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Dbda03

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Dbda03

  1. 1. Doing Baysian Data Analysis Chap 3. What Is This Stuff Called Probability? Yoshifumi Seki 2013/08/03 @Matsuo Lab. Summer Seminar
  2. 2. 3.1 The set of all possible events • 確率を考えるために起こりうるすべての結果を考えよう • コインを投げるとき – 表がでるのか – 裏がでるのか – 横になるのか • 事象は2つ – コインが均一に作られていれば表がでる確率: θ=0.5 • p(θ)を考える – コインが均一に作られている確率はいくらか – p(θ=0.5) = 0.99 – p(θ=0.1) = 0.0001 • 「どのような結果がでるか」も「その結果がどれぐらい信頼 できるか」も両方確率である.
  3. 3. 3.1.1 Coin Flips: Why You Should Care? • なんでそんなにコインを投げることにこだわる の???? • コインの表裏なんて人生に関係無いじゃん? – でも薬が効くか効かないかとかだったら重要だよね? – コインの表裏も薬の効果も本質的にはいっしょなんだよ! – だからみんながんばろうね!
  4. 4. 3.2 Probability: outside or inside the head • 確率の事象には outsideなものとinside the headなもの がある – outside • この世で起こっている誰もが観察できる事象 • コインの投げるとか、サイコロを振るとか • 一定数の試行によって一律に収束する – inside the head • 主観的な確率 ( subjective belief ) • ギャンブルの話 – 1月1日の大雪で道路が通行止めになると$100,コインが表 だと$100もらえるのいずれか » コインのほうを選ぶよね – 明日雨が振ると$100もらえる,サイコロが1をでると$100もら える » これだとどうだろうー » 雨を降る確率が50%はないだろうけど,10% ~ 20%ぐらい あるかもなーって思ってる感じ • こういった結果にたいする信頼性の度合いも確率で表現できる
  5. 5. 3.3 Probability Distributions • コインを投げるとか,サイコロを振るとかっていう話 – 離散的な数値 – 1が1/6で出るとか • 1日に成人男性が消費するカロリー – 連続値 – ex: 平均2000カロリー – あるときは2345.223かもしれない,あるときは1734.2かもしれ ない • 確率的な話だけど離散値みたいには表現できない – 区間に区切ってみよう • 1500未満, 1500以上2000未満, 2000以上2500未満, 2500以上 • これなら確率として表現できるね!
  6. 6. probability density • ボードゲームなどで使われる”spin”で針がど こを指すかを考える • SpinがFairであれば – 均等に2分割した時それぞれの領域が選ばれる確 率は0.5 – 均等にN分割したときそれぞれの領域が選ばれる 確率は1/N – Nを大きくしていくと確率はどんどん小さくなっ ていく • 人間にとってわかりにくい・計算しにくい • Spinの針が指す確率を考える代わりに幅を考 えることにしよう – これを確率密度という – 1/Nのsectorに止まる確率は1/N => (1/N)/(1/N) = 1 – そのsectorにおけるprobabilitu massを表す – FairなSpinにおいては確率密度はすべて1
  7. 7. probability density • scaleを0から0.5にする – 0から0.1を考える • probability : 0.2 • width : 0.1 • density : 0.2/0.1 = 2 – 一般化 • width: w • probabilty : 2w • density : 2 • scaleを1.0から100に,進 み方を対数にする – 図のような形になる
  8. 8. 3.3 Probability Distributions p([x,y]): xからyの確率 p(x) : 確率密度
  9. 9. 3.3.3 Mean and Variance of a Distribution • Mean – 期待値 – サイコロ • 1/6 * 1 + 1/6 * 2 + … + 1/6 * 6 = 3.5 – 連続値でどのように扱うか?
  10. 10. • Variance – 分散 – 期待値からどれだけ分布が離れているか? • Mean Squared Deviation (MSD) 3.3.3 Mean and Variance of a Distribution
  11. 11. 3.3.2.2 The Normal Probability Density Function • 一番有名な分布 – ガウス分布・正規分布 – E[p(x)] = μ – Var[p(x)] = σ^2
  12. 12. Variance as Uncertainty in Beliefs • p(θ) : θの信頼出来る程度を示 す • Variance – どれだけ分布が広がっているか • Varが大きいと正規分布だとよ こにでかくなる • Varが小さい=> certainである – ある領域に定まる • Highest Density Interval(HDI) – 分布のW%がどの範囲に収まる か?
  13. 13. 3.4 Two-Way Distribution • 同時確率分布 – コインを3こ同時になげたときにどうやって確率を表現する か? • 表が何回でるか? • 何回結果が変わるか?
  14. 14. 同時分布と周辺確率 • 同時確率 – P( S, H ) • S回入れ替わった時にH回表がでる確率 – S = 0, H = 0 : P(S, H) = 0.0 ( 裏裏裏 ) – S = 1. H = 1 : P(S, H) = 0.25 (裏裏表, 表裏裏) • 周辺確率 ( marginal ) – P(S), P(H) • P(S=0) = 2/8
  15. 15. 3.4.1 Marginal Probability • 離散値 • 連続値
  16. 16. 3.4.2 Conditional Probability • 条件付き確率 • コインを3回投げるとき – 表が2回でる時に,何回表裏が入れ替わるのか? • 表が2回でる事象 – 表表裏: S=1 , 表裏表 : S=2, 裏表表: S=1 » P(S=1| H=2) = 2/3 » P(S=2| H=2) = 1/3 – P(H=2,S=1) = 2/8 , P(H=2) = 3/8 – P(S=1|H=2) = 2/3
  17. 17. 3.4.3 Independence of Attributes • 事象が独立であるときに,条件付き確率はどのようにな るか? – サイコロを2こふったとき、1こ目が1だった時に2個めいく つがでるか • まぁ関係ないよね! – なので以下のようになる

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