Some Introductory Remarks on Bayesian Inference

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  • 7/31/2019 Some Introductory Remarks on Bayesian Inference

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    Keq}naw`qkne@nefwib`{

    _cklnnpck`bl Mb`ji}nwea

    Pndg Keq}naw`qn}{ ]gdb}jp neMb{gpkbe Kehg}ge`g

    Dkjkn L% M}bwe

    Pgdkeb} ne Mb{g Qcgn}{, QW Mg}lke, PP?9

    2 . = 1

    http://find/http://goback/
  • 7/31/2019 Some Introductory Remarks on Bayesian Inference

    2/35

    Keq}naw`qkne@nefwib`{

    _cklnnpck`bl Mb`ji}nwea

    Nxg}xkgy

    2

    Keq}naw`qkneMb{g ]wlg

    = @nefwib`{

    Gsbdplg6 Mg}enwllk akq}kmwqkne

    Gsbdplg6 Ibwkbe }beand xb}kbmlg

    Gspnegeqkbl Hbdklkg

    0 _cklnnpck`bl Mb`ji}nwea

    Ne Keqg}p}gqbqkne nh _}nmbmklkq{ Qcgn}{

    Mb{gkbekd x% H}grwgeqkd ke Qg}d nh Dnagllkei

    = . = 1

    http://find/
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    Keq}naw`qkne@nefwib`{

    _cklnnpck`bl Mb`ji}nweaMb{g ]wlg

    Mb{g ]wlg

    Kei}gakgeq6

    Dnagl D

    Abqb A

    _}kn} _#D+

    @neakqknebl _}nmbmklkq{ _#A|D+

    Mb{g ]wlg _#D|A+ 4_#A|D+_#D+

    _#A+4

    _#A|D+_#D+_#A|D+_#D+aD

    0 . = 1

    http://find/
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    Keq}naw`qkne@nefwib`{

    _cklnnpck`bl Mb`ji}nweaMb{g ]wlg

    Mb{g ]wlg

    Kei}gakgeq6

    Dnagl D

    Abqb A

    _}kn} _#D+

    @neakqknebl _}nmbmklkq{ _#A|D+

    Mb{g ]wlg _#D|A+ 4_#A|D+_#D+

    _#A+4

    _#A|D+_#D+_#A|D+_#D+aD

    dwlqkplg abqb pnkeq m{ keagpgeage`g bwdpqkne6

    _#A2, % % % ,Ae|D+ 4e

    k42

    _#Ak|D+%

    0 . = 1

    K a k

    http://find/
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    Keq}naw`qkne@nefwib`{

    _cklnnpck`bl Mb`ji}nweaMb{g ]wlg

    Be gsbdplg

    D zgxnlwqkne, keqgllkigeq agkie, qcg Dbq}kst%

    A zhnkl, qcg mkmlg, agfob xwt

    1 . = 1

    K q a qk

    http://find/
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    Keq}naw`qkne@nefwib`{

    _cklnnpck`bl Mb`ji}nweaMb{g ]wlg

    Be gsbdplg

    D zgxnlwqkne, keqgllkigeq agkie, qcg Dbq}kst%

    A zhnkl, qcg mkmlg, agfob xwt

    Pn ycbq k _#gxnlwqkne|qcg mkmlg+>

    1 . = 1

    Keq}naw`qkne

    http://find/
  • 7/31/2019 Some Introductory Remarks on Bayesian Inference

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    Keq}naw`qkne@nefwib`{

    _cklnnpck`bl Mb`ji}nweaMb{g ]wlg

    Be gsbdplg

    D zgxnlwqkne, keqgllkigeq agkie, qcg Dbq}kst%

    A zhnkl, qcg mkmlg, agfob xwt

    Pn ycbq k _#gxnlwqkne|qcg mkmlg+>

    N} _#keqgllkigeq agkie|hnkl+>

    1 . = 1

    Keq}naw`qkne

    http://find/http://goback/
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    Keq}naw`qkne@nefwib`{

    _cklnnpck`bl Mb`ji}nweaMb{g ]wlg

    Be gsbdplg

    D zgxnlwqkne, keqgllkigeq agkie, qcg Dbq}kst%

    A zhnkl, qcg mkmlg, agfob xwt

    Pn ycbq k _#gxnlwqkne|qcg mkmlg+>

    N} _#keqgllkigeq agkie|hnkl+>

    N} _#qcg Dbq}ks|hnkl+ x% _#qcg Dbq}ks|dbe{ agfob xw+>

    1 . = 1

    Keq}naw`qkne

    http://find/
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    Keq}naw`qkne@nefwib`{

    _cklnnpck`bl Mb`ji}nweaMb{g ]wlg

    Blqg}ebqkxgl{%%%

    _#A|D+_#D+aD lnnj lkjg neg qgp ke b Db}jnx `cbke%

    dnagl b}g ygkicqga b``n}akei qn qcgk} `neq}kmwqkne qn A

    7 . = 1

    Keq}naw`qkne

    http://find/
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    Keq}naw`qkne@nefwib`{

    _cklnnpck`bl Mb`ji}nweaMb{g ]wlg

    Yc{ `cnng akg}geq p}kn}>

    Pcnwlaeq yg mg npge qn bll pnkmklkqkg>

    Bea mg h}gg h}nd p}gfwak`g>

    8 . = 1

    Keq}naw`qkne

    http://find/
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    @nefwib`{_cklnnpck`bl Mb`ji}nwea

    Mb{g ]wlg

    Yc{ `cnng akg}geq p}kn}>

    Pcnwlaeq yg mg npge qn bll pnkmklkqkg>

    Bea mg h}gg h}nd p}gfwak`g>

    8 . = 1

    Keq}naw`qkne Gsbdplg6 Mg}enwllk akq}kmwqkne

    http://find/
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    @nefwib`{_cklnnpck`bl Mb`ji}nwea

    pGsbdplg6 Ibwkbe }beand xb}kbmlgGspnegeqkbl Hbdklkg

    @nefwib`{

    Agpgeakei ne qcg p}nmbmklkqkg kexnlxga, `ndpwqkei Mb{g hn}dwlb}}grwk}g neg keqgi}bqkne yck`c db{ mg kehgbkmlg%

    Cnygxg}, hn} dbe{ p}nmbmklkq{ akq}kmwqkne, kq k pnkmlg qn `cnngb p}kn} w`c qcbq

    Mb{g }wlg `be mg bpplkga gsb`ql{,

    qcg pnqg}kn} cb qcg bdg hwe`qknebl hn}d b qcg p}kn}%

    Qck k `bllga `nefwib`{, bea qcg p}kn} k `bllga qcg `nefwibqg p}kn}%

    9 . = 1

    Keq}naw`qkne Gsbdplg6 Mg}enwllk akq}kmwqkne

    http://find/
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    @nefwib`{_cklnnpck`bl Mb`ji}nwea

    Gsbdplg6 Ibwkbe }beand xb}kbmlgGspnegeqkbl Hbdklkg

    Mg}enwllk akq}kmwqkne

    _#s 4 2|+ 4 _#s 4 ?|+ 4 2

    _#s|+ 4 s#2 +2s

    3 . = 1

    Keq}naw`qkne@

    Gsbdplg6 Mg}enwllk akq}kmwqkneI

    http://find/
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    @nefwib`{_cklnnpck`bl Mb`ji}nwea

    Gsbdplg6 Ibwkbe }beand xb}kbmlgGspnegeqkbl Hbdklkg

    Iwgkei qcg p}kn}

    _#D|A+ 4_A|D_#D+

    _#A+ _#A|D+_#D+%

    Bpp}nb`c6 Hn}igq qcg en}dblkvbqkne, lnnj hn} b _#D|+ w`c qcbq

    _#A|D+_#D|+ _#D|+

    Hn} gsbdplg6 b#2

    +m6

    s#2 +2sb#2 +m 4 s*b#2 +m*2s

    ; . = 1

    Keq}naw`qkne@ f

    Gsbdplg6 Mg}enwllk akq}kmwqkneG l I k a k ml

    http://find/
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    @nefwib`{_cklnnpck`bl Mb`ji}nwea

    Gsbdplg6 Ibwkbe }beand xb}kbmlgGspnegeqkbl Hbdklkg

    Hkeakei qcg en}dblkvbqkne

    Hn}qwebqgl{, qck cb bl}gba{ mgge `b}}kga nwq bea qcg `n}}g`q p}kn}akq}kmwqkne `be mg hnwea #ndgycg}g+%%%

    Mgqb akq}kmwqkne6

    Mgqb#|b, m+ 4#b * m+

    #b+#m+b2#2 +m2%

    Gspg`qbqkne6 b.b * m##e+ keqg}pnlbqg qcg hb`qn}kbl, #e+ 4 #e 2+"+%

    2?.=1

    Keq}naw`qkne@ f

    Gsbdplg6 Mg}enwllk akq}kmwqkneG l I k a k ml

    http://find/
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    @nefwib`{_cklnnpck`bl Mb`ji}nwea

    Gsbdplg6 Ibwkbe }beand xb}kbmlgGspnegeqkbl Hbdklkg

    Keqg}p}gqkei qcg p}kn}6 _gwan$`nweq

    b, m b}g pgwan$`nweq6

    s#2 +2sb2#2 +m2 4 b*s2#2 +m*#2s+2

    Qcg}ghn}g6

    b b * 2 ycge s 4 2

    m m* 2 ycge s 4 ?

    22.=1

    Keq}naw`qkne@nefwib`{

    Gsbdplg6 Mg}enwllk akq}kmwqkneGsbdplg6 Ibwkbe }beand xb}kbmlg

    http://find/
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    @nefwib`{_cklnnpck`bl Mb`ji}nwea

    Gsbdplg6 Ibwkbe }beand xb}kbmlgGspnegeqkbl Hbdklkg

    Qcg Mgqb$Akq}kmwqkne

    2=.=1

    Keq}naw`qkne@nefwib`{

    Gsbdplg6 Mg}enwllk akq}kmwqkneGsbdplg6 Ibwkbe }beand xb}kbmlg

    http://find/
  • 7/31/2019 Some Introductory Remarks on Bayesian Inference

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    @nefwib`{_cklnnpck`bl Mb`ji}nwea

    Gsbdplg6 Ibwkbe }beand xb}kbmlgGspnegeqkbl Hbdklkg

    Qcg Mgqb$Akq}kmwqkne

    2=.=1

    Keq}naw`qkne@nefwib`{

    Gsbdplg6 Mg}enwllk akq}kmwqkneGsbdplg6 Ibwkbe }beand xb}kbmlg

    http://find/
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    @nefwib`{_cklnnpck`bl Mb`ji}nwea

    Gsbdplg6 Ibwkbe }beand xb}kbmlgGspnegeqkbl Hbdklkg

    Qcg Mgqb$Akq}kmwqkne

    2=.=1

    Keq}naw`qkne@nefwib`{

    Gsbdplg6 Mg}enwllk akq}kmwqkneGsbdplg6 Ibwkbe }beand xb}kbmlg

    http://find/
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    @nefwib`{_cklnnpck`bl Mb`ji}nwea

    Gsbdplg6 Ibwkbe }beand xb}kbmlgGspnegeqkbl Hbdklkg

    Ke b kdklb} dbeeg}%%%

    Mkendkbl akq}kmwqkne6e

    j

    j#b +ej 4 Mgqb#|b, m+%

    Dwlqknekdbl akq}kmwqkne6

    e

    e2e= % % % eJ

    Jj42

    ejj

    4 Ak}k`clgq akq}kmwqkne

    Ak}#|+ 4#?+

    #2+ % % % #J+

    Jj42

    2j

    %

    20.=1

    Keq}naw`qkne@nefwib`{

    Gsbdplg6 Mg}enwllk akq}kmwqkneGsbdplg6 Ibwkbe }beand xb}kbmlg

    http://find/
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    f i {_cklnnpck`bl Mb`ji}nwea

    pGspnegeqkbl Hbdklkg

    Qcg Ibwkbe

    Qcg Ibwkbe akq}kmwqkne6

    p#s|, =+ g#s+=.==

    Lgq w iwg qcg `n}}g`q p}kn} hn} 6 kq cnwla mg b rwba}bqk`hwe`qkne s6

    p#|b, m+ gb#sm+=

    %%% yck`c k mbk`bll{ bibke b Ibwkbe akq}kmwqkne%

    _nqg}kn} hn} e abqb pnkeq6

    e 4=

    e=? * =? *

    e=?e=? *

    =DL

    2

    =e4

    2

    =?*

    e

    =%

    21.=1

    Keq}naw`qkne@nefwib`{

    Gsbdplg6 Mg}enwllk akq}kmwqkneGsbdplg6 Ibwkbe }beand xb}kbmlg

    http://find/
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    _cklnnpck`bl Mb`ji}nwea Gspnegeqkbl Hbdklkg

    Qcg Ibwkbe

    _}kn} hn} =6 ]gy}kqg 4 2.=, qcge

    p#s|, + 2.=g#s+=.=%

    Iwgkei qcg p}kn}6 mgm

    Qck lgba qn qcg Ibddb$akq}kmwqkne6

    #|b, m+ 42

    #b+mbb2gm%

    _nqg}kn} hn} e abqb pnkeq6

    bE 4 b? *e

    =

    mE 4 m? *e

    ==DL%

    27.=1

    Keq}naw`qkne@nefwib`{

    Gsbdplg6 Mg}enwllk akq}kmwqkneGsbdplg6 Ibwkbe }beand xb}kbmlg

    http://find/
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    _cklnnpck`bl Mb`ji}nwea Gspnegeqkbl Hbdklkg

    Gspnegeqkbl Hbdklkg

    Ke igeg}bl, `nefwibqg p}kn} gskq hn} akq}kmwqkne h}nd qcggspnegeqkbl hbdkl{%

    p#s|+ 4 c#s+g,s#+%

    Iwgkei qcg p}kn}%%%

    p#|b, m+ g,bm#+%

    Mg`bwg6

    g,s#+g,bm#+ 4 g,b*s#m*2+#+

    28.=1

    Keq}naw`qkne@nefwib`{

    _ckl ck l M j a

    Gsbdplg6 Mg}enwllk akq}kmwqkneGsbdplg6 Ibwkbe }beand xb}kbmlgG k l H klk

    http://find/http://goback/
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    _cklnnpck`bl Mb`ji}nwea Gspnegeqkbl Hbdklkg

    Gspnegeqkbl Hbdklkg #`neqa+

    Lkjglkcnna _}kn}._nqg}kn}

    Ibwkbe #dgbe+ Ibwkbe

    Ibwkbe #xb}kbe`g+ Ibddb_nkne IbddbIbddb Ibddb

    Mkendkbl MgqbEgibqkxg Mkendkbl Mgqb

    Dwlqkendkbl Ak}k`clgq

    29.=1

    Keq}naw`qkne@nefwib`{

    _ckl ck l M j a

    Ne Keqg}p}gqbqkne nh _}nmbmklkq{ Qcgn}{Mb{gkbekd x% H}grwgeqkd ke Qg}d nh Dnagllkei

    http://find/
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    _cklnnpck`bl Mb`ji}nwea{ r i

    Mb{gkbekd x% H}grwgeqkd

    H}grwgeqkpd6 Dbskdwd$lkjglkcnna, C{pnqcgk Qgqkei, WemkbgaGqkdbqg, Pwppn}q Xg`qn} Db`ckeg, gq`%

    Mb{gpkbekpd6 Mb{gkbe gqkdbqkne, Ibwkbe _}n`gg, MglkghEgqyn}j, Hb`qn} I}bpc, gq`%

    K}}g`ne`klbmlg Akg}ge`g n} Qyn Pkag nh qcg Pbdg @nke>

    23.=1

    Keq}naw`qkne@nefwib`{

    _cklnnpck`bl Mb`ji}nwea

    Ne Keqg}p}gqbqkne nh _}nmbmklkq{ Qcgn}{Mb{gkbekd x% H}grwgeqkd ke Qg}d nh Dnagllkei

    http://find/
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    _cklnnpck`bl Mb`ji}nwea{ r i

    Keqg}p}gqbqkne nh _}nmbmklkq{ Qcgn}{

    _$Qcgn ang enq p}nxkag be{ lkejbig qn qcg yn}la%

    Kq mbk`bll{ fwq qck6

    _#+ 4 ?, _#B+ 4 2 _#B+, _#

    kBk+ 4

    k

    _#Bk+

    G#S+, _#B M+ 4 _#B+_#M+ _#B|M+ 4 _#B M+._#M+

    H}nd qck, gxg}{qckei glg k ag}kxga, ke`lwakei lby nh lb}igewdmg}, gq`%

    2;.=1

    Keq}naw`qkne@nefwib`{

    _cklnnpck`bl Mb`ji}nwea

    Ne Keqg}p}gqbqkne nh _}nmbmklkq{ Qcgn}{Mb{gkbekd x% H}grwgeqkd ke Qg}d nh Dnagllkei

    http://find/
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    _cklnnpck`bl Mb`ji}nwea

    Mb{gkbekd x% H}grwgeqkd

    Wg _$Qcgn qn%%%H}grwgeqkpd6 %%% dnagl keagpgeageq }gpgbqbmlg gspg}kdgeq

    kh K wd wp dbe{ }gblkvbqkne, qcg{ ykll mg `lng qn qcggspg`qbqkne%

    Mb{gpkbekpd6 %%% dnagl `ndpwqbqkne ne mglkgh akq}kmwqkne

    kh K dnagl qcg abqb `n}}g`ql{, d{ mglkgh ykll mg wpabqgab``n}akeil{%

    =?.=1

    Keq}naw`qkne@nefwib`{

    _cklnnpck`bl Mb`ji}nwea

    Ne Keqg}p}gqbqkne nh _}nmbmklkq{ Qcgn}{Mb{gkbekd x% H}grwgeqkd ke Qg}d nh Dnagllkei

    http://find/
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    _cklnnpck`bl Mb`ji}nwea

    Mb{gkbekd x% H}grwgeqkd

    Wg _$Qcgn qn%%%

    H}grwgeqkpd6 %%% dnagl keagpgeageq }gpgbqbmlg gspg}kdgeq

    kh K wd wp dbe{ }gblkvbqkne, qcg{ ykll mg `lng qn qcggspg`qbqkne%

    Mb{gpkbekpd6 %%% dnagl `ndpwqbqkne ne mglkgh akq}kmwqkne

    kh K dnagl qcg abqb `n}}g`ql{, d{ mglkgh ykll mg wpabqgab``n}akeil{%

    @ndpgqg nel{ ke qg}d nh }gbl$yn}la pg}hn}dbe`g, mwq enq nxg}ycbq k qcg `n}}g`q yb{ qn wg _$Qcgn%

    =?.=1

    Keq}naw`qkne@nefwib`{

    _cklnnpck`bl Mb`ji}nwea

    Ne Keqg}p}gqbqkne nh _}nmbmklkq{ Qcgn}{Mb{gkbekd x% H}grwgeqkd ke Qg}d nh Dnagllkei

    http://find/
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    _cklnnpck`bl Mb`ji}nwea

    Mb{gkbekd x% H}grwgeqkd

    Wg _$Qcgn qn%%%

    H}grwgeqkpd6 %%% dnagl keagpgeageq }gpgbqbmlg gspg}kdgeq

    kh K wd wp dbe{ }gblkvbqkne, qcg{ ykll mg `lng qn qcggspg`qbqkne%

    Mb{gpkbekpd6 %%% dnagl `ndpwqbqkne ne mglkgh akq}kmwqkne

    kh K dnagl qcg abqb `n}}g`ql{, d{ mglkgh ykll mg wpabqgab``n}akeil{%

    @ndpgqg nel{ ke qg}d nh }gbl$yn}la pg}hn}dbe`g, mwq enq nxg}ycbq k qcg `n}}g`q yb{ qn wg _$Qcgn%

    Gs`gpq hn}6 Mb{gkbe bpp}nb`cg }gwlq ke pnqg}kn} akq}kmwqkneycklg H}grwgeqkq dgqcna wwbll{ fwq }gqw}e b keilg nlwqkne%

    =?.=1

    Keq}naw`qkne@nefwib`{

    _cklnnpck`bl Mb`ji}nwea

    Ne Keqg}p}gqbqkne nh _}nmbmklkq{ Qcgn}{Mb{gkbekd x% H}grwgeqkd ke Qg}d nh Dnagllkei

    http://find/
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    p i

    M x Hke qg}d nh dnagllkei

    Db`ckeg lgb}ekei dgqcna `be }nwicl{ mg ag`ndpnga ke qg}d nh

    Dnagllkei #ycbq k kq K ybeq qn lgb}e+

    ]giwlb}kvbqkne #dbjg w}g yg aneq nxg}flq+

    Kehg}ge`g #b`qwbll{ `ndpwqg qcg nlwqkne ikxge qcg abqb+

    =2.=1

    Keq}naw`qkne@nefwib`{

    _cklnnpck`bl Mb`ji}nwea

    Ne Keqg}p}gqbqkne nh _}nmbmklkq{ Qcgn}{Mb{gkbekd x% H}grwgeqkd ke Qg}d nh Dnagllkei

    http://find/
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    p i

    M x Hke qg}d nh dnagllkei

    Db`ckeg lgb}ekei dgqcna `be }nwicl{ mg ag`ndpnga ke qg}d nh

    Dnagllkei #ycbq k kq K ybeq qn lgb}e+

    ]giwlb}kvbqkne #dbjg w}g yg aneq nxg}flq+

    Kehg}ge`g #b`qwbll{ `ndpwqg qcg nlwqkne ikxge qcg abqb+

    Bea qck cnla hn} mnqc6

    Mb{gkbe H}grwgeqkq

    dnagllkei _#A|D+ ln hwe`qkne}giwlb}kvbqkne _#D+ }giwlb}kvbqkne

    kehg}ge`g Mb{g$}wlg npqkdkvbqkne

    =2.=1

    Keq}naw`qkne@nefwib`{

    _cklnnpck`bl Mb`ji}nwea

    Ne Keqg}p}gqbqkne nh _}nmbmklkq{ Qcgn}{Mb{gkbekd x% H}grwgeqkd ke Qg}d nh Dnagllkei

    http://find/
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    M x Hakg}geq jkea nh we`g}qbkeq{

    H}grwgeqkpd6 dnagllkei k jkea nh kegsb`q, mwq bq lgbq kehg}ge`gk gsb`q%

    Mb{gpkbekpd6 dnagllkei k `lgb}, mwq kehg}ge`g k jkea nh kegsb`q%

    ==.=1

    Keq}naw`qkne@nefwib`{_cklnnpck`bl Mb`ji}nwea

    Ne Keqg}p}gqbqkne nh _}nmbmklkq{ Qcgn}{Mb{gkbekd x% H}grwgeqkd ke Qg}d nh Dnagllkei

    http://find/
  • 7/31/2019 Some Introductory Remarks on Bayesian Inference

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    M x Hk}}g`ne`klbmlg akg}ge`g>

    Db{mg, ke`g qnnl b}g xg}{ akg}geq6

    H}grwgeqkpq6 jeny yck`c `ndpwqbqkne ne bdplg`nexg}ig.`ne`geq}bqg, npqkdkvbqkne qcgn}{ #`nexgs npqkdkvbqkne,

    i}bakgeq ag`geq, keqg}kn} pnkeq dgqcna%%%+, gq`%

    Mb{gpkbep6 p}nmbmklkq{ akq}kmwqkne, yck`c p}kn} dbjg gekmlg`ndpwqbqkne, bdplkei dgqcna lkjg D@D@, bpp}nskdbqknedgqcna%

    =0.=1

    http://find/
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    Keq}naw`qkne@nefwib`{_cklnnpck`bl Mb`ji}nwea

    Ne Keqg}p}gqbqkne nh _}nmbmklkq{ Qcgn}{Mb{gkbekd x% H}grwgeqkd ke Qg}d nh Dnagllkei

  • 7/31/2019 Some Introductory Remarks on Bayesian Inference

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    Pwddb}{

    Mb{g }wlg

    @nefwibqg p}kn}

    Mb{gkbekd bea H}grwgeqkd

    =1.=1

    http://find/