TSSB上の状態遷移モデル (2)TSSB

l 各ノードが、次の状態への確率分布(TSSB)を持っている93 / 124 Summer School 数理物理2021 Lecture1

TSSB上の状態遷移モデル (3)lは独立ではない!
– [1 2 4] =「名詞-固有名詞-一般」からの遷移確率は、[1 2] =「名詞-固有名詞」を引き継いでいる– [1 2] は [1] を、[1] は [] に影響されている階層モデル!
TSSB

94 / 124 Summer School 数理物理2021 Lecture1

階層的木構造Stick-breaking過程 (HTSSB)lを構成するSBP(=ディリクレ過程)が、親のの対応するディリクレ過程から生成されている→階層ディリクレ過程 (HDP)
HDP

TSSB

TSSB

HDP
HDP

95 / 124 Summer School 数理物理2021 Lecture1

Random draws from HTSSB
親TSSB

···
<latexit sha1_base64="2TZXmgjNO9vjxU4OGFWNeCp6ob0=">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</latexit>

子TSSB

96 / 124 Summer School 数理物理2021 Lecture1

HTSSB (2)l HDPのStick-breaking表現より、データDの下で–は親の でのの値
l 親のはさらにその親のに依存！
à (すさまじい)再帰的な計算が必要

97 / 124 Summer School 数理物理2021 Lecture1

HTSSB (3)がわかれば、 の各要素sでの確率

lは98 / 124 Summer School 数理物理2021 Lecture1

HTSSBの学習
l TSSB ßà CDPで事後分布
l HTSSB ßà HCDP

(階層的Chinese District Process)で事後分布– HDPに対する階層的CRPと同様– 詳細は、論文を参照ください

99 / 124 Summer School 数理物理2021 Lecture1

無限木構造HMM (iTHMM)l HTSSBにより、無限木構造上の状態遷移確率とその事後分布が計算できる→ HTSSB-HMM = Infinite Tree HMM (iTHMM)

TSSB

100 / 124 Summer School 数理物理2021 Lecture1

iTHMMの単語出力確率l 親子関係にあるとは独立ではない– [2 1]=“動詞-動作” 〜 [2]=“動詞”
l 本研究では、階層Pitman-Yor過程 (Teh 2006)を用いる• ハイパーパラメータd,θも自動推定• カウントの追加/削除
で、木構造上の分布が自動的に更新101 / 124 Summer School 数理物理2021 Lecture1

無限木構造HMMの生成モデル
l iTHMMの生成モデル(1) TSSBを生成.
(2) 無限木構造の各ノードsについて、(a) 状態遷移確率
を親の
から
と生成.
(b) 単語出力確率分布

を親の

と生成.
l BOSから始めて、隠れ状態列を生成.

102 / 124 Summer School 数理物理2021 Lecture1

から

と単語列

iTHMMの学習

l Gibbsサンプリング (Goldwater+2007)–

を次々とサンプリングà正しい値に収束

103 / 124 Summer School 数理物理2021 Lecture1

iTHMMの学習 (2)l 問題:を数え上げられない!
–=[], [1 1], [1 1 2], [2 4 3], [17 5 3], ‥‥と無限に候補が存在– iHMMのように、確率的に右側を切り落とすことはできない– どうするか?

104 / 124 Summer School 数理物理2021 Lecture1

iTHMMの学習：注意l TSSBの全てのノードは、[0,1) に含まれる区間のどれかに対応している→[0,1)の乱数rでノードを選べる
10r<latexit sha1_base64="N8UiF8I7aRKBFnQwEchSntCV4D8=">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</latexit>

105 / 124 Summer School 数理物理2021 Lecture1

iTHMMの学習 (3)尤度l 基本的な考え方:– からをランダムにサンプルするには、まずからを一様に選び、それを尤度に従って選べばよい

106 / 124 Summer School 数理物理2021 Lecture1

iTHMMの学習 (4)
尤度

l 解法:

–

から一様にサンプリングするには、先に一様乱数を決め、対応するノードを選べばよい(Retrospective sampling; Papaspiliopoulos 2008)– 次に、に比例してスライスサンプリング

107 / 124 Summer School 数理物理2021 Lecture1

iTHMMの学習 (5)(1) 現在の確率から、スライスを作る(2) 一様乱数 r〜Unif(0,1] を引いて、対応するノードを求める- が存在しなければ作成(3)
なら をaccept(4) そうでなければ、乱数の範囲を左右に変更して(一種の二分探索)、(2)に戻る

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実装
l C++で7000行程度– boost::serializationのお蔭– 現在, 数1000単語/秒のサンプリング速度l 無限木構造を必要に応じて実体化– ノード
からの遷移を表すTSSBで新しいノードが作られた際、もとの木構造自体を拡張– 各ノードsのTSSBが、もとの木構造自体と自己同型になっている (ポインタが張られている)
l 状態の参照カウントを管理して、Gibbsのiteration
毎に不要な状態を削除して全体をリナンバー

109 / 124 Summer School 数理物理2021 Lecture1

実験 (1)l 教師なし学習: “Alice in Wonderland”, 学習1200文,

テスト231文

110 / 124 Summer School 数理物理2021 Lecture1

実験 (1)l 教師なし学習: “Alice in Wonderland”, 学習1200文,

テスト231文

111 / 124 Summer School 数理物理2021 Lecture1

実験 (1)l 教師なし学習: “Alice in Wonderland”, 学習1200文,

テスト231文

112 / 124 Summer School 数理物理2021 Lecture1

実験 (1)l 教師なし学習: “Alice in Wonderland”, 学習1200文,

テスト231文– 学習が終われば、尤度の計算は通常の前向きアルゴ
リズム

113 / 124 Summer School 数理物理2021 Lecture1

実験 (2)l 半教師あり学習: 京大コーパスから10000文の品詞を教師ありデータとして固定、37400文をサンプル

114 / 124 Summer School 数理物理2021 Lecture1

実験 (2)l 半教師あり学習: 京大コーパスから10000文の品詞を教師ありデータとして固定、37400文をサンプル

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実験 (2)l 半教師あり学習: 京大コーパスから10000文の品詞を教師ありデータとして固定、37400文をサンプル

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実験 (2)l 半教師あり学習: 京大コーパスから10000文の品詞を教師ありデータとして固定、37400文をサンプル

117 / 124 Summer School 数理物理2021 Lecture1

実験 (3)l “未知の言語” : クリンゴン語、Star Trekの宇宙人語
l クリンゴン語「ハムレット」– 3733行, 19927語
Qo'noS ta'puq Hamlet lotlut
lutvaD ghotvam luDalu'
Qo'noS ta' ghaH
ben ta' puqloD; DaHjaj ta' loDnI'puqloD je ghaH
Qang ghaH
Hamlet jup ghaHpolonyuS puqloD ghaH
toy'wI'pu' chaH

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実験 (3)l 1＝副詞&呼びかけ?

– tugh=“soon”, DaH=“now”, vaj=“then”

119 / 124 Summer School 数理物理2021 Lecture1

実験 (3)l 2＝動詞?

– ‘el=“go”, mej=“leave”, vaj=“then”, Ha’=“let’s go”

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測度の空間と分割
l 通常のHMMは、出力確率測度全体の空間を分割して、各クラスタの間の遷移を考えている
ことと等価

121 / 124 Summer School 数理物理2021 Lecture1

再帰的分割とiTHMM
l iTHMMは、状態空間を再帰的に分割して、より細かい遷移を表現– カウントの多さに応じた階層ベイズスムージング

122 / 124 Summer School 数理物理2021 Lecture1

まとめ
l 木構造Stick-breaking過程 (Adams+ 2010)をそれ自体、無限木構造上で階層化した
階層的木構造Stick-breaking過程を提案＝ Infinite Tree HMM– 自然言語処理や品詞推定に限らない、HMMの本質的な拡張l HMMの状態空間の再帰的な分割+ベイズ推定
l 「品詞体系」の教師なし学習が初めて可能に– ハイパーパラメータの推定など、学習にはまだ課題がある

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課題
l Forward-backward– 通常の方法では無理だが、状態はすべて[0,1)の範囲で表せるため、Embedded HMM (Neal 2004)が使える可能性が高い
l 行列式点過程 (Determinantal point process)による、
重複した状態の抑制
l トピックモデルへの適用 (研究中)

124 / 124 Summer School 数理物理2021 Lecture1