Statistical Rethinking - Lecture 16 (part 1)

Lecture 16 (part 1) - Mixture models, zero-inflated Poisson - Statistical Rethinking: A Bayesian Course with R Examples

Transcript

1. Week 8: Monsters & Mixtures Richard McElreath Statistical Rethinking
2. Mixtures • Some outcomes mix different processes • replace parameter of likelihood with distribution of its own • Over-dispersion: counts often more variable than expected, because probabilities/rates are variable • beta-binomial, gamma-Poisson (negative-binomial) • Zero-inflated mixtures
3. Monastery Mystery • Monks copy manuscripts • They also get drunk • Data: num manuscripts completed each day • Can infer number of days they got drunk?
4. Analyze? • Zero-inflated Poisson observations • Hidden state: drunk or sober • Can estimate probability of drinking and rate of production when sober • Must build a new likelihood, a mixture of stochastic processes p 1 – p observe y = 0 observe y > 0 Drink Work 'ĶĴłĿIJ ƉƉƌ -Fę 4USVDUVSF PG UIF [FSP HJOOJOH BU UIF UPQ UIF NPOLT ESJOL Q P UIF UJNF %SJOLJOH NPOLT BMXBZT QSPEV NPOLT NBZ QSPEVDF FJUIFS Z =  PS Z > [FSPJOĘBUFE PCTFSWBUJPOT ćF CMVF MJOF PCTFSWBUJPOT UIBU BSPTF GSPN ESJOLJOH *
5. Analyze? p 1 – p observe y = 0 observe y > 0 Drink Work 'ĶĴłĿIJ ƉƉƌ -Fę 4USVDUVSF PG UIF [FSP HJOOJOH BU UIF UPQ UIF NPOLT ESJOL Q P UIF UJNF %SJOLJOH NPOLT BMXBZT QSPEV NPOLT NBZ QSPEVDF FJUIFS Z =  PS Z > [FSPJOĘBUFE PCTFSWBUJPOT ćF CMVF MJOF PCTFSWBUJPOT UIBU BSPTF GSPN ESJOLJOH *  .0/45&34 "/% .*9563&4 1 – p observe y > 0 Work 0 1 2 3 4 5 0 50 100 150 manuscripts completed Frequency VSF PG UIF [FSPJOĘBUFE MJLFMJIPPE DBMDVMBUJPO #F POLT ESJOL Q PG UIF UJNF PS JOTUFBE XPSL  − Q PG T BMXBZT QSPEVDF BO PCTFSWBUJPO Z =  8PSLJOH drunk zeros
6. p 1 – p observe zero Poisson process FYQ(−λ) TJ ∼ #JOPNJBM(OJ, QJ) MPHJU QJ = α TJ ∼ #FUB#JOPNJBM(OJ, ¯ QJ, θ) MPHJU ¯ QJ = α + β1 1J MPH θ = τ observe n λO FYQ(−λ) O! TJ ∼ #JOPNJBM(OJ, QJ) MPHJU QJ = α TJ ∼ #FUB#JOPNJBM(OJ, ¯ QJ, θ) MPHJU ¯ QJ = α + β1 1J MPH θ = τ Binomial process
7. p 1 – p observe zero Poisson process FYQ(−λ) TJ ∼ #JOPNJBM(OJ, QJ) MPHJU QJ = α TJ ∼ #FUB#JOPNJBM(OJ, ¯ QJ, θ) MPHJU ¯ QJ = α + β1 1J MPH θ = τ observe n λO FYQ(−λ) O! TJ ∼ #JOPNJBM(OJ, QJ) MPHJU QJ = α TJ ∼ #FUB#JOPNJBM(OJ, ¯ QJ, θ) MPHJU ¯ QJ = α + β1 1J MPH θ = τ Binomial process 1S(|Q, λ) = Q + ( − Q) FYQ(−λ) TJ ∼ #JOPNJBM(OJ, QJ) MPHJU QJ = α TJ ∼ #FUB#JOPNJBM(OJ, ¯ QJ, θ) MPHJU ¯ QJ = α + β1 1J MPH θ = τ
8. p 1 – p observe zero Poisson process FYQ(−λ) TJ ∼ #JOPNJBM(OJ, QJ) MPHJU QJ = α TJ ∼ #FUB#JOPNJBM(OJ, ¯ QJ, θ) MPHJU ¯ QJ = α + β1 1J MPH θ = τ observe n λO FYQ(−λ) O! TJ ∼ #JOPNJBM(OJ, QJ) MPHJU QJ = α TJ ∼ #FUB#JOPNJBM(OJ, ¯ QJ, θ) MPHJU ¯ QJ = α + β1 1J MPH θ = τ Binomial process 1S(O|Q, λ) = ( − Q) λO FYQ(−λ) O! TJ ∼ #JOPNJBM(OJ, QJ) MPHJU QJ = α TJ ∼ #FUB#JOPNJBM(OJ, ¯ QJ, θ) MPHJU ¯ QJ = α + β1 1J 1S(|Q, λ) = Q + ( − Q) FYQ(−λ) TJ ∼ #JOPNJBM(OJ, QJ) MPHJU QJ = α TJ ∼ #FUB#JOPNJBM(OJ, ¯ QJ, θ) MPHJU ¯ QJ = α + β1 1J MPH θ = τ
9. Zero-inflated Poisson model NF GSPN XIJDI QSPDFTT FWFS QSPEVDF Z >  UIF FYQSFTTJPO BCPWF JT KVTU UIF DIBODF UIF BOE ĕOJTI Z NBOVTDSJQUT UIF EJTUSJCVUJPO BCPWF XJUI QBSBNFUFST Q QSPCBCJMJUZ PG B [FSP BOE FTDSJCF JUT TIBQF ćFO B [FSPJOĘBUFE 1PJTTPO SFHSFTTJPO UBLFT UIF ZJ ∼ ;*1PJTTPO(QJ, λJ) MPHJU(QJ) = αQ + βQ YJ MPH(λJ) = αλ + βλ YJ P MJOFBS NPEFMT BOE UXP MJOL GVODUJPOT POF GPS FBDI QSPDFTT JO UIF FST PG UIF MJOFBS NPEFMT EJČFS CFDBVTF BOZ QSFEJDUPS TVDI BT Y NBZ XJUI FBDI QBSU PG UIF NJYUVSF *O GBDU ZPV EPOU FWFO IBWF UP VTF PUI NPEFMT‰ZPV DBO DPOTUSVDU UIF UXP MJOFBS NPEFMT IPXFWFS ZPV PVS IZQPUIFTJT XF OFFE OPX FYDFQU GPS TPNF BDUVBM EBUB 4P MFUT TJNVMBUF UIF SLJOH ćFO ZPVMM TFF UIF DPEF VTFE UP SFDPWFS UIF QBSBNFUFS WBMVFT p 1 – p observe y = 0 observe y > 0 Drink Work 'ĶĴłĿIJ ƉƉƌ -Fę 4USVDUVSF PG UIF [FSP HJOOJOH BU UIF UPQ UIF NPOLT ESJOL Q P UIF UJNF %SJOLJOH NPOLT BMXBZT QSPEV NPOLT NBZ QSPEVDF FJUIFS Z =  PS Z > [FSPJOĘBUFE PCTFSWBUJPOT ćF CMVF MJOF PCTFSWBUJPOT UIBU BSPTF GSPN ESJOLJOH * Linear models are independent
10. Simulate, validate, cromulate • As models get more complicated, no guarantees you can • specify model correctly • estimate actual process reliably • Bayes not magic, just logic • Simulate “dummy data” • recover estimates • understand the model • Try parameter combinations hostile to estimation, so you know limits of the golem
11. Simulated manuscripts CF BTTPDJBUFE EJČFSFOUMZ XJUI FBDI QBSU PG UIF NJYUVSF *O GBDU ZPV EPOU FWFO IBWF UP VTF UIF TBNF QSFEJDUPST JO CPUI NPEFMT‰ZPV DBO DPOTUSVDU UIF UXP MJOFBS NPEFMT IPXFWFS ZPV XJTI EFQFOEJOH VQPO ZPVS IZQPUIFTJT 8F IBWF FWFSZUIJOH XF OFFE OPX FYDFQU GPS TPNF BDUVBM EBUB 4P MFUT TJNVMBUF UIF NPOLT ESJOLJOH BOE XPSLJOH ćFO ZPVMM TFF UIF DPEF VTFE UP SFDPWFS UIF QBSBNFUFS WBMVFT VTFE JO UIF TJNVMBUJPO 3 DPEF  ȃ  !$) +-( / -. +-*Ǭ-$)& ʄǤ ƻǏƽ ȃ ƽƻɳ *! 4. -/ Ǭ2*-& ʄǤ Ƽ ȃ 1 -" Ƽ ()0.-$+/ + - 4 ȃ .(+' *) 4 - *! +-*0/$*)  ʄǤ ƾǁǀ  ;&30*/'-"5&% 065$0.&4  ȃ .$(0'/ 4. (*)&. -$)& -$)& ʄǤ -$)*(ǭ  ǐ Ƽ ǐ +-*Ǭ-$)& Ǯ ȃ .$(0'/ ()0.-$+/. *(+' /  4 ʄǤ ǭƼǤ-$)&ǮǷ-+*$.ǭ  ǐ -/ Ǭ2*-& Ǯ ćF PVUDPNF WBSJBCMF XF HFU UP PCTFSWF JT 4 XIJDI JT KVTU B MJTU PG DPVOUT PG DPNQMFUFE NBOVTDSJQUT POF DPVOU GPS FBDI EBZ PG UIF ZFBS 5BLF B MPPL BU UIF PVUDPNF WBSJBCMF 3 DPEF  .$(+' #$./ǭ 4 ǐ 3'ʃǙ()0.-$+/. *(+' / Ǚ ǐ '2ʃƿ Ǯ 5 -*.Ǭ-$)& ʄǤ .0(ǭ-$)&Ǯ 5 -*.Ǭ2*-& ʄǤ .0(ǭ4ʃʃƻ ƺ -$)&ʃʃƻǮ 5 -*.Ǭ/*/' ʄǤ .0(ǭ4ʃʃƻǮ '$) .ǭ ǭƻǐƻǮ ǐ ǭ5 -*.Ǭ2*-&ǐ5 -*.Ǭ/*/'Ǯ ǐ '2ʃƿ ǐ *'ʃ-)"$ƽ Ǯ ćJT QMPU JT TIPXO PO UIF SJHIUIBOE TJEF PG 'ĶĴłĿIJ ƉƉƌ ćF [FSPT QSPEVDFE CZ ESJOLJOH BSF   .0/45&34 "/% .*9563&4 p 1 – p observe y = 0 observe y > 0 Drink Work 0 1 2 3 4 5 0 50 100 150 manuscripts completed Frequency drunk zeros
12. Fit model to dummy data ćF PVUDPNF WBSJBCMF XF HFU UP PCTFSWF JT 4 XIJDI JT KVTU B MJTU PG DPVOUT PG DPNQMFUFE NBOVTDSJQUT POF DPVOU GPS FBDI EBZ PG UIF ZFBS 5BLF B MPPL BU UIF PVUDPNF WBSJBCMF 3 DPEF  .$(+' #$./ǭ 4 ǐ 3'ʃǙ()0.-$+/. *(+' / Ǚ ǐ '2ʃƿ Ǯ 5 -*.Ǭ-$)& ʄǤ .0(ǭ-$)&Ǯ 5 -*.Ǭ2*-& ʄǤ .0(ǭ4ʃʃƻ ƺ -$)&ʃʃƻǮ 5 -*.Ǭ/*/' ʄǤ .0(ǭ4ʃʃƻǮ '$) .ǭ ǭƻǐƻǮ ǐ ǭ5 -*.Ǭ2*-&ǐ5 -*.Ǭ/*/'Ǯ ǐ '2ʃƿ ǐ *'ʃ-)"$ƽ Ǯ ćJT QMPU JT TIPXO PO UIF SJHIUIBOE TJEF PG 'ĶĴłĿIJ ƉƉƌ ćF [FSPT QSPEVDFE CZ ESJOLJOH BSF TIPXO JO CMVF ćPTF GSPN XPSL BSF TIPXO JO CMBDL ćF UPUBM OVNCFS PG [FSPT JT JOĘBUFE SFMBUJWF UP B UZQJDBM 1PJTTPO EJTUSJCVUJPO "OE UP ĕU UIF NPEFM UIF - /#$)&$)" QBDLBHF QSPWJEFT UIF [FSPJOĘBUFE 1PJTTPO MJLF MJIPPE BT 5$+*$. 'PS NPSF EFUBJM PO IPX JU SFMBUFT UP UIF NBUIFNBUJDT BCPWF TFF UIF 0WFSUIJOLJOH CPY BU UIF FOE PG UIJT TFDUJPO 6TJOH 5$+*$. JT TUSBJHIUGPSXBSE 3 DPEF  (ƼƼǏƿ ʄǤ (+ǭ '$./ǭ 4 ʋ 5$+*$.ǭ + ǐ '( Ǯǐ '*"$/ǭ+Ǯ ʄǤ +ǐ '*"ǭ'(Ǯ ʄǤ 'ǐ + ʋ )*-(ǭƻǐƼǮǐ ' ʋ )*-(ǭƻǐƼƻǮ Ǯ ǐ /ʃ'$./ǭ4ʃ4Ǯ Ǯ +- $.ǭ(ƼƼǏƿǮ  ) / 1 ƽǏǀɳ DŽǂǏǀɳ + ǤƼǏƾDŽ ƻǏƾƼ ǤƽǏƻ ǤƻǏǂǃ ' ƻǏƻǀ ƻǏƻǃ ǤƻǏƼ ƻǏƽƼ JOLJOH NPOLT OFWFS QSPEVDF Z >  UIF FYQSFTTJPO BCPWF JT KVTU UIF D PUI XPSL  − Q BOE ĕOJTI Z NBOVTDSJQUT OF ;*1PJTTPO BT UIF EJTUSJCVUJPO BCPWF XJUI QBSBNFUFST Q QSPCBCJMJUZ PG B PG 1PJTTPO UP EFTDSJCF JUT TIBQF ćFO B [FSPJOĘBUFE 1PJTTPO SFHSFTTJPO ZJ ∼ ;*1PJTTPO(QJ, λJ) MPHJU(QJ) = αQ + βQ YJ MPH(λJ) = αλ + βλ YJ IBU UIFSF BSF UXP MJOFBS NPEFMT BOE UXP MJOL GVODUJPOT POF GPS FBDI QSPD O ćF QBSBNFUFST PG UIF MJOFBS NPEFMT EJČFS CFDBVTF BOZ QSFEJDUPS TVDI JBUFE EJČFSFOUMZ XJUI FBDI QBSU PG UIF NJYUVSF *O GBDU ZPV EPOU FWFO IB QSFEJDUPST JO CPUI NPEFMT‰ZPV DBO DPOTUSVDU UIF UXP MJOFBS NPEFMT IPX QFOEJOH VQPO ZPVS IZQPUIFTJT IBWF FWFSZUIJOH XF OFFE OPX FYDFQU GPS TPNF BDUVBM EBUB 4P MFUT TJN ESJOLJOH BOE XPSLJOH ćFO ZPVMM TFF UIF DPEF VTFE UP SFDPWFS UIF QBSBNF IF TJNVMBUJPO
13. "OE UP ĕU UIF NPEFM UIF - /#$)&$)" QBDLBHF QSPWJEFT UIF [FSPJOĘBUFE 1PJTTPO MJLF MJIPPE BT 5$+*$. 'PS NPSF EFUBJM PO IPX JU SFMBUFT UP UIF NBUIFNBUJDT BCPWF TFF UIF 0WFSUIJOLJOH CPY BU UIF FOE PG UIJT TFDUJPO 6TJOH 5$+*$. JT TUSBJHIUGPSXBSE 3 DPEF  (ƼƼǏƿ ʄǤ (+ǭ '$./ǭ 4 ʋ 5$+*$.ǭ + ǐ '( Ǯǐ '*"$/ǭ+Ǯ ʄǤ +ǐ '*"ǭ'(Ǯ ʄǤ 'ǐ + ʋ )*-(ǭƻǐƼǮǐ ' ʋ )*-(ǭƻǐƼƻǮ Ǯ ǐ /ʃ'$./ǭ4ʃ4Ǯ Ǯ +- $.ǭ(ƼƼǏƿǮ  ) / 1 ƽǏǀɳ DŽǂǏǀɳ + ǤƼǏƾDŽ ƻǏƾƼ ǤƽǏƻ ǤƻǏǂǃ ' ƻǏƻǀ ƻǏƻǃ ǤƻǏƼ ƻǏƽƼ 0O UIF OBUVSBM TDBMF UIPTF ."1 FTUJNBUFT BSF 3 DPEF  '*"$./$ǭǤƼǏƾDŽǮ ȃ +-*$'$/4 -$)& 3+ǭƻǏƻǀǮ ȃ -/ !$)$.# ()0.-$+/.ǐ 2# ) )*/ -$)&$)" ǯƼǰ ƻǏƼDŽDŽƿƻǂǃ ǯƼǰ ƼǏƻǀƼƽǂƼ /PUJDF UIBU XF DBO HFU BO BDDVSBUF FTUJNBUF PG UIF QSPQPSUJPO PG EBZT UIF NPOLT ESJOL FWFO UIPVHI XF DBOU TBZ GPS BOZ QBSUJDVMBS EBZ XIFUIFS PS OPU UIFZ ESBOL ćJT FYBNQMF JT UIF TJNQMFTU QPTTJCMF *O SFBM QSPCMFNT ZPV NJHIU IBWF QSFEJDUPS WBSJ BCMFT UIBU BSF BTTPDJBUFE POF PS CPUI QSPDFTTFT JOTJEF UIF [FSPJOĘBUFE 1PJTTPO NJYUVSF *O "OE UP ĕU UIF NPEFM UIF - /#$)&$)" QBDLBHF QSPWJEFT UIF [FSPJOĘBUFE 1PJTTPO MJLF MJIPPE BT 5$+*$. 'PS NPSF EFUBJM PO IPX JU SFMBUFT UP UIF NBUIFNBUJDT BCPWF TFF UIF 0WFSUIJOLJOH CPY BU UIF FOE PG UIJT TFDUJPO 6TJOH 5$+*$. JT TUSBJHIUGPSXBSE 3 DPEF  (ƼƼǏƿ ʄǤ (+ǭ '$./ǭ 4 ʋ 5$+*$.ǭ + ǐ '( Ǯǐ '*"$/ǭ+Ǯ ʄǤ +ǐ '*"ǭ'(Ǯ ʄǤ 'ǐ + ʋ )*-(ǭƻǐƼǮǐ ' ʋ )*-(ǭƻǐƼƻǮ Ǯ ǐ /ʃ'$./ǭ4ʃ4Ǯ Ǯ +- $.ǭ(ƼƼǏƿǮ  ) / 1 ƽǏǀɳ DŽǂǏǀɳ + ǤƼǏƾDŽ ƻǏƾƼ ǤƽǏƻ ǤƻǏǂǃ ' ƻǏƻǀ ƻǏƻǃ ǤƻǏƼ ƻǏƽƼ 0O UIF OBUVSBM TDBMF UIPTF ."1 FTUJNBUFT BSF 3 DPEF  '*"$./$ǭǤƼǏƾDŽǮ ȃ +-*$'$/4 -$)& 3+ǭƻǏƻǀǮ ȃ -/ !$)$.# ()0.-$+/.ǐ 2# ) )*/ -$)&$)" ǯƼǰ ƻǏƼDŽDŽƿƻǂǃ ǯƼǰ ƼǏƻǀƼƽǂƼ /PUJDF UIBU XF DBO HFU BO BDDVSBUF FTUJNBUF PG UIF QSPQPSUJPO PG EBZT UIF NPOLT ESJOL FWFO UIPVHI XF DBOU TBZ GPS BOZ QBSUJDVMBS EBZ XIFUIFS PS OPU UIFZ ESBOL ćJT FYBNQMF JT UIF TJNQMFTU QPTTJCMF *O SFBM QSPCMFNT ZPV NJHIU IBWF QSFEJDUPS WBSJ BCMFT UIBU BSF BTTPDJBUFE POF PS CPUI QSPDFTTFT JOTJEF UIF [FSPJOĘBUFE 1PJTTPO NJYUVSF *O SFMBUJWF UP B UZQJDBM 1PJTTPO EJTUSJCVUJPO "OE UP ĕU UIF NPEFM UIF - /#$)&$)" QBDLBHF QSPWJEFT UIF [FSPJOĘBUFE 1PJTTPO MJLF MJIPPE BT 5$+*$. 'PS NPSF EFUBJM PO IPX JU SFMBUFT UP UIF NBUIFNBUJDT BCPWF TFF UIF 0WFSUIJOLJOH CPY BU UIF FOE PG UIJT TFDUJPO 6TJOH 5$+*$. JT TUSBJHIUGPSXBSE 3 DPEF  (ƼƼǏƿ ʄǤ (+ǭ '$./ǭ 4 ʋ 5$+*$.ǭ + ǐ '( Ǯǐ '*"$/ǭ+Ǯ ʄǤ +ǐ '*"ǭ'(Ǯ ʄǤ 'ǐ + ʋ )*-(ǭƻǐƼǮǐ ' ʋ )*-(ǭƻǐƼƻǮ Ǯ ǐ /ʃ'$./ǭ4ʃ4Ǯ Ǯ +- $.ǭ(ƼƼǏƿǮ  ) / 1 ƽǏǀɳ DŽǂǏǀɳ + ǤƼǏƾDŽ ƻǏƾƼ ǤƽǏƻ ǤƻǏǂǃ ' ƻǏƻǀ ƻǏƻǃ ǤƻǏƼ ƻǏƽƼ 0O UIF OBUVSBM TDBMF UIPTF ."1 FTUJNBUFT BSF 3 DPEF  '*"$./$ǭǤƼǏƾDŽǮ ȃ +-*$'$/4 -$)& 3+ǭƻǏƻǀǮ ȃ -/ !$)$.# ()0.-$+/.ǐ 2# ) )*/ -$)&$)" ǯƼǰ ƻǏƼDŽDŽƿƻǂǃ ǯƼǰ ƼǏƻǀƼƽǂƼ /PUJDF UIBU XF DBO HFU BO BDDVSBUF FTUJNBUF PG UIF QSPQPSUJPO PG EBZT UIF NPOLT ESJOL FWFO UIPVHI XF DBOU TBZ GPS BOZ QBSUJDVMBS EBZ XIFUIFS PS OPU UIFZ ESBOL ćJT FYBNQMF JT UIF TJNQMFTU QPTTJCMF *O SFBM QSPCMFNT ZPV NJHIU IBWF QSFEJDUPS WBSJ BCMFT UIBU BSF BTTPDJBUFE POF PS CPUI QSPDFTTFT JOTJEF UIF [FSPJOĘBUFE 1PJTTPO NJYUVSF *O observe y = 0 observe y > 0 Drink Work 0 1 2 3 4 5 0 manuscripts completed 'ĶĴłĿIJ ƉƉƌ -Fę 4USVDUVSF PG UIF [FSPJOĘBUFE MJLFMJIPPE DBMDVMBUJPO #F HJOOJOH BU UIF UPQ UIF NPOLT ESJOL Q PG UIF UJNF PS JOTUFBE XPSL  − Q PG UIF UJNF %SJOLJOH NPOLT BMXBZT QSPEVDF BO PCTFSWBUJPO Z =  8PSLJOH NPOLT NBZ QSPEVDF FJUIFS Z =  PS Z >  3JHIU 'SFRVFODZ EJTUSJCVUJPO PG [FSPJOĘBUFE PCTFSWBUJPOT ćF CMVF MJOF TFHNFOU PWFS [FSP TIPXT UIF Z =  PCTFSWBUJPOT UIBU BSPTF GSPN ESJOLJOH *O SFBM EBUB XF UZQJDBMMZ DBOOPU TFF XIJDI [FSPT DPNF GSPN XIJDI QSPDFTT 4JODF ESJOLJOH NPOLT OFWFS QSPEVDF Z >  UIF FYQSFTTJPO BCPWF JT KVTU UIF DIBODF UIF NPOLT CPUI XPSL  − Q BOE ĕOJTI Z NBOVTDSJQUT %FĕOF ;*1PJTTPO BT UIF EJTUSJCVUJPO BCPWF XJUI QBSBNFUFST Q QSPCBCJMJUZ PG B [FSP BOE λ NFBO PG 1PJTTPO UP EFTDSJCF JUT TIBQF ćFO B [FSPJOĘBUFE 1PJTTPO SFHSFTTJPO UBLFT UIF GPSN ZJ ∼ ;*1PJTTPO(QJ, λJ) MPHJU(QJ) = αQ + βQ YJ MPH(λJ) = αλ + βλ YJ /PUJDF UIBU UIFSF BSF UXP MJOFBS NPEFMT BOE UXP MJOL GVODUJPOT POF GPS FBDI QSPDFTT JO UIF ;*1PJTTPO ćF QBSBNFUFST PG UIF MJOFBS NPEFMT EJČFS CFDBVTF BOZ QSFEJDUPS TVDI BT Y NBZ
14. Other mixtures • Can ZIBinomial, too • Also “hurdle” models, aka zero-augmented • zero-augmented gamma, example in text • Continuous mixtures for overdispersed counts • beta-binomial • gamma-Poisson • We’ll focus on multilevel models instead