L16 Statistical Rethinking Winter 2019

L16 Statistical Rethinking Winter 2019

Lecture 16 of the Dec 2018 through March 2019 edition of Statistical Rethinking. Covers Chapter 13, intro to multilevel modeling.

A0f2f64b2e58f3bfa48296fb9ed73853?s=128

Richard McElreath

February 15, 2019
Tweet

Transcript

  1. Multilevel Models Statistical Rethinking Winter 2019 Lecture 16 / Week

    8
  2. Prosocial chimpanzees partner focal  #*/0.*"- 3&(3&44*0/ 'ĶĴłĿIJ ƉƈƉ $IJNQBO[FF

    QSPTPD FYQFSJNFOU BT TFFO GSPN UIF QFSTQF PG UIF GPDBM BOJNBM ćF MFę BOE MFWFST BSF JOEJDBUFE JO UIF GPSFHSP 1VMMJOH FJUIFS FYQBOET BO BDDPSEJPO WJDF JO UIF DFOUFS QVTIJOH UIF GPPE UPXBSET CPUI FOET PG UIF UBCMF #PUI USBZT DMPTF UP UIF GPDBM BOJNBM IBWF JO UIFN 0OMZ POF PG UIF GPPE USBZ UIF PUIFS TJEF DPOUBJOT GPPE ćF QBS DPOEJUJPO NFBOT BOPUIFS BOJNBM BT UVSFE TJUT PO UIF PUIFS FOE PG UIF U 0UIFSXJTF UIF PUIFS FOE XBT FNQUZ
  3. Cross-classification • Can use more than one cluster type •

    Chimpanzee experiment data • Pulls in chimpanzees • Pulls in blocks • Each chimp in each block • Not nested, but cross-classified 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 1 2 3 4 5 6 block row in data 504 400 300 200 100 1
  4. Multilevel chimpanzees varying intercepts on actor DMVTUFS UZQF 5P BEE

    UIF TFDPOE DMVTUFS UZQF '*& XF NFSFMZ SFQMJDBUF UIF TUSV /*- DMVTUFS ćJT NFBOT UIF MJOFBS NPEFM HFUT ZFU BOPUIFS WBSZJOH JOUFSDFQU α UIF NPEFM HFUT BOPUIFS BEBQUJWF QSJPS BOE ZFU BOPUIFS TUBOEBSE EFWJBUJPO QBSBN )FSF JT UIF NBUIFNBUJDBM GPSN PG UIF NPEFM XJUI UIF OFX QJFDFT PG UIF N MJHIUFE JO CMVF -J ∼ #JOPNJBM(, QJ) MPHJU(QJ) = αĮİŁļĿ[J] + γįĹļİĸ[J] + βŁĿIJĮŁĺIJĻŁ[J] βK ∼ /PSNBM(, .) GPS K = .. αK ∼ /PSNBM(¯ α, σα) GPS K = .. γK ∼ /PSNBM(, σγ) GPS K = .. ¯ α ∼ /PSNBM(, .) σα ∼ &YQPOFOUJBM() σγ ∼ &YQPOFOUJBM() &BDI DMVTUFS HFUT JUT PXO WFDUPS PG QBSBNFUFST 'PS BDUPST UIF WFDUPS JT α BOE JU I CFDBVTF UIFSF BSF  DIJNQBO[FFT JO UIF TBNQMF 'PS CMPDLT UIF WFDUPS JT γ BOE JU CFDBVTF UIFSF BSF  CMPDLT &BDI DMVTUFS WBSJBCMF OFFET JUT PXO TUBOEBSE EFWJBUJP
  5. varying intercepts on actor DMVTUFS UZQF 5P BEE UIF TFDPOE

    DMVTUFS UZQF '*& XF NFSFMZ SFQMJDBUF UIF TUSV /*- DMVTUFS ćJT NFBOT UIF MJOFBS NPEFM HFUT ZFU BOPUIFS WBSZJOH JOUFSDFQU α UIF NPEFM HFUT BOPUIFS BEBQUJWF QSJPS BOE ZFU BOPUIFS TUBOEBSE EFWJBUJPO QBSBN )FSF JT UIF NBUIFNBUJDBM GPSN PG UIF NPEFM XJUI UIF OFX QJFDFT PG UIF N MJHIUFE JO CMVF -J ∼ #JOPNJBM(, QJ) MPHJU(QJ) = αĮİŁļĿ[J] + γįĹļİĸ[J] + βŁĿIJĮŁĺIJĻŁ[J] βK ∼ /PSNBM(, .) GPS K = .. αK ∼ /PSNBM(¯ α, σα) GPS K = .. γK ∼ /PSNBM(, σγ) GPS K = .. ¯ α ∼ /PSNBM(, .) σα ∼ &YQPOFOUJBM() σγ ∼ &YQPOFOUJBM() &BDI DMVTUFS HFUT JUT PXO WFDUPS PG QBSBNFUFST 'PS BDUPST UIF WFDUPS JT α BOE JU I CFDBVTF UIFSF BSF  DIJNQBO[FFT JO UIF TBNQMF 'PS CMPDLT UIF WFDUPS JT γ BOE JU CFDBVTF UIFSF BSF  CMPDLT &BDI DMVTUFS WBSJBCMF OFFET JUT PXO TUBOEBSE EFWJBUJP Multilevel chimpanzees varying intercepts on block
  6. Multilevel chimpanzees m13.4 <- ulam( alist( pulled_left ~ dbinom( 1

    , p ) , logit(p) <- a[actor] + g[block_id] + b[treatment] , b[treatment] ~ dnorm( 0 , 0.5 ), # adaptive priors a[actor] ~ dnorm( a_bar , sigma_a ), g[block_id] ~ dnorm( 0 , sigma_g ), # hyper-priors a_bar ~ dnorm( 0 , 1.5 ), sigma_a ~ dexp(1), sigma_g ~ dexp(1) ) , data=dat_list , chains=4 , cores=4 , log_lik=TRUE )
  7. Multilevel chimpanzees m13.4 <- ulam( alist( pulled_left ~ dbinom( 1

    , p ) , logit(p) <- a[actor] + g[block_id] + b[treatment] , b[treatment] ~ dnorm( 0 , 0.5 ), # adaptive priors a[actor] ~ dnorm( a_bar , sigma_a ), g[block_id] ~ dnorm( 0 , sigma_g ), # hyper-priors a_bar ~ dnorm( 0 , 1.5 ), sigma_a ~ dexp(1), sigma_g ~ dexp(1) ) , data=dat_list , chains=4 , cores=4 , log_lik=TRUE )
  8. Multilevel chimpanzees m13.4 <- ulam( alist( pulled_left ~ dbinom( 1

    , p ) , logit(p) <- a[actor] + g[block_id] + b[treatment] , b[treatment] ~ dnorm( 0 , 0.5 ), # adaptive priors a[actor] ~ dnorm( a_bar , sigma_a ), g[block_id] ~ dnorm( 0 , sigma_g ), # hyper-priors a_bar ~ dnorm( 0 , 1.5 ), sigma_a ~ dexp(1), sigma_g ~ dexp(1) ) , data=dat_list , chains=4 , cores=4 , log_lik=TRUE )
  9. Cross-classified chimpanzees   .0%&-4 8*5)065 "./&4*" sigma_g sigma_a a_bar

    g[6] g[5] g[4] g[3] g[2] g[1] a[7] a[6] a[5] a[4] a[3] a[2] a[1] b[4] b[3] b[2] b[1] 0 2 4 6 Value 0 1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 standard deviation Density actor block
  10. Cross-classified chimpanzees • Incorporating block: no benefits; little cost sigma_g

    sigma_a a_bar g[6] g[5] g[4] g[3] g[2] g[1] a[7] a[6] 0 2 4 6 Value 0 1 2 3 0.0 0.5 1.0 1.5 2.0 2.5 standard deviation Density actor block 'ĶĴłĿIJ ƉƋƌ -Fę 1PTUFSJPS NFBOT BOE  DPNQBUJCJMJUZ JOUFSWBMT (ǎǐǡǑ ćF HSFBUFS WBSJBUJPO BDSPTT BDUPST UIBO CMPDLT DBO CF TFFO JN EJBUFMZ JO UIF  BOE " EJTUSJCVUJPOT 3JHIU 1PTUFSJPS EJTUSJCVUJPOT PG TUBOEBSE EFWJBUJPOT PG WBSZJOH JOUFSDFQUT CZ BDUPS CMBDL BOE CMPDL CM 3 DPEF  *(+- ǿ (ǎǐǡǑ Ǣ (ǎǐǡǒ Ȁ   +    2 $"#/   (ǎǐǡǒ ǒǐǎǡǍ Ǖǡǒ Ǎ Ǎǡǔǐ ǎǖǡǏǐ  (ǎǐǡǑ ǒǐǏǡǖ ǎǍǡǖ Ǐ ǍǡǏǔ ǎǖǡǐǖ ǎǡǓǑ -PPL BU UIF +  DPMVNO XIJDI SFQPSUT UIF iFČFDUJWF OVNCFS PG QBSBNFUFSTw 8 JG XF QMPU UIF NBSHJOBM QPTUFSJPS EJTUSJCVUJPOT PG UIFTF UXP QBSBNFUFST *WF EPXO UIJT PO UIF SJHIU JO 'ĶĴłĿIJ ƉƋƌ 8IJMF UIFSFT VODFSUBJOUZ BCPVU UIF WBSJBUJPO BNPOH BDUPST UIJT NPEFM JT DPOĕEFOU UIBU BDUPST WBSZ NPSF UIBO CMPDLT :PV DBO FBTJMZ TFF UIJT WBSJBUJPO JO UIF WBSZJOH JOUFSDFQU EJTUSJCVUJPOT UIF  EJTUSJCVUJPOT BSF NVDI NPSF TDBUUFSFE UIBO BSF UIF " EJTUSJCVUJPOT ćF DIJNQBO[FFT WBSZ CVU UIF CMPDLT BSF BMM UIF TBNF "T B DPOTFRVFODF BEEJOH '*& UP UIJT NPEFM IBTOU BEEFE B MPU PG PWFSĕUUJOH SJTL -FUT DPNQBSF UIF NPEFM XJUI POMZ WBSZJOH JOUFSDFQUT PO /*- UP UIF NPEFM XJUI CPUI LJOET PG WBSZJOH JOUFSDFQUT ćF NPEFM UIBU JHOPSFT CMPDL JT 3 DPEF  . /ǡ. ǿǎǑȀ (ǎǐǡǒ ʚǶ 0'(ǿ '$./ǿ +0'' Ǿ' !/ ʡ $)*(ǿ ǎ Ǣ + Ȁ Ǣ '*"$/ǿ+Ȁ ʚǶ ȁ/*-Ȃ ʔ ȁ/- /( )/Ȃ Ǣ ȁ/- /( )/Ȃ ʡ )*-(ǿ Ǎ Ǣ Ǎǡǒ ȀǢ ȁ/*-Ȃ ʡ )*-(ǿ Ǿ- Ǣ .$"(Ǿ ȀǢ Ǿ- ʡ )*-(ǿ Ǎ Ǣ ǎǡǒ ȀǢ .$"(Ǿ ʡ  3+ǿǎȀ Ȁ Ǣ /ʙ/Ǿ'$./ Ǣ #$).ʙǑ Ǣ *- .ʙǑ Ǣ '*"Ǿ'$&ʙ Ȁ $PNQBSJOH UP UIF NPEFM XJUI CPUI DMVTUFST
  11. Everything is random • “Random effects” have many definitions •

    One common & bad definition: • Random effects are not fixed by the experiment • Does not matter—we want pooling for estimation benefits • Consider e.g. treatments BCPVU UIF FYQFSJNFOU .PEFM DPNQBSJTPO JT PG WBMVF .PEFM TFMFDUJPO BU MFBTU XIFO XF BSF QVSTVJOH TDJFOUJĕD JOGFSFODF JT VTVBMMZ OPU  &WFO NPSF DMVTUFST :PV NJHIU OPUJDF UIBU UIF USFBUNFOU FČFDUT UIF  QBSBNFUFST MPPL B MPU MJLF UIF  BOE " QBSBNFUFST $PVME XF BMTP VTF QBSUJBM QPPMJOH PO UIF USFBUNFOU FČFDUT :FT XF DPVME 4PNF QFPQMF XJMM TDSFBN i/Pw BU UIJT TVHHFTUJPO CFDBVTF UIFZ IBWF CFFO UBVHIU UIBU WBSZJOH FČFDUT BSF POMZ GPS WBSJBCMFT UIBU XFSF OPU FYQFSJNFOUBMMZ DPOUSPMMFE 4JODF USFBUNFOU XBT iĕYFEw CZ UIF FYQFSJNFOU UIF UIJOLJOH HPFT XF TIPVME VTF VOQPPMFE iĕYFEw FČFDUT ćJT JT BMM XSPOH ćF SFBTPO UP VTF WBSZJOH FČFDUT JT CFDBVTF UIFZ QSPWJEF CFUUFS JOGFS FODFT *U EPFTOU NBUUFS IPX UIF DMVTUFST BSJTF *G UIF JOEJWJEVBM VOJUT BSF IJŅİĵĮĻĴĮįĹIJ‰ UIF JOEFY WBMVFT DPVME CF SFBTTJHOFE XJUIPVU DIBOHJOH UIF NFBOJOH PG UIF NPEFM‰UIFO QBSUJBM QPPMJOH DPVME IFMQ *O UIJT DBTF UIFSF BSF POMZ GPVS USFBUNFOUT BOE UIFSF JT B MPU PG EBUB PO FBDI USFBUNFOU 4P QBSUJBM QPPMJOH JTOU HPJOH UP NBLF BOZ EJČFSFODF BOZXBZ )FSF JT (ǎǐǡǑ CVU OPX XJUI QBSUJBM QPPMJOH PO UIF USFBUNFOUT 3 DPEF  . /ǡ. ǿǎǒȀ (ǎǐǡǓ ʚǶ 0'(ǿ '$./ǿ +0'' Ǿ' !/ ʡ $)*(ǿ ǎ Ǣ + Ȁ Ǣ '*"$/ǿ+Ȁ ʚǶ ȁ/*-Ȃ ʔ "ȁ'*&Ǿ$Ȃ ʔ ȁ/- /( )/Ȃ Ǣ ȁ/- /( )/Ȃ ʡ )*-(ǿ Ǎ Ǣ .$"(Ǿ ȀǢ ȁ/*-Ȃ ʡ )*-(ǿ Ǿ- Ǣ .$"(Ǿ ȀǢ "ȁ'*&Ǿ$Ȃ ʡ )*-(ǿ Ǎ Ǣ .$"(Ǿ" ȀǢ Ǿ- ʡ )*-(ǿ Ǎ Ǣ ǎǡǒ ȀǢ .$"(Ǿ ʡ  3+ǿǎȀǢ .$"(Ǿ" ʡ  3+ǿǎȀǢ
  12. Everything is random FODFT *U EPFTOU NBUUFS IPX UIF DMVTUFST

    BSJTF *G UIF JOEJWJEVBM VOJUT BSF IJŅİĵĮĻĴĮįĹIJ‰ UIF JOEFY WBMVFT DPVME CF SFBTTJHOFE XJUIPVU DIBOHJOH UIF NFBOJOH PG UIF NPEFM‰UIFO QBSUJBM QPPMJOH DPVME IFMQ *O UIJT DBTF UIFSF BSF POMZ GPVS USFBUNFOUT BOE UIFSF JT B MPU PG EBUB PO FBDI USFBUNFOU 4P QBSUJBM QPPMJOH JTOU HPJOH UP NBLF BOZ EJČFSFODF BOZXBZ )FSF JT (ǎǐǡǑ CVU OPX XJUI QBSUJBM QPPMJOH PO UIF USFBUNFOUT 3 DPEF  . /ǡ. ǿǎǒȀ (ǎǐǡǓ ʚǶ 0'(ǿ '$./ǿ +0'' Ǿ' !/ ʡ $)*(ǿ ǎ Ǣ + Ȁ Ǣ '*"$/ǿ+Ȁ ʚǶ ȁ/*-Ȃ ʔ "ȁ'*&Ǿ$Ȃ ʔ ȁ/- /( )/Ȃ Ǣ ȁ/- /( )/Ȃ ʡ )*-(ǿ Ǎ Ǣ .$"(Ǿ ȀǢ ȁ/*-Ȃ ʡ )*-(ǿ Ǿ- Ǣ .$"(Ǿ ȀǢ "ȁ'*&Ǿ$Ȃ ʡ )*-(ǿ Ǎ Ǣ .$"(Ǿ" ȀǢ Ǿ- ʡ )*-(ǿ Ǎ Ǣ ǎǡǒ ȀǢ .$"(Ǿ ʡ  3+ǿǎȀǢ .$"(Ǿ" ʡ  3+ǿǎȀǢ .$"(Ǿ ʡ  3+ǿǎȀ Ȁ Ǣ /ʙ/Ǿ'$./ Ǣ #$).ʙǑ Ǣ *- .ʙǑ Ǣ '*"Ǿ'$&ʙ Ȁ * !/ǿ(ǎǐǡǑǢ(ǎǐǡǓȀ (ǎǐǡǑ (ǎǐǡǓ ȁǎȂ ǶǍǡǎǎ ǶǍǡǎǒ ȁǏȂ ǍǡǑǎ ǍǡǐǑ ȁǐȂ ǶǍǡǑǓ ǶǍǡǑǔ ȁǑȂ Ǎǡǐǎ ǍǡǏǒ * DVU PČ UIF SFTU PG UIF * !/ PVUQVU 8FSF POMZ JOUFSFTUFE JO UIF  QBSBNFUFST SJHIU OPX
  13. None
  14. None
  15. Divergent transitions • HMC runs a physics simulation • Each

    transition is a sample path • In real physics, energy is conserved • If energy at end of transition is not equal to energy at start, transition is divergent • Indicates inaccurate approximation • Tends to happen in regions of strong curvature of log-posterior • Other sampling strategies also bad in these cases, but produce no warnings! -4 -2 0 2 4 -4 -2 0 2 4 v x
  16. Divergent transitions -4 -2 0 2 4 -4 -2 0

    2 4 v x v ⇠ Normal(0, 3) x ⇠ Normal(0, exp(v)) <latexit sha1_base64="8drtdwTmbbGw2w9Z1GpZEWgxz8c=">AAACMnicfVDLSgMxFM3UVx1foy7dBIvSgpQZK+iy6EY3UsE+oFNKJk3b0GRmSDKlZeg3ufFLBBe6UMStH2GmnYW24oHA4Zxzyb3HCxmVyrZfjMzS8srqWnbd3Njc2t6xdvdqMogEJlUcsEA0PCQJoz6pKqoYaYSCIO4xUvcGV4lfHxIhaeDfq3FIWhz1fNqlGCktta2bITx2JeXQ5Uj1BY9vA8ERm+ShDU9gCRZc1xz9F3HJKMwPC7DQtnJ20Z4CLhInJTmQotK2ntxOgCNOfIUZkrLp2KFqxUgoihmZmG4kSYjwAPVIU1MfcSJb8fTkCTzSSgd2A6Gfr+BU/TkRIy7lmHs6mWwt571E/MtrRqp70YqpH0aK+Hj2UTdiUAUw6Q92qCBYsbEmCAuqd4W4jwTCSrds6hKc+ZMXSe206JSK9t1ZrnyZ1pEFB+AQ5IEDzkEZXIMKqAIMHsAzeAPvxqPxanwYn7Noxkhn9sEvGF/f202m1g==</latexit>
  17. -4 -2 0 2 4 -4 -2 0 2 4

    v x -4 -2 0 2 4 -4 -2 0 2 4 v x large step size small step size Divergent transitions
  18. Divergent transitions • Two basic strategies: • (1) Increase Stan’s

    adapt_delta control parameter => better step size adaptation + slower exploration • (2) Re-parameterize! -4 -2 0 2 4 -4 -2 0 2 4 v x
  19. Re-parameterize! • Most any statistical model can be expressed in

    several mathematically identical ways [BUJPO " NPSF FČFDUJWF TUSBUFHZ JT UP ĿIJĽĮĿĮĺIJŁIJĿĶŇIJ UIF NPEFM DBM NPEFM JU DBO CF XSJUUFO JO TFWFSBM GPSNT UIBU BSF NBUIFNBUJDBMMZ BMMZ NPSF PS MFTT FďDJFOU 4XJUDIJOH B NPEFM GSPN POF GPSN UP BOPUIFS [BUJPO * LOPX UIJT TPVOET DSB[Z "OE SFBMMZ * UPP XPVME SBUIFS OPU VOL MJLF UIJT #VU SFQBSBNFUFSJ[JOH B NPEFM JT JODSFEJCMZ VTFGVM XIFO WFM NPEFMT :PV SFBMMZ DBOU JHOPSF JU ćBOLGVMMZ JUT OPU BDUVBMMZ IBSE IZ B TJOHMF NPEFM DBO IBWF NVMUJQMF GPSNT DPOTJEFS UIF FYBNQMF PG UI B (BVTTJBO QSJPS UIBU JUTFMG DPOUBJOT QBSBNFUFST GPS JUT NFBO BOE α ∼ /PSNBM(µ, σ) E σ IBWF UIFJS PXO QSJPST CVU XF DBO JHOPSF UIPTF GPS OPX ćF ĕSTU FBUF B EJČFOU CVU FRVJWBMFOU GPSN JT UP TVCUSBDU PVU UIF NFBO µ GSPN α = µ + β β ∼ /PSNBM(, σ) UJD GVODUJPO PG µ BOE B OFX QBSBNFUFS β #VU JU IBT UIF TBNF EJTUSJCV F UIF NFBO PG β JT [FSP 4P BEEJOH µ ZJFMET B NFBO PG µ ćJT JT SFBMMZ
  20. Re-parameterize! • Most any statistical model can be expressed in

    several mathematically identical ways [BUJPO " NPSF FČFDUJWF TUSBUFHZ JT UP ĿIJĽĮĿĮĺIJŁIJĿĶŇIJ UIF NPEFM DBM NPEFM JU DBO CF XSJUUFO JO TFWFSBM GPSNT UIBU BSF NBUIFNBUJDBMMZ BMMZ NPSF PS MFTT FďDJFOU 4XJUDIJOH B NPEFM GSPN POF GPSN UP BOPUIFS [BUJPO * LOPX UIJT TPVOET DSB[Z "OE SFBMMZ * UPP XPVME SBUIFS OPU VOL MJLF UIJT #VU SFQBSBNFUFSJ[JOH B NPEFM JT JODSFEJCMZ VTFGVM XIFO WFM NPEFMT :PV SFBMMZ DBOU JHOPSF JU ćBOLGVMMZ JUT OPU BDUVBMMZ IBSE IZ B TJOHMF NPEFM DBO IBWF NVMUJQMF GPSNT DPOTJEFS UIF FYBNQMF PG UI B (BVTTJBO QSJPS UIBU JUTFMG DPOUBJOT QBSBNFUFST GPS JUT NFBO BOE α ∼ /PSNBM(µ, σ) E σ IBWF UIFJS PXO QSJPST CVU XF DBO JHOPSF UIPTF GPS OPX ćF ĕSTU FBUF B EJČFOU CVU FRVJWBMFOU GPSN JT UP TVCUSBDU PVU UIF NFBO µ GSPN α = µ + β β ∼ /PSNBM(, σ) UJD GVODUJPO PG µ BOE B OFX QBSBNFUFS β #VU JU IBT UIF TBNF EJTUSJCV F UIF NFBO PG β JT [FSP 4P BEEJOH µ ZJFMET B NFBO PG µ ćJT JT SFBMMZ BUJPO * LOPX UIJT TPVOET DSB[Z "OE SFBMMZ * UPP XPVME SBUIFS OPU OL MJLF UIJT #VU SFQBSBNFUFSJ[JOH B NPEFM JT JODSFEJCMZ VTFGVM XIFO M NPEFMT :PV SFBMMZ DBOU JHOPSF JU ćBOLGVMMZ JUT OPU BDUVBMMZ IBSE Z B TJOHMF NPEFM DBO IBWF NVMUJQMF GPSNT DPOTJEFS UIF FYBNQMF PG I B (BVTTJBO QSJPS UIBU JUTFMG DPOUBJOT QBSBNFUFST GPS JUT NFBO BOE α ∼ /PSNBM(µ, σ) σ IBWF UIFJS PXO QSJPST CVU XF DBO JHOPSF UIPTF GPS OPX ćF ĕSTU BUF B EJČFOU CVU FRVJWBMFOU GPSN JT UP TVCUSBDU PVU UIF NFBO µ GSPN α = µ + β β ∼ /PSNBM(, σ) JD GVODUJPO PG µ BOE B OFX QBSBNFUFS β #VU JU IBT UIF TBNF EJTUSJCV UIF NFBO PG β JT [FSP 4P BEEJOH µ ZJFMET B NFBO PG µ ćJT JT SFBMMZ JBCMF CZ TVCUSBDUJOH JUT NFBO GSPN FWFSZ WBMVF 6OGPSUVOBUFMZ JO UIF BUVSF UIJT GPSN XJUI [FSP JO UIF QSJPS JT LOPXO BT UIF ĻļĻİIJĻŁIJĿIJı
  21. Re-parameterize! • Most any statistical model can be expressed in

    several mathematically identical ways [BUJPO " NPSF FČFDUJWF TUSBUFHZ JT UP ĿIJĽĮĿĮĺIJŁIJĿĶŇIJ UIF NPEFM DBM NPEFM JU DBO CF XSJUUFO JO TFWFSBM GPSNT UIBU BSF NBUIFNBUJDBMMZ BMMZ NPSF PS MFTT FďDJFOU 4XJUDIJOH B NPEFM GSPN POF GPSN UP BOPUIFS [BUJPO * LOPX UIJT TPVOET DSB[Z "OE SFBMMZ * UPP XPVME SBUIFS OPU VOL MJLF UIJT #VU SFQBSBNFUFSJ[JOH B NPEFM JT JODSFEJCMZ VTFGVM XIFO WFM NPEFMT :PV SFBMMZ DBOU JHOPSF JU ćBOLGVMMZ JUT OPU BDUVBMMZ IBSE IZ B TJOHMF NPEFM DBO IBWF NVMUJQMF GPSNT DPOTJEFS UIF FYBNQMF PG UI B (BVTTJBO QSJPS UIBU JUTFMG DPOUBJOT QBSBNFUFST GPS JUT NFBO BOE α ∼ /PSNBM(µ, σ) E σ IBWF UIFJS PXO QSJPST CVU XF DBO JHOPSF UIPTF GPS OPX ćF ĕSTU FBUF B EJČFOU CVU FRVJWBMFOU GPSN JT UP TVCUSBDU PVU UIF NFBO µ GSPN α = µ + β β ∼ /PSNBM(, σ) UJD GVODUJPO PG µ BOE B OFX QBSBNFUFS β #VU JU IBT UIF TBNF EJTUSJCV F UIF NFBO PG β JT [FSP 4P BEEJOH µ ZJFMET B NFBO PG µ ćJT JT SFBMMZ BUJPO * LOPX UIJT TPVOET DSB[Z "OE SFBMMZ * UPP XPVME SBUIFS OPU OL MJLF UIJT #VU SFQBSBNFUFSJ[JOH B NPEFM JT JODSFEJCMZ VTFGVM XIFO M NPEFMT :PV SFBMMZ DBOU JHOPSF JU ćBOLGVMMZ JUT OPU BDUVBMMZ IBSE Z B TJOHMF NPEFM DBO IBWF NVMUJQMF GPSNT DPOTJEFS UIF FYBNQMF PG I B (BVTTJBO QSJPS UIBU JUTFMG DPOUBJOT QBSBNFUFST GPS JUT NFBO BOE α ∼ /PSNBM(µ, σ) σ IBWF UIFJS PXO QSJPST CVU XF DBO JHOPSF UIPTF GPS OPX ćF ĕSTU BUF B EJČFOU CVU FRVJWBMFOU GPSN JT UP TVCUSBDU PVU UIF NFBO µ GSPN α = µ + β β ∼ /PSNBM(, σ) JD GVODUJPO PG µ BOE B OFX QBSBNFUFS β #VU JU IBT UIF TBNF EJTUSJCV UIF NFBO PG β JT [FSP 4P BEEJOH µ ZJFMET B NFBO PG µ ćJT JT SFBMMZ JBCMF CZ TVCUSBDUJOH JUT NFBO GSPN FWFSZ WBMVF 6OGPSUVOBUFMZ JO UIF BUVSF UIJT GPSN XJUI [FSP JO UIF QSJPS JT LOPXO BT UIF ĻļĻİIJĻŁIJĿIJı α ∼ /PSNBM(µ, σ) σ IBWF UIFJS PXO QSJPST CVU XF DBO JHOPSF UIPTF GPS OPX ćF ĕSTU UF B EJČFOU CVU FRVJWBMFOU GPSN JT UP TVCUSBDU PVU UIF NFBO µ GSPN α = µ + β β ∼ /PSNBM(, σ) D GVODUJPO PG µ BOE B OFX QBSBNFUFS β #VU JU IBT UIF TBNF EJTUSJCV UIF NFBO PG β JT [FSP 4P BEEJOH µ ZJFMET B NFBO PG µ ćJT JT SFBMMZ JBCMF CZ TVCUSBDUJOH JUT NFBO GSPN FWFSZ WBMVF 6OGPSUVOBUFMZ JO UIF UVSF UIJT GPSN XJUI [FSP JO UIF QSJPS JT LOPXO BT UIF ĻļĻİIJĻŁIJĿIJı BSHPO JT OFWFS LJOE 8F DBO NBLF BOPUIFS GPSN PG UIJT EFĕOJUJPO CZ BMTP TNVHHMJOH PVU -JLF UIJT α = µ + [σ [ ∼ /PSNBM(, ) Centered Non-centered
  22. Re-parameterize! • Why would this madness help with sampling? •

    HMC sees a different geometry! v ⇠ Normal(0, 3) x ⇠ Normal(0, exp(v)) <latexit sha1_base64="8drtdwTmbbGw2w9Z1GpZEWgxz8c=">AAACMnicfVDLSgMxFM3UVx1foy7dBIvSgpQZK+iy6EY3UsE+oFNKJk3b0GRmSDKlZeg3ufFLBBe6UMStH2GmnYW24oHA4Zxzyb3HCxmVyrZfjMzS8srqWnbd3Njc2t6xdvdqMogEJlUcsEA0PCQJoz6pKqoYaYSCIO4xUvcGV4lfHxIhaeDfq3FIWhz1fNqlGCktta2bITx2JeXQ5Uj1BY9vA8ERm+ShDU9gCRZc1xz9F3HJKMwPC7DQtnJ20Z4CLhInJTmQotK2ntxOgCNOfIUZkrLp2KFqxUgoihmZmG4kSYjwAPVIU1MfcSJb8fTkCTzSSgd2A6Gfr+BU/TkRIy7lmHs6mWwt571E/MtrRqp70YqpH0aK+Hj2UTdiUAUw6Q92qCBYsbEmCAuqd4W4jwTCSrds6hKc+ZMXSe206JSK9t1ZrnyZ1pEFB+AQ5IEDzkEZXIMKqAIMHsAzeAPvxqPxanwYn7Noxkhn9sEvGF/f202m1g==</latexit> v ⇠ Normal(0, 3) x = z exp(v) z ⇠ Normal(0, 1) <latexit sha1_base64="NNyGHlVuqkzFbLoemgZLRuhyL0Q=">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</latexit> Centered Non-centered
  23. v ⇠ Normal(0, 3) x ⇠ Normal(0, exp(v)) <latexit sha1_base64="8drtdwTmbbGw2w9Z1GpZEWgxz8c=">AAACMnicfVDLSgMxFM3UVx1foy7dBIvSgpQZK+iy6EY3UsE+oFNKJk3b0GRmSDKlZeg3ufFLBBe6UMStH2GmnYW24oHA4Zxzyb3HCxmVyrZfjMzS8srqWnbd3Njc2t6xdvdqMogEJlUcsEA0PCQJoz6pKqoYaYSCIO4xUvcGV4lfHxIhaeDfq3FIWhz1fNqlGCktta2bITx2JeXQ5Uj1BY9vA8ERm+ShDU9gCRZc1xz9F3HJKMwPC7DQtnJ20Z4CLhInJTmQotK2ntxOgCNOfIUZkrLp2KFqxUgoihmZmG4kSYjwAPVIU1MfcSJb8fTkCTzSSgd2A6Gfr+BU/TkRIy7lmHs6mWwt571E/MtrRqp70YqpH0aK+Hj2UTdiUAUw6Q92qCBYsbEmCAuqd4W4jwTCSrds6hKc+ZMXSe206JSK9t1ZrnyZ1pEFB+AQ5IEDzkEZXIMKqAIMHsAzeAPvxqPxanwYn7Noxkhn9sEvGF/f202m1g==</latexit>

    v ⇠ Normal(0, 3) x = z exp(v) z ⇠ Normal(0, 1) <latexit sha1_base64="NNyGHlVuqkzFbLoemgZLRuhyL0Q=">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</latexit> -6 -4 -2 0 2 4 6 -3 -2 -1 0 1 2 3 v z -4 -2 0 2 4 -4 -2 0 2 4 v x
  24. v ⇠ Normal(0, 3) x ⇠ Normal(0, exp(v)) <latexit sha1_base64="8drtdwTmbbGw2w9Z1GpZEWgxz8c=">AAACMnicfVDLSgMxFM3UVx1foy7dBIvSgpQZK+iy6EY3UsE+oFNKJk3b0GRmSDKlZeg3ufFLBBe6UMStH2GmnYW24oHA4Zxzyb3HCxmVyrZfjMzS8srqWnbd3Njc2t6xdvdqMogEJlUcsEA0PCQJoz6pKqoYaYSCIO4xUvcGV4lfHxIhaeDfq3FIWhz1fNqlGCktta2bITx2JeXQ5Uj1BY9vA8ERm+ShDU9gCRZc1xz9F3HJKMwPC7DQtnJ20Z4CLhInJTmQotK2ntxOgCNOfIUZkrLp2KFqxUgoihmZmG4kSYjwAPVIU1MfcSJb8fTkCTzSSgd2A6Gfr+BU/TkRIy7lmHs6mWwt571E/MtrRqp70YqpH0aK+Hj2UTdiUAUw6Q92qCBYsbEmCAuqd4W4jwTCSrds6hKc+ZMXSe206JSK9t1ZrnyZ1pEFB+AQ5IEDzkEZXIMKqAIMHsAzeAPvxqPxanwYn7Noxkhn9sEvGF/f202m1g==</latexit>

    v ⇠ Normal(0, 3) x = z exp(v) z ⇠ Normal(0, 1) <latexit sha1_base64="NNyGHlVuqkzFbLoemgZLRuhyL0Q=">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</latexit> -6 -4 -2 0 2 4 6 -3 -2 -1 0 1 2 3 v z -4 -2 0 2 4 -4 -2 0 2 4 v x
  25. Re-parameterize! KVTU SFQMBDF UIF ĕYFE SFHVMBSJ[JOH QSJPS XJUI BO BEBQUJWF

    QSJPS 8FMM BMTP BEE B TUFS UZQF 5P BEE UIF TFDPOE DMVTUFS UZQF '*& XF NFSFMZ SFQMJDBUF UIF TUSVDUVSF /*- DMVTUFS ćJT NFBOT UIF MJOFBS NPEFM HFUT ZFU BOPUIFS WBSZJOH JOUFSDFQU αįĹļİĸ NPEFM HFUT BOPUIFS BEBQUJWF QSJPS BOE ZFU BOPUIFS TUBOEBSE EFWJBUJPO QBSBNFUFS )FSF JT UIF NBUIFNBUJDBM GPSN PG UIF NPEFM XJUI UIF OFX QJFDFT PG UIF NBDIJO IUFE JO CMVF -J ∼ #JOPNJBM(, QJ) MPHJU(QJ) = αĮİŁļĿ[J] + γįĹļİĸ[J] + βŁĿIJĮŁĺIJĻŁ[J] βK ∼ /PSNBM(, .) GPS K = .. αK ∼ /PSNBM(¯ α, σα) GPS K = .. γK ∼ /PSNBM(, σγ) GPS K = .. ¯ α ∼ /PSNBM(, .) σα ∼ &YQPOFOUJBM() σγ ∼ &YQPOFOUJBM() DI DMVTUFS HFUT JUT PXO WFDUPS PG QBSBNFUFST 'PS BDUPST UIF WFDUPS JT α BOE JU IBT MF DBVTF UIFSF BSF  DIJNQBO[FFT JO UIF TBNQMF 'PS CMPDLT UIF WFDUPS JT γ BOE JU IBT MF DBVTF UIFSF BSF  CMPDLT &BDI DMVTUFS WBSJBCMF OFFET JUT PXO TUBOEBSE EFWJBUJPO QBS
  26.   .0%&-4 8*5)065 "./&4*" FTU IPQF 8IBU XF XBOU

    JT B WFSTJPO PG (ǎǐǡǑ JO XIJDI XF HFU UIF QBSBNFUF EBQUJWF QSJPST BOE JOTUFBE JOUP UIF MJOFBS NPEFM *U MPPLT MJLF UIJT -J ∼ #JOPNJBM(, QJ) MPHJU(QJ) = ¯ α + [ĮİŁļĿ[J] σα + YįĹļİĸ[J] σγ + βŁĿIJĮŁĺIJĻŁ[J] βK ∼ /PSNBM(, .) GPS K = .. [K ∼ /PSNBM(, ) GPS K = .. YK ∼ /PSNBM(, ) GPS K = .. ¯ α ∼ /PSNBM(, .) σα ∼ &YQPOFOUJBM() σγ ∼ &YQPOFOUJBM() ćJT JT JOEFFE UIF TBNF NPEFM BT CFGPSF B NVMUJMFWFM NPEF XJUI WBSZJOH JOUFSDF )FSF JT UIF NBUIFNBUJDBM GPSN PG UIF NPEFM XJUI UIF OFX QJFDFT PG UIF NBDIJOF FE JO CMVF -J ∼ #JOPNJBM(, QJ) MPHJU(QJ) = αĮİŁļĿ[J] + γįĹļİĸ[J] + βŁĿIJĮŁĺIJĻŁ[J] βK ∼ /PSNBM(, .) GPS K = .. αK ∼ /PSNBM(¯ α, σα) GPS K = .. γK ∼ /PSNBM(, σγ) GPS K = .. ¯ α ∼ /PSNBM(, .) σα ∼ &YQPOFOUJBM() σγ ∼ &YQPOFOUJBM() DMVTUFS HFUT JUT PXO WFDUPS PG QBSBNFUFST 'PS BDUPST UIF WFDUPS JT α BOE JU IBT MFOH VTF UIFSF BSF  DIJNQBO[FFT JO UIF TBNQMF 'PS CMPDLT UIF WFDUPS JT γ BOE JU IBT MFOH VTF UIFSF BSF  CMPDLT &BDI DMVTUFS WBSJBCMF OFFET JUT PXO TUBOEBSE EFWJBUJPO QBSBN
  27.   .0%&-4 8*5)065 "./&4*" FTU IPQF 8IBU XF XBOU

    JT B WFSTJPO PG (ǎǐǡǑ JO XIJDI XF HFU UIF QBSBNFUF EBQUJWF QSJPST BOE JOTUFBE JOUP UIF MJOFBS NPEFM *U MPPLT MJLF UIJT -J ∼ #JOPNJBM(, QJ) MPHJU(QJ) = ¯ α + [ĮİŁļĿ[J] σα + YįĹļİĸ[J] σγ + βŁĿIJĮŁĺIJĻŁ[J] βK ∼ /PSNBM(, .) GPS K = .. [K ∼ /PSNBM(, ) GPS K = .. YK ∼ /PSNBM(, ) GPS K = .. ¯ α ∼ /PSNBM(, .) σα ∼ &YQPOFOUJBM() σγ ∼ &YQPOFOUJBM() ćJT JT JOEFFE UIF TBNF NPEFM BT CFGPSF B NVMUJMFWFM NPEF XJUI WBSZJOH JOUFSDF )FSF JT UIF NBUIFNBUJDBM GPSN PG UIF NPEFM XJUI UIF OFX QJFDFT PG UIF NBDIJOF FE JO CMVF -J ∼ #JOPNJBM(, QJ) MPHJU(QJ) = αĮİŁļĿ[J] + γįĹļİĸ[J] + βŁĿIJĮŁĺIJĻŁ[J] βK ∼ /PSNBM(, .) GPS K = .. αK ∼ /PSNBM(¯ α, σα) GPS K = .. γK ∼ /PSNBM(, σγ) GPS K = .. ¯ α ∼ /PSNBM(, .) σα ∼ &YQPOFOUJBM() σγ ∼ &YQPOFOUJBM() DMVTUFS HFUT JUT PXO WFDUPS PG QBSBNFUFST 'PS BDUPST UIF WFDUPS JT α BOE JU IBT MFOH VTF UIFSF BSF  DIJNQBO[FFT JO UIF TBNQMF 'PS CMPDLT UIF WFDUPS JT γ BOE JU IBT MFOH VTF UIFSF BSF  CMPDLT &BDI DMVTUFS WBSJBCMF OFFET JUT PXO TUBOEBSE EFWJBUJPO QBSBN
  28.   .0%&-4 8*5)065 "./&4*" FTU IPQF 8IBU XF XBOU

    JT B WFSTJPO PG (ǎǐǡǑ JO XIJDI XF HFU UIF QBSBNFUF EBQUJWF QSJPST BOE JOTUFBE JOUP UIF MJOFBS NPEFM *U MPPLT MJLF UIJT -J ∼ #JOPNJBM(, QJ) MPHJU(QJ) = ¯ α + [ĮİŁļĿ[J] σα + YįĹļİĸ[J] σγ + βŁĿIJĮŁĺIJĻŁ[J] βK ∼ /PSNBM(, .) GPS K = .. [K ∼ /PSNBM(, ) GPS K = .. YK ∼ /PSNBM(, ) GPS K = .. ¯ α ∼ /PSNBM(, .) σα ∼ &YQPOFOUJBM() σγ ∼ &YQPOFOUJBM() ćJT JT JOEFFE UIF TBNF NPEFM BT CFGPSF B NVMUJMFWFM NPEF XJUI WBSZJOH JOUFSDF )FSF JT UIF NBUIFNBUJDBM GPSN PG UIF NPEFM XJUI UIF OFX QJFDFT PG UIF NBDIJOF FE JO CMVF -J ∼ #JOPNJBM(, QJ) MPHJU(QJ) = αĮİŁļĿ[J] + γįĹļİĸ[J] + βŁĿIJĮŁĺIJĻŁ[J] βK ∼ /PSNBM(, .) GPS K = .. αK ∼ /PSNBM(¯ α, σα) GPS K = .. γK ∼ /PSNBM(, σγ) GPS K = .. ¯ α ∼ /PSNBM(, .) σα ∼ &YQPOFOUJBM() σγ ∼ &YQPOFOUJBM() DMVTUFS HFUT JUT PXO WFDUPS PG QBSBNFUFST 'PS BDUPST UIF WFDUPS JT α BOE JU IBT MFOH VTF UIFSF BSF  DIJNQBO[FFT JO UIF TBNQMF 'PS CMPDLT UIF WFDUPS JT γ BOE JU IBT MFOH VTF UIFSF BSF  CMPDLT &BDI DMVTUFS WBSJBCMF OFFET JUT PXO TUBOEBSE EFWJBUJPO QBSBN
  29. WF QSJPST BOE JOTUFBE JOUP UIF MJOFBS NPEFM *U MPPLT

    MJLF UIJT -J ∼ #JOPNJBM(, QJ) MPHJU(QJ) = ¯ α + [ĮİŁļĿ[J] σα + YįĹļİĸ[J] σγ + βŁĿIJĮŁĺIJĻŁ[J] βK ∼ /PSNBM(, .) GPS K = .. [K ∼ /PSNBM(, ) GPS K = .. YK ∼ /PSNBM(, ) GPS K = .. ¯ α ∼ /PSNBM(, .) σα ∼ &YQPOFOUJBM() σγ ∼ &YQPOFOUJBM() JOEFFE UIF TBNF NPEFM BT CFGPSF B NVMUJMFWFM NPEF XJUI WBSZJOH JOUFSDFQUT P BOE CMPDL #VU JU EPFTOU MPPL UIBU XBZ BU ĕSTU CFDBVTF OPX UIF BEBQUJWF QS HFUIFS JOTJEF UIF MJOFBS NPEFM ćF TUBUFE QSJPST CFMPX UIF MJOFBS NPEFM GPS U FUFS WFDUPST [ BOE Y BSF KVTU TUBOEBSEJ[FE OPSNBMT UT TBNQMF GSPN UIJT QPTUFSJPS OPX BOE TFF XIBU UIF SFQBSBNFUFSJ[BUJPO HBJOT ǿǎǐȀ σγ ∼ &YQPOFOUJBM() ćJT JT JOEFFE UIF TBNF NPEFM BT CFGPSF B NVMUJMFWFM NPEF XJUI WBSZJOH JOUFSDFQUT PO CPUI BDUPS BOE CMPDL #VU JU EPFTOU MPPL UIBU XBZ BU ĕSTU CFDBVTF OPX UIF BEBQUJWF QSJPST HFU QVU UPHFUIFS JOTJEF UIF MJOFBS NPEFM ćF TUBUFE QSJPST CFMPX UIF MJOFBS NPEFM GPS UIF OFX QBSBNFUFS WFDUPST [ BOE Y BSF KVTU TUBOEBSEJ[FE OPSNBMT -FUT TBNQMF GSPN UIJT QPTUFSJPS OPX BOE TFF XIBU UIF SFQBSBNFUFSJ[BUJPO HBJOT VT 3 DPEF  . /ǡ. ǿǎǐȀ (ǎǐǡǑ) ʚǶ 0'(ǿ '$./ǿ +0'' Ǿ' !/ ʡ $)*(ǿ ǎ Ǣ + Ȁ Ǣ '*"$/ǿ+Ȁ ʚǶ Ǿ- ʔ 5ȁ/*-Ȃȉ.$"(Ǿ ʔ 3ȁ'*&Ǿ$Ȃȉ.$"(Ǿ" ʔ ȁ/- /( )/Ȃ Ǣ ȁ/- /( )/Ȃ ʡ )*-(ǿ Ǎ Ǣ Ǎǡǒ ȀǢ 5ȁ/*-Ȃ ʡ )*-(ǿ Ǎ Ǣ ǎ ȀǢ 3ȁ'*&Ǿ$Ȃ ʡ )*-(ǿ Ǎ Ǣ ǎ ȀǢ Ǿ- ʡ )*-(ǿ Ǎ Ǣ ǎǡǒ ȀǢ .$"(Ǿ ʡ  3+ǿǎȀǢ .$"(Ǿ" ʡ  3+ǿǎȀ Ȁ Ǣ /ʙ/Ǿ'$./ Ǣ #$).ʙǑ Ǣ *- .ʙǑ Ȁ /PX MFUT DPNQBSF UIF )Ǿ !! OVNCFST PG FČFDUJWF TBNQMFT GPS UIFTF UXP GPSNT 3 DPEF  ) !!Ǿ ʚǶ +- $.ǿ (ǎǐǡǑ Ǣ  +/#ʙǏ ȀȁȁǪ)Ǿ !!ǪȂȂ ) !!Ǿ) ʚǶ +- $.ǿ (ǎǐǡǑ) Ǣ  +/#ʙǏ ȀȁȁǪ)Ǿ !!ǪȂȂ +-Ǿ)( . ʚǶ -*2)( .ǿ +- $.ǿ (ǎǐǡǑ Ǣ  +/#ʙǏ Ȁ Ȁ ) !!Ǿ/' ʚǶ $)ǿ ) !!Ǿ Ǣ ) !!Ǿ) Ȁ -*2)( .ǿ) !!Ǿ/' Ȁ ʚǶ +-Ǿ)( .
  30. Non-centered vs centered 3ȁ'*&Ǿ$Ȃ ʡ )*-(ǿ Ǎ Ǣ ǎ ȀǢ

    Ǿ- ʡ )*-(ǿ Ǎ Ǣ ǎǡǒ ȀǢ .$"(Ǿ ʡ  3+ǿǎȀǢ .$"(Ǿ" ʡ  3+ǿǎȀ Ȁ Ǣ /ʙ/Ǿ'$./ Ǣ #$).ʙǑ Ǣ *- .ʙǑ Ȁ /PX MFUT DPNQBSF UIF )Ǿ !! OVNCFST PG FČFDUJWF TBNQMFT GPS UIFTF UXP GPSNT 3 DPEF  ) !!Ǿ ʚǶ +- $.ǿ (ǎǐǡǑ Ǣ  +/#ʙǏ ȀȁȁǪ)Ǿ !!ǪȂȂ ) !!Ǿ) ʚǶ +- $.ǿ (ǎǐǡǑ) Ǣ  +/#ʙǏ ȀȁȁǪ)Ǿ !!ǪȂȂ +-Ǿ)( . ʚǶ -*2)( .ǿ +- $.ǿ (ǎǐǡǑ Ǣ  +/#ʙǏ Ȁ Ȁ ) !!Ǿ/' ʚǶ $)ǿ ) !!Ǿ Ǣ ) !!Ǿ) Ȁ -*2)( .ǿ) !!Ǿ/' Ȁ ʚǶ +-Ǿ)( . -*0)ǿ/ǿ) !!Ǿ/' ȀȀ ȁǎȂ ȁǏȂ ȁǐȂ ȁǑȂ ȁǎȂ ȁǏȂ ȁǐȂ ȁǑȂ ȁǒȂ ȁǓȂ ȁǔȂ "ȁǎȂ "ȁǏȂ "ȁǐȂ ) !!Ǿ ǒǕǑ ǒǔǏ ǒǕǔ ǒǏǐ ǒǓǏ ǔǒǖ ǒǑǎ ǒǑǔ ǒǕǏ ǓǍǎ Ǔǔǔ ǑǎǕ ǔǖǐ ǔǕǑ ) !!Ǿ) ǎǎǑǑ ǎǏǐǐ ǎǎǑǑ ǎǎǎǒ ǒǖǏ ǖǓǏ ǓǎǑ ǓǎǑ Ǔǎǎ ǓǏǔ ǔǔǏ ǎǕǎǎ Ǐǎǐǔ ǎǔǑǕ "ȁǑȂ "ȁǒȂ "ȁǓȂ Ǿ- .$"(Ǿ .$"(Ǿ" ) !!Ǿ ǓǍǒ ǖǐǑ ǓǍǐ ǕǏǑ ǕǖǕ ǏǑǐ ) !!Ǿ) ǎǒǔǒ ǏǎǏǔ ǎǑǏǖ ǒǔǐ ǔǔǔ ǖǔǍ
  31. Posterior predictions • Predictions more subtle: Same clusters or new

    clusters? • Same clusters: proceed as usual • New clusters: should average over distribution of varying effects • In this case: • Same clusters: Predictions for these chimpanzees • New clusters: Prediction for a new chimpanzee or rather for population of chimpanzees
  32. Same clusters, new clusters • Same actors: • Really same

    as before: varying effects are just parameters; you know the model; push samples back through the model • link() and sim() obey this rule • New actors (counterfactual): • which actor (cluster) to use for counterfactual predictions? • average actor • marginal of actor • show sample of actors from posterior
  33. Average actor • “average actor” means actor with population average

    intercept, “alpha” • Strategy: • replace varying intercept samples with zeros => all actors have average intercept now • compute predictions as usual BWFSBHF BDUPS #Z iBWFSBHF w * NFBO BO JOEJWJEVBM DIJNQBO[FF XJUI BO JOUFSDFQU FYBDUMZ BU  α UIF QPQVMBUJPO NFBO ćJT TJNVMUBOFPVTMZ JNQMJFT B WBSZJOH JOUFSDFQU PG [FSP 4JODF UIFSF JT VODFSUBJOUZ BCPVU UIF QPQVMBUJPO NFBO UIFSF JT TUJMM VODFSUBJOUZ BCPVU UIJT BWFSBHF JOEJWJEVBMT JOUFSDFQU #VU BT ZPVMM TFF UIF VODFSUBJOUZ JT NVDI TNBMMFS UIBO JU SFBMMZ TIPVME CF JG XF XJTI UP IPOFTUMZ SFQSFTFOU UIF QSPCMFN PG XIBU UP FYQFDU GSPN B OFX JOEJWJEVBM ćF ĕSTU TUFQ JT UP NBLF B OFX EBUB MJTU UP DPNQVUF QSFEJDUJPOT PWFS :PVWF EPOF UIJT JO QSFWJPVT DIBQUFST )FSF JT PVS OFX MJTU SFQSFTFOUJOH UIF GPVS EJČFSFOU USFBUNFOUT 3 DPEF  ǡ+-  ʚǶ '$./ǿ +-*.*Ǿ' !/ ʙ ǿǍǢǎǢǍǢǎȀǢ ȕ -$"#/ȅ' !/ȅ-$"#/ȅ' !/ *)$/$*) ʙ ǿǍǢǍǢǎǢǎȀǢ ȕ *)/-*'ȅ*)/-*'ȅ+-/) -ȅ+-/) - /*- ʙ - +ǿǏǢǑȀ ȕ +' #*' - Ȁ /FYU XFSF HPJOH UP NBLF B NBUSJY PG [FSPT UP SFQMBDF UIF WBSZJOH JOUFSDFQU TBNQMFT XJUI *UT FBTJFTU UP KVTU LFFQ UIF TBNF EJNFOTJPO BT UIF PSJHJOBM NBUSJY *O UIJT DBTF UIBU NFBOT VTJOH  TBNQMFT GPS FBDI PG  BDUPST #VU BMM PG UIF TBNQMFT XJMM CF TFU UP [FSP 3 DPEF  ȕ - +' 1-4$)" $)/ - +/ .(+' . 2$/# 5 -*. ȕ ǎǍǍǍ .(+' . 4 ǔ /*-. Ǿ/*-Ǿ5 -*. ʚǶ (/-$3ǿǍǢǎǍǍǍǢǔȀ ćBUT UIF POMZ OFX USJDL /PX XF KVTU QBTT UIJT OFX NBUSJY UP '$)& VTJOH UIF PQUJPOBM - +' BSHVNFOU .BLF TVSF UIF OFX NBUSJY JT OBNFE UIF TBNF BT UIF WBSZJOH JOUFSDFQU NBUSJY Ǿ/*- 0UIFSXJTF JU XPOU SFQMBDF BOZUIJOH UIBU BQQFBST JO UIF NPEFM 3 DPEF  ȕ !$- 0+ '$)& ȕ )*/ 0. *! - +' '$./ '$)&ǡ(ǎǏǡǑ ʚǶ '$)&ǿ (ǎǏǡǑ Ǣ )ʙǎǍǍǍ Ǣ /ʙǡ+-  Ǣ - +' ʙ'$./ǿǾ/*-ʙǾ/*-Ǿ5 -*.Ȁ Ȁ CF JG XF XJTI UP IPOFTUMZ SFQSFTFOU UIF QSPCMFN PG XIBU UP FYQFDU GSPN B OFX JOEJWJEVBM ćF ĕSTU TUFQ JT UP NBLF B OFX EBUB MJTU UP DPNQVUF QSFEJDUJPOT PWFS :PVWF EPOF UIJT JO QSFWJPVT DIBQUFST )FSF JT PVS OFX MJTU SFQSFTFOUJOH UIF GPVS EJČFSFOU USFBUNFOUT 3 DPEF  ǡ+-  ʚǶ '$./ǿ +-*.*Ǿ' !/ ʙ ǿǍǢǎǢǍǢǎȀǢ ȕ -$"#/ȅ' !/ȅ-$"#/ȅ' !/ *)$/$*) ʙ ǿǍǢǍǢǎǢǎȀǢ ȕ *)/-*'ȅ*)/-*'ȅ+-/) -ȅ+-/) - /*- ʙ - +ǿǏǢǑȀ ȕ +' #*' - Ȁ /FYU XFSF HPJOH UP NBLF B NBUSJY PG [FSPT UP SFQMBDF UIF WBSZJOH JOUFSDFQU TBNQMFT XJUI *UT FBTJFTU UP KVTU LFFQ UIF TBNF EJNFOTJPO BT UIF PSJHJOBM NBUSJY *O UIJT DBTF UIBU NFBOT VTJOH  TBNQMFT GPS FBDI PG  BDUPST #VU BMM PG UIF TBNQMFT XJMM CF TFU UP [FSP 3 DPEF  ȕ - +' 1-4$)" $)/ - +/ .(+' . 2$/# 5 -*. ȕ ǎǍǍǍ .(+' . 4 ǔ /*-. Ǿ/*-Ǿ5 -*. ʚǶ (/-$3ǿǍǢǎǍǍǍǢǔȀ ćBUT UIF POMZ OFX USJDL /PX XF KVTU QBTT UIJT OFX NBUSJY UP '$)& VTJOH UIF PQUJPOBM - +' BSHVNFOU .BLF TVSF UIF OFX NBUSJY JT OBNFE UIF TBNF BT UIF WBSZJOH JOUFSDFQU NBUSJY Ǿ/*- 0UIFSXJTF JU XPOU SFQMBDF BOZUIJOH UIBU BQQFBST JO UIF NPEFM 3 DPEF  ȕ !$- 0+ '$)& ȕ )*/ 0. *! - +' '$./ '$)&ǡ(ǎǏǡǑ ʚǶ '$)&ǿ (ǎǏǡǑ Ǣ )ʙǎǍǍǍ Ǣ /ʙǡ+-  Ǣ - +' ʙ'$./ǿǾ/*-ʙǾ/*-Ǿ5 -*.Ȁ Ȁ ȕ .0((-$5 ) +'*/ +- ǡ+ǡ( ) ʚǶ ++'4ǿ '$)&ǡ(ǎǏǡǑ Ǣ Ǐ Ǣ ( ) Ȁ
  34.  TBNQMFT GPS FBDI PG  BDUPST #VU BMM PG

    UIF TBNQMFT XJMM CF TFU UP [FSP 3 DPEF  ȕ - +' 1-4$)" $)/ - +/ .(+' . 2$/# 5 -*. ȕ ǎǍǍǍ .(+' . 4 ǔ /*-. Ǿ/*-Ǿ5 -*. ʚǶ (/-$3ǿǍǢǎǍǍǍǢǔȀ ćBUT UIF POMZ OFX USJDL /PX XF KVTU QBTT UIJT OFX NBUSJY UP '$)& VTJOH UIF PQUJPOBM - +' BSHVNFOU .BLF TVSF UIF OFX NBUSJY JT OBNFE UIF TBNF BT UIF WBSZJOH JOUFSDFQU NBUSJY Ǿ/*- 0UIFSXJTF JU XPOU SFQMBDF BOZUIJOH UIBU BQQFBST JO UIF NPEFM 3 DPEF  ȕ !$- 0+ '$)& ȕ )*/ 0. *! - +' '$./ '$)&ǡ(ǎǏǡǑ ʚǶ '$)&ǿ (ǎǏǡǑ Ǣ )ʙǎǍǍǍ Ǣ /ʙǡ+-  Ǣ - +' ʙ'$./ǿǾ/*-ʙǾ/*-Ǿ5 -*.Ȁ Ȁ ȕ .0((-$5 ) +'*/ +- ǡ+ǡ( ) ʚǶ ++'4ǿ '$)&ǡ(ǎǏǡǑ Ǣ Ǐ Ǣ ( ) Ȁ +- ǡ+ǡ ʚǶ ++'4ǿ '$)&ǡ(ǎǏǡǑ Ǣ Ǐ Ǣ  Ǣ +-*ʙǍǡǕ Ȁ +'*/ǿ Ǎ Ǣ Ǎ Ǣ /4+ ʙǫ)ǫ Ǣ 3'ʙǫ+-*.*Ǿ' !/ȅ*)$/$*)ǫ Ǣ 4'ʙǫ+-*+*-/$*) +0''  ' !/ǫ Ǣ 4'$(ʙǿǍǢǎȀ Ǣ 33/ʙǫ)ǫ Ǣ 3'$(ʙǿǎǢǑȀ Ȁ 3$.ǿ ǎ Ǣ /ʙǎǣǑ Ǣ ' '.ʙǿǫǍȅǍǫǢǫǎȅǍǫǢǫǍȅǎǫǢǫǎȅǎǫȀ Ȁ Ǿ/*-Ǿ5 -*. ʚǶ (/-$3ǿǍǢǎǍǍǍǢǔȀ ćBUT UIF POMZ OFX USJDL /PX XF KVTU QBTT UIJT OFX NBUSJY UP '$)& VTJOH UIF PQUJPOBM - +' BSHVNFOU .BLF TVSF UIF OFX NBUSJY JT OBNFE UIF TBNF BT UIF WBSZJOH JOUFSDFQU NBUSJY Ǿ/*- 0UIFSXJTF JU XPOU SFQMBDF BOZUIJOH UIBU BQQFBST JO UIF NPEFM 3 DPEF  ȕ !$- 0+ '$)& ȕ )*/ 0. *! - +' '$./ '$)&ǡ(ǎǏǡǑ ʚǶ '$)&ǿ (ǎǏǡǑ Ǣ )ʙǎǍǍǍ Ǣ /ʙǡ+-  Ǣ - +' ʙ'$./ǿǾ/*-ʙǾ/*-Ǿ5 -*.Ȁ Ȁ ȕ .0((-$5 ) +'*/ +- ǡ+ǡ( ) ʚǶ ++'4ǿ '$)&ǡ(ǎǏǡǑ Ǣ Ǐ Ǣ ( ) Ȁ +- ǡ+ǡ ʚǶ ++'4ǿ '$)&ǡ(ǎǏǡǑ Ǣ Ǐ Ǣ  Ǣ +-*ʙǍǡǕ Ȁ +'*/ǿ Ǎ Ǣ Ǎ Ǣ /4+ ʙǫ)ǫ Ǣ 3'ʙǫ+-*.*Ǿ' !/ȅ*)$/$*)ǫ Ǣ 4'ʙǫ+-*+*-/$*) +0''  ' !/ǫ Ǣ 4'$(ʙǿǍǢǎȀ Ǣ 33/ʙǫ)ǫ Ǣ 3'$(ʙǿǎǢǑȀ Ȁ 3$.ǿ ǎ Ǣ /ʙǎǣǑ Ǣ ' '.ʙǿǫǍȅǍǫǢǫǎȅǍǫǢǫǍȅǎǫǢǫǎȅǎǫȀ Ȁ  .6-5*-&7&- 1045&3*03 13&%*$5*0/4  0.0 0.2 0.4 0.6 0.8 1.0 prosoc_left/condition proportion pulled left 0/0 1/0 0/1 1/1 average actor 0.0 0.2 0.4 0.6 0.8 1.0 prosoc_left/condition proportion pulled left 0/0 1/0 0/1 1/1 marginal of actor 0.0 0.2 0.4 0.6 0.8 1.0 prosoc_left/condition proportion pulled left 0/0 1/0 0/1 1/1 50 simulated actors
  35. Marginal of actor • “Marginal of” means “averaging over variation

    in actors” => shows variation arising from variation across actors • Strategy: • Extract samples for sigma_actor • Simulate new varying intercepts • Use simulated intercepts to simulate predictions VMU JT EJTQMBZFE JO 'ĶĴłĿIJ ƉƊƍ PO UIF MFę ćF HSBZ SFHJPO TIPXT UIF  JOUF S XJUI BO BWFSBHF JOUFSDFQU ćJT LJOE PG DBMDVMBUJPO NBLFT JU FBTZ UP TFF UIF .*Ǿ' !/ BT XFMM BT VODFSUBJOUZ BCPVU XIFSF UIF BWFSBHF JT CVU JU EPFTOU TI O BNPOH BDUPST TIPX UIF WBSJBUJPO BNPOH BDUPST XFMM OFFE UP VTF .$"(Ǿ/*- JO UIF DBMD BHBJO TNVHHMF UIJT JOUP '$)& CZ VTJOH UIF - +' BSHVNFOU ćJT UJNF I NVMBUF B NBUSJY PG OFX WBSZJOH JOUFSDFQUT GSPN B (BVTTJBO EJTUSJCVUJPO EFĕOFE F QSJPS JO UIF NPEFM JUTFMG αĮİŁļĿ ∼ /PSNBM(, σĮİŁļĿ) QMJFT UIBU PODF XF IBWF TBNQMFT GPS σĮİŁļĿ XF DBO TJNVMBUFE OFX BDUPS JOU JT EJTUSJCVUJPO )FSFT UIF DPEF UP EP KVTU UIBU VTJOH -)*-( BOE UIFO QBTT UI UFSDFQUT JOUP '$)&  1-4$)" $)/ - +/ .(+' . 2$/# .$(0'/$*). Ƕ 3/-/ǡ.(+' .ǿ(ǎǏǡǑȀ -Ǿ.$(. ʚǶ -)*-(ǿǔǍǍǍǢǍǢ+*./ɶ.$"(Ǿ/*-Ȁ -Ǿ.$(. ʚǶ (/-$3ǿǾ/*-Ǿ.$(.ǢǎǍǍǍǢǔȀ
  36. MBUFE JOUFSDFQUT JOUP '$)& 3 DPEF  ȕ - +'

    1-4$)" $)/ - +/ .(+' . 2$/# .$(0'/$*). +*./ ʚǶ 3/-/ǡ.(+' .ǿ(ǎǏǡǑȀ Ǿ/*-Ǿ.$(. ʚǶ -)*-(ǿǔǍǍǍǢǍǢ+*./ɶ.$"(Ǿ/*-Ȁ Ǿ/*-Ǿ.$(. ʚǶ (/-$3ǿǾ/*-Ǿ.$(.ǢǎǍǍǍǢǔȀ ȕ !$- 0+ '$)& ȕ )*/ 0. *! - +' '$./ '$)&ǡ(ǎǏǡǑ ʚǶ '$)&ǿ (ǎǏǡǑ Ǣ )ʙǎǍǍǍ Ǣ /ʙǡ+-  Ǣ - +' ʙ'$./ǿǾ/*-ʙǾ/*-Ǿ.$(.Ȁ Ȁ 4VNNBSJ[JOH BOE QMPUUJOH JT FYBDUMZ BT CFGPSF BOE UIF SFTVMU JT EJTQMBZFE JO UIF NJEEMF PG 'ĶĴłĿIJ ƉƊƍ ćFTF QPTUFSJPS QSFEJDUJPOT BSF NBSHJOBM PG BDUPS XIJDI NFBOT UIBU UIFZ BW FSBHF PWFS UIF VODFSUBJOUZ BNPOH BDUPST *O DPOUSBTU UIF QSFEJDUJPOT PO UIF MFę KVTU TFU UIF BDUPS UP UIF BWFSBHF JHOPSJOH WBSJBUJPO BNPOH BDUPST "U UIJT QPJOU TUVEFOUT VTVBMMZ BTL i4P XIJDI POF TIPVME * VTF w ćF BOTXFS JT i*U EF QFOETw #PUI BSF VTFGVM EFQFOEJOH VQPO UIF RVFTUJPO ćF QSFEJDUJPOT GPS BO BWFSBHF BDUPS IFMQ UP WJTVBMJ[F UIF JNQBDU PG USFBUNFOU ćF QSFEJDUJPOT UIBU BSF NBSHJOBM PG BDUPS JMMVT USBUF IPX WBSJBCMF EJČFSFOU DIJNQBO[FFT BSF BDDPSEJOH UP UIF NPEFM :PV QSPCBCMZ XBOU UP DPNQVUF CPUI GPS ZPVSTFMG XIFO USZJOH UP VOEFSTUBOE B NPEFM #VU XIJDI ZPV JODMVEF JO B SFQPSU XJMM EFQFOE VQPO DPOUFYU *O UIJT DBTF XF DBO EP CFUUFS CZ NBLJOH B QMPU UIBU EJTQMBZT CPUI UIF USFBUNFOU FČFDU BOE UIF WBSJBUJPO BNPOH BDUPST 8F DBO EP UIJT CZ GPSHFUUJOH BCPVU JOUFSWBMT BOE JOTUFBE TJNV MBUJOH B TFSJFT PG OFX BDUPST JO FBDI PG UIF GPVS USFBUNFOUT #Z ESBXJOH B MJOF GPS FBDI BDUPS BDSPTT BMM GPVS USFBUNFOUT XFMM CF BCMF UP WJTVBMJ[F CPUI UIF [JH[BH JNQBDU PG +-*.*Ǿ' !/ BT XFMM BT UIF WBSJBUJPO BNPOH JOEJWJEVBMT 8IBU XFMM EP OPX JT XSJUF B OFX GVODUJPO UIBU TJNVMBUFT B OFX BDUPS GSPN UIF FTUJNBUFE  .6-5*-&7&- 1045&3*03 13&%*$5*0/4  0.0 0.2 0.4 0.6 0.8 1.0 prosoc_left/condition proportion pulled left 0/0 1/0 0/1 1/1 average actor 0.0 0.2 0.4 0.6 0.8 1.0 prosoc_left/condition proportion pulled left 0/0 1/0 0/1 1/1 marginal of actor 0.0 0.2 0.4 0.6 0.8 1.0 prosoc_left/condition proportion pulled left 0/0 1/0 0/1 1/1 50 simulated actors
  37. 0.0 0.2 0.4 0.6 0.8 1.0 prosoc_left/condition proportion pulled left

    0/0 1/0 0/1 1/1 average actor 0.0 0.2 0.4 0.6 0.8 1.0 prosoc_left/condition proportion pulled left 0/0 1/0 0/1 1/1 marginal of actor 0.0 0.2 0.4 0.6 0.8 1.0 prosoc_left/condition proportion pulled left 0/0 1/0 0/1 1/1 50 simulated actors 'ĶĴłĿIJ ƉƊƍ 1PTUFSJPS QSFEJDUJWF EJTUSJCVUJPOT GPS UIF DIJNQBO[FFT WBSZ JOH JOUFSDFQU NPEFM (ǎǏǡǑ ćF TPMJE MJOFT BSF QPTUFSJPS NFBOT BOE UIF TIBEFE SFHJPOT BSF  QFSDFOUJMF JOUFSWBMT -Fę 4FUUJOH UIF WBSZJOH JO UFSDFQU Ǿ/*- UP [FSP QSPEVDFT QSFEJDUJPOT GPS BO BWFSBHF BDUPS ćFTF QSFEJDUJPOT JHOPSF VODFSUBJOUZ BSJTJOH GSPN WBSJBUJPO BNPOH BDUPST .JE EMF 4JNVMBUJOH WBSZJOH JOUFSDFQUT VTJOH UIF QPTUFSJPS TUBOEBSE EFWJBUJPO
  38. Homework • Frogs & contraception • Next week: Chapter 14

    — varying slopes and other wonders