Bayesian statistics Tokyo.R#94

第94回Tokyo.Rでトークした際のスライド資料です。

Transcript

1. #94
   @kilometer00
   2021.09.11
   BeginneR Session
   -- Bayesian statistics --
   

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2. Who！？
   Who?
   

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3. Who！？
   ・ @kilometer
   ・Postdoc Researcher (Ph.D. Eng.)
   ・Neuroscience
   ・Computational Behavior
   ・Functional brain imaging
   ・R： ~ 10 years
   

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4. 宣伝!!（書籍の翻訳に参加しました。）
   絶賛販売中！
   

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5. 宣伝!!（筆頭論⽂が出版されました！！）
   

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6. BeginneR Session
   

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7. -FU`TTUBSU3
   ɾ'SFF
   ɾ -PXJOTUBMMBUJPODPTUGPSCBTJDFOWJSPONFOU
   ɾ'VMMSBOHFPGGVODUJPOTGPSEBUBTDJFODF
   ɾ.BOZFYUFOTJPOT QBDLBHFT
   
   ɾ4USPOHDPNNVOJUZˡ QPTJUJPOUBML
   
   

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8. -FU`TTUBSU3
   ɾ'SFF
   ɾ -PXJOTUBMMBUJPODPTUGPSCBTJDFOWJSPONFOU
   ɾ'VMMSBOHFPGGVODUJPOTGPSEBUBTDJFODF
   ɾ.BOZFYUFOTJPOT QBDLBHFT
   
   ɾ4USPOHDPNNVOJUZˡ QPTJUJPOUBML
   
   https://tokyor.connpass.com/
   

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9. -FU`TTUBSU3
   ɾ'SFF
   ɾ -PXJOTUBMMBUJPODPTUGPSCBTJDFOWJSPONFOU
   ɾ'VMMSBOHFPGGVODUJPOTGPSEBUBTDJFODF
   ɾ.BOZFYUFOTJPOT QBDLBHFT
   
   ɾ4USPOHDPNNVOJUZˡ QPTJUJPOUBML
   
   h0ps://tokyor.connpass.com/
   SXBLBMBOH
   TMBDLXPSLTQBDF
   
   .FNCFSਓ
   

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10. 3Λ࢝ΊΑ͏
    【Step】
    1. Install R
    2. Install RStudio
    
    
    
    
    

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11. *OTUBMM3
    ☝
    

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12. *OTUBMM34UVEJP
    ౷߹։ൃ؀ڥ JOUFHSBUFEEFWFMPQNFOUFOWJSPONFOU *%&
    
    ☝
    

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13. ☝
    *OTUBMM34UVEJP
    ౷߹։ൃ؀ڥ JOUFHSBUFEEFWFMPQNFOUFOWJSPONFOU *%&
    
    

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14. )PXUPVTF34UVEJP
    4DSJQUFEJUPS
    $POTPMF
    &OWJSPONFOU
    QMPU FUD
    1 write
    2 select
    3 run(⌘ + ↩)
    output
    

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15. )PXUPVTF34UVEJP
    

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16. )PXUPVTF34UVEJP
    

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17. 
    
    
    
    
    > x + y
    [1] 3
    4DSJQUFEJUPS
    $POTPMFPVUQVU
    )PXUPVTF34UVEJP
    

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18. 
    
    
    
    
    
    > x + y
    [1] 4
    ಉ͡ม਺໊ʹ୅ೖ͢Δͱ্ॻ͖͞ΕΔ
    DPNNFOUPVU
    4DSJQUFEJUPS
    $POTPMFPVUQVU
    )PXUPVTF34UVEJP
    

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19. QBDLBHFT
    $3"/ 5IF$PNQSFIFOTJWF3"SDIJWF/FUXPSL
    
    0GGJDJBM3QBDLBHFSFQPTJUPSZ h0ps://cran.r-project.org/
    2021.09.04
    

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20. 
    $dyverse: データサイエンス関連パッケージ群をまとめたパッケージ
    ・dplyr: テーブルデータの加⼯・集計
    ・ggplot2: グラフの描画
    ・stringr: ⽂字列加⼯
    ・$dyr: データの整形や変形
    ・purrrr: 関数型プログラミング⽤
    ・magri7r: パイプ演算⼦%>%を提供
    *OTUBMMQBDLBHFGSPN$3"/
    QBDLBHFT
    $3"/ 5IF$PNQSFIFOTJWF3"SDIJWF/FUXPSL
    
    0⒏DJBM3QBDLBHFSFQPTJUPSZ https://cran.r-project.org/
    

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21. 0367*22(4*,1*/.6&41/6 ) $70-98.56.$'
    20+5*59&4*,1*/. ) $70-98.56.$'
    20+5*59&70-98.56.'###%# !"
    "UUBDIUIFQBDLBHF
    QBDLBHFT
    $3"/ 5IF$PNQSFIFOTJWF3"SDIJWF/FUXPSL
    
    0GGJDJBM3QBDLBHFSFQPTJUPSZ h0ps://cran.r-project.org/
    *OTUBMMQBDLBHFGSPN$3"/
    

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22. Stan
    A state-of-the-art platform
    for statistical modeling
    R
    A free so4ware environment
    for sta7s7cal compu7ng and graphics.
    {rstan} package
    A pla:orm using stan from R
    

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23. View Slide

24. BeginneR
    

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25. Before After
    BeginneR Session
    BeginneR BeginneR
    

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26. BeginneR Advanced Hoxo_m
    If I have seen further it is by standing on the
    shoulders of Giants.
    -- Sir Isaac Newton, 1676
    

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27. #94
    @kilometer00
    BeginneR Session
    -- Bayesian statistics --
    

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28. Experiment
    hypothesis observation
    principle phenotype
    model data
    Truth
    Knowledge f X
    (unknown)
    

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29. “Hypothesis driven” “Data driven”
    Experimental design
    A
    B
    Front
    Back
    Right
    Left
    VerAcal Up
    A B
    

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30. Strong hypothesis obs.
    principle phenotype
    f
    Weak hypothesis obs.
    principle phenotype
    model Complex data
    f
    model Simple data
    “Hypothesis driven” “Data driven”
    Experimental design
    X X
    

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31. Strong hypothesis obs.
    principle phenotype
    f X
    Weak hypothesis obs.
    principle phenotype
    model Complex data
    f X
    model Simple data
    “Hypothesis driven” “Data driven”
    Experimental design ここが気になる（気になりだす）
    

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32. Hypothesis ObservaEon
    Truth
    Knowledge
    principle phenotype
    model data
    Dice with
    α
    faces
    (regular polyhedron)
    ! = 5
    ?
    

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33. Dice with
    α
    faces ! = 5
    $ % = ! α = 4 = 0
    $ % = ! α = 6 =
    1
    6
    $ % = ! α = 8 =
    1
    8
    $ % = ! α = 12 =
    1
    12
    $ % = ! α = 20 =
    1
    20
    likelihood
    
    maximum
    likelihood
    

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34. Dice with
    α
    faces ! = {5, 4, 3, 4, 2, 1, 2, 3, 1, 4}
    $ % = ! α = 4 = 0
    $ % = ! α = 6 =
    1
    6!"
    $ % = ! α = 8 =
    1
    8!"
    $ % = ! α = 12 =
    1
    12!"
    $ % = ! α = 20 =
    1
    20!"
    likelihood
    maximum
    likelihood
    
    

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35. 
    
    
    Could you find
    α
    ?
    Yes, yes, yes.
    αis 6!!
    Why do you think so?
    Because, arg max!
    - . α = 6 !!
    Hmmm......, so......, how about ?
    $(α = 6)
    Oh, it is
    "
    #!"!! ......nnNNNNO!!! WHAT!!????
    

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36. 
    
    
    Hmmm......,
    so, how about ?
    $(α = 6)
    Dice with
    α
    faces ! = {5, 4, 3, 4, 2, 1, 2, 3, 1, 4}
    $ % = ! α = 6 =
    1
    6!"
    maximum
    likelihood
    
    ! α = 6 % = & !!??
    

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37. Probability distribution
    $(% = !)
    ! %
    $(% = !|α = 6)
    #(% = '|α)
    parameter
    data
    

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38. Probability distribution
    $(%)
    ! %
    arg max!
    -(2|α)
    1
    6!"
    α = 6
    α = 8
    α = 12
    $(4)
    α 4
    -(5 = α|2 = .)
    ! = #
    α = 20
    

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39. Probability distribuEon
    $#(%)
    ! %
    arg max!
    -$
    (2|α)
    1
    6!"
    $$(4)
    α 4
    -!
    (5 = α|2 = .)
    ! = #
    α = 6
    α = 8
    α = 12
    α = 20
    

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40. Probability distribuEon
    $#(%)
    ! %
    arg max!
    -$
    (2|α)
    1
    6!"
    $$(4)
    α 4
    -!
    (5 = α|2 = .)
    ! = #
    '
    5
    : α → & '
    6
    : & → α
    α = 6
    α = 8
    α = 12
    α = 20
    

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41. CondiEonal probability
    "($) "(&)
    " $ ∩ & = "(& ∩ $)
    

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42. CondiEonal probability
    "($) "(&)
    "!
    $ ∩ & = ""
    (& ∩ $)
    

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43. CondiEonal probability
    "($) "(&)
    !
    7
    * ∗ !
    8
    , * = !
    7
    *|, ∗ !
    8
    ,
    

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44. Bayes’ theorem
    !
    7
    *|, =
    !
    8
    , * ∗ !
    7
    (*)
    !
    8
    ,
    "!
    $ ∩ & = ""
    (& ∩ $)
    !
    7
    * ∗ !
    8
    , * = !
    7
    *|, ∗ !
    8
    ,
    

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45. !
    7
    *|, =
    !
    8
    , * ∗ !
    7
    (*)
    !
    8
    ,
    $!
    ) = α|+ = ! =
    $"
    + = ! ) = α ∗ $!
    (α)
    $"
    !
    '
    5
    : α → &
    '
    6
    : & → α
    Bayes’ theorem
    

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46. !
    7
    *|, =
    !
    8
    , * ∗ !
    7
    (*)
    !
    8
    ,
    $!
    ) = α|+ = ! =
    $"
    + = ! ) = α ∗ $!
    (α)
    $"
    !
    '
    5
    : α → &
    '
    6
    : & → α
    likelihood
    Bayes’ theorem
    

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47. !
    7
    *|, =
    !
    8
    , * ∗ !
    7
    (*)
    !
    8
    ,
    $!
    α|! =
    $"
    ! α ∗ $!
    (α)
    $"
    !
    '
    5
    : α → &
    '
    6
    : & → α
    likelihood
    Bayes’ theorem
    

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48. !
    7
    *|, =
    !
    8
    , * ∗ !
    7
    (*)
    !
    8
    ,
    $!
    α|! =
    $"
    ! α ∗ $!
    () = α)
    $"
    + = !
    '
    5
    : α → &
    '
    6
    : & → α
    likelihood
    Bayes’ theorem
    

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49. $!
    α|! =
    $"
    ! α ∗ $!
    () = α)
    $"
    + = !
    '
    5
    : α → &
    '
    6
    : & → α
    likelihood
    $$ 4 = α = $$ 4 = α|1
    = $$ 4 = α|% = 9
    %: 9 → !
    sample space
    

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50. $!
    α|! =
    $"
    ! α ∗ $!
    () = α)
    $"
    + = !
    '
    5
    : α → &
    '
    6
    : & → α
    likelihood
    $$ 4 = α = $$ 4 = α|1
    = $$ 4 = α|% = 9
    %: 9 → !
    sample space
    $# % = ! = $# % = !|1
    = $# % = !|4 = <
    4: < → α
    sample space
    

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51. $!
    α|! =
    $"
    ! α ∗ $!
    () = α)
    $"
    + = !
    '
    5
    : α → &
    '
    6
    : & → α
    likelihood
    $$ 4 = α = $$ 4 = α|% = 9
    $# % = ! = $# % = !|4 = <
    = =
    ∀$
    $# % = !|4 = α ∗ $$ 4 = α|% = 9
    marginaliza7on
    α ∈ {4, 6, 8, 12, 20}
    

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52. $!
    α|! =
    $"
    ! α ∗ $!
    () = α)
    $"
    + = !
    '
    5
    : α → &
    '
    6
    : & → α
    likelihood
    = =
    ∀$
    $# !|α ∗ $$ α|9
    marginalization
    α ∈ {4, 6, 8, 12, 20}
    $$ 4 = α = $$ α|9
    $# % = ! = $# !|<
    

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53. $!
    α|! =
    $"
    ! α ∗ $!
    () = α)
    $"
    + = !
    '
    5
    : α → &
    '
    6
    : & → α
    likelihood
    = =
    ∀$
    $# !|α ∗ $$ α|9
    marginaliza7on
    α ∈ {4, 6, 8, 12, 20}
    likelihood
    $$ 4 = α = $$ α|9
    $# % = ! = $# !|<
    

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54. $!
    α|! =
    $"
    ! α ∗ $!
    () = α)
    $"
    + = !
    '
    5
    : α → &
    '
    6
    : & → α
    likelihood
    $$ 4 = α = $$ α|9
    $# % = ! = $# !|<
    = =
    ∀$
    $# !|α ∗ $$ α|9
    marginalization
    α ∈ {4, 6, 8, 12, 20}
    likelihood
    

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55. $!
    α|! =
    $"
    ! α ∗ $!
    (α)
    $"
    !
    '
    5
    : α → &
    '
    6
    : & → α
    likelihood
    =
    $"
    ! α ∗ $!
    (α|-)
    Σ∀!
    $"
    !|α ∗ $!
    α|-
    

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56. Dice with
    α
    faces ! = {5, 4, 3, 4, 2, 1, 2, 3, 1, 4}
    $ % = ! α = 4 = 0
    $ % = ! α = 6 =
    1
    6!"
    $ % = ! α = 8 =
    1
    8!"
    $ % = ! α = 12 =
    1
    12!"
    $ % = ! α = 20 =
    1
    20!"
    likelihood
    
    

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57. $!
    α|! =
    $"
    ! α ∗ $!
    (α|-)
    Σ∀!
    $"
    !|α ∗ $!
    α|-
    '
    5
    : α → &
    '
    6
    : & → α
    likelihood
    $!
    () = α|+ = -)
    

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58. $!
    α|! =
    $"
    ! α ∗ $!
    (α|-)
    Σ∀!
    $"
    !|α ∗ $!
    α|-
    '
    5
    : α → &
    '
    6
    : & → α
    likelihood
    $!
    () = α|+ = -)
    %: 9 → !
    9 : sample space of data ! (20!"= 1,024,000,000,000 pa+ern)
    

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59. $!
    α|! =
    $"
    ! α ∗ $!
    (α|-)
    Σ∀!
    $"
    !|α ∗ $!
    α|-
    '
    5
    : α → &
    '
    6
    : & → α
    likelihood
    $!
    () = α|+ = -)
    %: 9 → !
    9 : sample space of data !
    (20$%= 1,024,000,000,000 paHern)
    

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60. View Slide

61. $!
    α|! =
    $"
    ! α ∗ $!
    (α|-)
    Σ∀!
    $"
    !|α ∗ $!
    α|-
    '
    5
    : α → &
    '
    6
    : & → α
    likelihood
    $!
    () = α|+ = -)
    + ≅ +′
    approximation
    $!
    ) = ∀α + = -& =
    1
    5
    α ∈ {4, 6, 8, 12, 20}
    

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62. $!
    α|! ≅
    $"
    ! α ∗ $!
    (α|-′)
    Σ∀!
    $"
    !|α ∗ $!
    α|-′
    '
    5
    : α → &
    '
    6
    : & → α
    likelihood
    =
    -$
    . α
    Σ∀!
    -$
    .|α
    =
    -$
    . α
    -$
    . 4 + -$
    . 6 + -$
    . 8 + -$
    . 12 + -$
    . 20
    ≈
    -$
    . α
    1.7485A − 08
    &!
    ∀α (" =
    1
    5
    

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63. 
    
    Hmmm......,
    so, how many ?
    $(α = 6)
    Dice with
    α
    faces ! = {5, 4, 3, 4, 2, 1, 2, 3, 1, 4}
    $ % = ! α = 6 =
    1
    6!"
    maximum
    likelihood
    $$ 4 = 6|! ≅
    $# % = ! 4 = 6
    1.7485C − 08
    ≈ 94.58%
    

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64. $$ 6|! ≈ 94.58%
    $$ 6|9′ = 20%
    $$ 8|! ≈ 5.32%
    $$ 8|9′ = 20%
    $$ 12|! ≈ 0.09%
    $$ 12|9′ = 20%
    $$ 20|! ≈ 0.0005%
    $$ 20|9′ = 20%
    $$ 4|! = 0%
    $$ 4|9′ = 20%
    prior probability posterior probability
    Maximum a posteriori (MAP) estimation
    arg max!
    $!
    α ! = 6
    

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65. 
    
    Hmmm......,
    so, how many ?
    $(α = 6)
    Dice with
    α
    faces ! = {5, 4, 3, 4, 2, 1, 2, 3, 1, 4}
    $ % = ! α = 6 =
    1
    6!"
    maximum
    likelihood
    $$ 4 = 6|! ≈ 94.58% maximum
    posteriori prob.
    

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66. 
    
    Hmmm......,
    so, how about ?
    $(α = 6)
    Dice with
    α
    faces ! = {5, 4, 3, 4, 2, 1, 2, 3, 1, 4}
    $ % = ! α = 6 =
    1
    6!"
    maximum
    likelihood
    $$ 4 = 6|! ≈ 94.58% maximum
    posteriori prob.
    Could you predict &
    II?
    

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67. Dice with
    α
    faces ! = {5, 4, 3, 4, 2, 1, 2, 3, 1, 4}
    $# !!! ≤ 6|4 ∗ $$ 4|! = 0%
    $# !!! ≤ 6|6 ∗ $$ 6|! ≈ 94.58%
    $# !!! ≤ 6|8 ∗ $$ 8|! ≈ 3.99%
    $# !!! ≤ 6|12 ∗ $$ 12|! ≈ 0.046%
    $# !!! ≤ 6|20 ∗ $$ 20|! ≈ 0.0001%
    $# !!! ≤ 6 = =
    ∀$
    {$# !!! ≤ 6|α ∗ $$ α|! }
    ≈ 98.62%
    predic$ve probability
    

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68. 
    
    Could you predict &
    II?
    $ ) = 6 ! ≈ 94.58%
    $ !$$
    ≤ 6 ! ≈ 98.62%
    and
    Dice with
    α
    faces ! = {5, 4, 3, 4, 2, 1, 2, 3, 1, 4}
    

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69. 
    
    Could you predict &
    II?
    $ ) = 6 ! ≈ 94.58%
    $ !$$
    ≤ 6 ! ≈ 98.62%
    and
    Dice with
    α
    faces ! = {5, 4, 3, 4, 2, 1, 2, 3, 1, 4}
    OK, let’s try !!!!!
    

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70. 
    
    !!!
    = 8
    Dice with
    α
    faces ! = {5, 4, 3, 4, 2, 1, 2, 3, 1, 4}
    

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71. 
    
    $ ) = 6 ! ≈ 94.58%
    $ !$$
    ≤ 6 ! ≈ 98.62%
    Dice with
    α
    faces ! = {5, 4, 3, 4, 2, 1, 2, 3, 1, 4}
    OK, let’s try "!!!!
    !))
    = 8
    " $ = 6 {,, ,##
    } = 0%
    

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72. "$
    α|, ≅
    "%
    , α ∗ "$
    (α|4′)
    "%
    (,)
    Dice with
    α
    faces ! = {5, 4, 3, 4, 2, 1, 2, 3, 1, 4}
    prior
    likelihood
    posterior
    /(
    ∀α 1) =
    1
    5
    

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73. "$
    α|, ≅
    "%
    , α ∗ "$
    (α|4′)
    "%
    (,)
    Dice with
    α
    faces ! = {5, 4, 3, 4, 2, 1, 2, 3, 1, 4}
    prior
    likelihood
    posterior
    /(
    ∀α 1) =
    1
    5
    "$
    α| ́
    , ≅
    "%
    ́
    , α ∗ "$
    (α|4′′)
    "%
    ( ́
    ,)
    Dice with
    α
    faces ́
    ! = {!, 8}
    

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74. "$
    α|, ≅
    "%
    , α ∗ "$
    (α|4′)
    "%
    (,)
    Dice with
    α
    faces ! = {5, 4, 3, 4, 2, 1, 2, 3, 1, 4}
    prior
    likelihood
    posterior
    /(
    ∀α 1) =
    1
    5
    "$
    α| ́
    , ≅
    "%
    ́
    , α ∗ "$
    (α|4′′)
    "%
    ( ́
    ,)
    Dice with
    α
    faces ́
    ! = {!, 8}
    

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75. "$
    α|, ≅
    "%
    , α ∗ "$
    (α|4′)
    "%
    (,)
    Dice with
    α
    faces ! = {5, 4, 3, 4, 2, 1, 2, 3, 1, 4}
    prior
    likelihood
    posterior
    /(
    ∀α 1) =
    1
    5
    "$
    α| ́
    , ≅
    "%
    ́
    , α ∗ "$
    (α|,)
    "%
    ( ́
    ,)
    Dice with
    α
    faces ́
    ! = {!, 8}
    

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76. Dice with
    α
    faces
    ! = {5, 4, 3, 4, 2, 1, 2, 3, 1, 4}
    ́
    ! = {!, 8}
    Non-informa$ve prior distribu$on
    20% 20% 20% 20% 20%
    0% 94.58% 5.32% 0.09% 0.005%
    0% 0% 99.98% 0.02% 0.000004%
    -!
    (α|C′)
    -!
    (α|.)
    -!
    (α| ́
    .)
    

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77. 
    
    $ ) = 8 ́
    ! ≈ 99.98%
    $ !$'
    ≤ 8 ́
    ! ≈ 99.98%
    Dice with
    α
    faces ́
    ! = {5, 4, 3, 4, 2, 1, 2, 3, 1, 4, 8}
    OK!!
    Let’s try !!"!!
    COME OOON
    

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78. No one knows
    what happened to them......
    

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79. Hypothesis ObservaEon
    Truth
    Knowledge
    principle phenotype
    model data
    Dice with
    α
    faces
    (regular polyhedron)
    ! = 5
    ?
    

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80. 
    
    
    Hmmm......,
    so, how about ?
    $(α = 6)
    Dice with
    α
    faces ! = {5, 4, 3, 4, 2, 1, 2, 3, 1, 4}
    $ % = ! α = 6 =
    1
    6!"
    maximum
    likelihood
    
    ! α = 6 % = & !!??
    

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81. !
    7
    *|, =
    !
    8
    , * ∗ !
    7
    (*)
    !
    8
    ,
    $!
    ) = α|+ = ! =
    $"
    + = ! ) = α ∗ $!
    (α)
    $"
    !
    '
    5
    : α → &
    '
    6
    : & → α
    likelihood
    Bayes’ theorem
    

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82. $!
    α|! ≅
    $"
    ! α ∗ $!
    (α|-′)
    Σ∀!
    $"
    !|α ∗ $!
    α|-′
    '
    5
    : α → &
    '
    6
    : & → α
    likelihood
    =
    -$
    . α
    Σ∀!
    -$
    .|α
    =
    -$
    . α
    -$
    . 4 + -$
    . 6 + -$
    . 8 + -$
    . 12 + -$
    . 20
    ≈
    -$
    . α
    1.7485A − 08
    &!
    ∀α (" =
    1
    5
    

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83. $$ 6|! ≈ 94.58%
    $$ 6|9′ = 20%
    $$ 8|! ≈ 5.32%
    $$ 8|9′ = 20%
    $$ 12|! ≈ 0.09%
    $$ 12|9′ = 20%
    $$ 20|! ≈ 0.0005%
    $$ 20|9′ = 20%
    $$ 4|! = 0%
    $$ 4|9′ = 20%
    prior probability posterior probability
    Maximum a posteriori probability (MAP) estimation
    arg max!
    $!
    α ! = 6
    

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84. Dice with
    α
    faces ! = {5, 4, 3, 4, 2, 1, 2, 3, 1, 4}
    $# !!! ≤ 6|4 ∗ $$ 4|! = 0%
    $# !!! ≤ 6|6 ∗ $$ 6|! ≈ 94.58%
    $# !!! ≤ 6|8 ∗ $$ 8|! ≈ 3.99%
    $# !!! ≤ 6|12 ∗ $$ 12|! ≈ 0.046%
    $# !!! ≤ 6|20 ∗ $$ 20|! ≈ 0.0001%
    $# !!! ≤ 6 = =
    ∀$
    {$# !!! ≤ 6|α ∗ $$ α|! }
    ≈ 98.62%
    predic$ve probability
    

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85. "$
    α|, ≅
    "%
    , α ∗ "$
    (α|4′)
    "%
    (,)
    Dice with
    α
    faces ! = {5, 4, 3, 4, 2, 1, 2, 3, 1, 4}
    prior
    likelihood
    posterior
    /(
    ∀α 1) =
    1
    5
    "$
    α| ́
    , ≅
    "%
    ́
    , α ∗ "$
    (α|4′′)
    "%
    ( ́
    ,)
    Dice with
    α
    faces ́
    ! = {!, 8}
    

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86. Experiment
    hypothesis observa$on
    principle phenotype
    model data
    Truth
    Knowledge f X
    (unknown)
    

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87. Strong hypothesis obs.
    principle phenotype
    f
    Weak hypothesis obs.
    principle phenotype
    model Complex data
    f
    model Simple data
    “Hypothesis driven” “Data driven”
    Experimental design
    X X
    

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88. α '
    -(.|α)
    α |'
    -(α|.)
    %|'
    -(2|α)- α .
    prior
    distribution
    posterior
    distribuBon
    data predictive
    distribution
    $!
    α ∗ $"
    ! α
    $"
    !
    = $!
    α|!
    likelihood
    prior
    posterior
    Bayes’ theorem
    

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89. α '
    -(.|α)
    α |'
    -(α|.)
    %|'
    -(2|α)- α .
    prior
    distribution
    posterior
    distribuBon
    data predictive
    distribution
    $!
    α ∗ $"
    ! α
    $"
    !
    = $!
    α|!
    likelihood
    prior
    posterior
    Bayes’ theorem
    Truth
    

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90. α '
    -(.|α)
    α |'
    -(α|.)
    %|'
    -(2|α)- α .
    prior
    distribuBon
    posterior
    distribuBon
    data predicBve
    distribuBon
    $!
    α ∗ $"
    ! α
    $"
    !
    = $!
    α|!
    likelihood
    prior
    posterior
    Bayes’ theorem
    #(%|')
    .(%)
    Truth
    L&'(M| $
    Kullback-Leibler
    divergence
    

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91. α '
    -(.|α)
    α |'
    -(α|.)
    %|'
    -(2|α)- α .
    prior
    distribuBon
    posterior
    distribuBon
    data predicBve
    distribuBon
    $!
    α ∗ $"
    ! α
    $"
    !
    = $!
    α|!
    likelihood
    prior
    posterior
    Bayes’ theorem
    #(%|')
    .(%)
    Truth
    L&'(M| $ = −N( + P
    KL divergence Entropy
    Generalization
    error
    

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92. /!"
    (.| #
    = Q[S $ − S(M)]
    = Q[(−log $ ) − (−log M )]
    = Q log
    (
    )
    = ∫ M % ∗ log
    ((#)
    )(,|#)
    Y%
    = ∫ M % ∗ log M(!) Y% − ∫ M % ∗ log $ % ! Y%
    = −Q S M − ∫ M % ∗ log $ % ! Y%
    B(
    C
    Entropy Generaliza$on error
    

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93. α '
    -(.|α)
    α |'
    -(α|.)
    %|'
    -(2|α)- α .
    prior
    distribuBon
    posterior
    distribution
    data predictive
    distribution
    $!
    α ∗ $"
    ! α
    $"
    !
    = $!
    α|!
    likelihood
    prior
    posterior
    Bayes’ theorem
    #(%|')
    .(%)
    Truth
    L&'(M| $ = −N( + P
    KL divergence Entropy GeneralizaBon error
    arg min) L&'(M| $ ⟺ arg min) P
    P ≅ WAIC Watanabe Akaike InformaAon Criterion
    

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94. Experiment
    hypothesis observa$on
    principle phenotype
    model data
    Truth
    Knowledge f X
    (unknown)
    

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95. Anaïs Nin –
    “Life shrinks or expands
    in proporRon to one’s courage.”
    h0ps://images.gr-assets.com
    

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96. Before ABer
    BeginneR Session
    BeginneR BeginneR
    

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97. Enjoy!!
    

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