Statistical Rethinking 3. Geocentric Models 2023 

Mars: June 2020 until Feb 2021 — Tunç Tezel — https://vimeo.com/user48630149 

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MARS EARTH Prediction Without Explanation Geocentric Model 

MARS EARTH 

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Giuseppe Piazzi 1746–1826 Palermo Circle 1.1.1801 

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Carl Friedrich Gauss 1777–1855 (portrait in 1803) 

1809 Bayesian argument for normal error and least-squares estimation 

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Linear Regression Geocentric: Describes associations, makes predictions, but mechanistically wrong Gaussian: Abstracts from generative error model, replaces with normal distribution, mechanistically silent Useful when handled with care Many special cases: ANOVA, ANCOVA, t-test, others From Breath of Bones: A Tale of the Golem 

Positions Distribution 

Positions Distribution 

Why Normal? Two arguments (1) Generative: Summed fluctuations tend towards normal distribution (2) Inferential: For estimating mean and variance, normal distribution is least informative distribution (maxent) Variable does not have to be normally distributed for normal model to be useful. It’s a machine for estimating mean/variance. 

Making Geocentric Models Skill development goals: (1) Language for representing models (2) Calculate posterior distributions with multiple unknowns (3) Constructing & understanding linear models 

FLOW 

Owl-drawing workflow (1) State a clear question (2) Sketch your causal assumptions (3) Use the sketch to define a generative model (4) Use generative model to build estimator (5) Profit 

Linear Regression Drawing the Owl (1) Question/goal/estimand (2) Scientific model (3) Statistical model(s) (4) Validate model (5) Analyze data 

Linear Regression Drawing the Owl (1) Question/goal/estimand (2) Scientific model (3) Statistical model(s) (4) Validate model (5) Analyze data library(rethinking) data(Howell1) 60 80 100 140 180 10 20 30 40 50 60 height (cm) weight (kg) 

Linear Regression Drawing the Owl (1) Question/goal/estimand Describe association between weight and height library(rethinking) data(Howell1) 60 80 100 140 180 10 20 30 40 50 60 height (cm) weight (kg) 

Linear Regression Drawing the Owl (1) Question/goal/estimand Describe association between ADULT weight and height data(Howell1) d <- Howell1[Howell1$age>=18,] 140 150 160 170 180 30 35 40 45 50 55 60 height (cm) weight (kg) 

Linear Regression (2) Scientific model How does height influence weight? 140 150 160 170 180 30 35 40 45 50 55 60 height (cm) weight (kg) data(Howell1) d <- Howell1[Howell1$age>=18,] H W W = f(H) “Weight is some function of height” 

Generative models Options (1) Dynamic: Incremental growth of organism; both mass and height (length) derive from growth pattern; Gaussian variation result of summed fluctuations (2) Static: Changes in height result in changes in weight, but no mechanism; Gaussian variation result of growth history 

(2) Scientific model How does height influence weight? H W W = f(H,U) “Weight is some function of height and unobserved stuff” U unobserved 

For adults, weight is a proportion of height plus the influence of unobserved causes: Generative model: H → W H W U UIF QSPQPSUJPOBMJUZ DPOTUBOU *G JUT  GPS FYBNQMF UIFO B QFSTPO XIP LH VTMZ OPU FWFSZPOF XJUI UIF TBNF IFJHIU IBT FYBDUMZ UIF TBNF XFJHIU TIPVME SFĘFDU UIJT 4P XF OFFE UP JOUSPEVDF TPNF WBSJBUJPO 8FMM VTF ćF XBZ *MM EP UIJT JT UP TJNVMBUF PVS VOPCTFSWFE 6 WBSJBCMF GSPN UI ćFO XF DBO DPNQVUF B QFSTPOT XFJHIU BT 8 = β) + 6 F DPEF UP EP UIJT n to simulate weights of individuals from height t <- function(H,b,sd) { 

Generative model: H → W ćF XBZ *MM EP UIJT JT UP TJNVMBUF PVS VOPCTFSWFE 6 WBSJBCMF GSPN UI ćFO XF DBO DPNQVUF B QFSTPOT XFJHIU BT 8 = β) + 6 F DPEF UP EP UIJT n to simulate weights of individuals from height t <- function(H,b,sd) { rnorm( length(H) , 0 , sd ) b*H + U n(W) F E XIFSF β JT UIF QSPQPSUJPOBMJUZ DPOTUBOU *G JUT  GPS FYBNQMF UIFO B QFSTPO XIP JT DN UBMM XFJHIT LH 0CWJPVTMZ OPU FWFSZPOF XJUI UIF TBNF IFJHIU IBT FYBDUMZ UIF TBNF XFJHIU "OE UIF TJNVMBUJPO TIPVME SFĘFDU UIJT 4P XF OFFE UP JOUSPEVDF TPNF WBSJBUJPO 8FMM VTF (BVTTJBO WBSJBUJPO ćF XBZ *MM EP UIJT JT UP TJNVMBUF PVS VOPCTFSWFE 6 WBSJBCMF GSPN UIF QSFWJPVT TFDUJPO ćFO XF DBO DPNQVUF B QFSTPOT XFJHIU BT 8 = β) + 6 )FSFT TPNF DPEF UP EP UIJT 3 DPEF  # function to simulate weights of individuals from height sim_weight <- function(H,b,sd) { U <- rnorm( length(H) , 0 , sd ) W <- b*H + U return(W) } ćF BSHVNFOU sd JT UIF TUBOEBSE EFWJBUJPO PG 6 *U EFUFSNJOFT IPX NVDI WBSJBUJPO UIFSF JT BSPVOE UIF FYQFDUFE WBMVF PG 8 GPS FBDI ) WBMVF 5P NBLF UIJT TJNVMBUJPO XPSL XF BMTP OFFE UP TJNVMBUF IFJHIU *MM KVTU VTF B VOJGPSN Generative code: 

F )FSFT TPNF DPEF UP EP UIJT 3 DPEF  # function to simulate weights of individuals from height sim_weight <- function(H,b,sd) { U <- rnorm( length(H) , 0 , sd ) W <- b*H + U return(W) } ćF BSHVNFOU sd JT UIF TUBOEBSE EFWJBUJPO PG 6 *U EFUFSNJOFT IPX NVDI WBSJBUJPO UIFSF JT BSPVOE UIF FYQFDUFE WBMVF PG 8 GPS FBDI ) WBMVF 5P NBLF UIJT TJNVMBUJPO XPSL XF BMTP OFFE UP TJNVMBUF IFJHIU *MM KVTU VTF B VOJGPSN EJTUSJCVUJPO PG BEVMU IFJHIU GSPN DN UP DN BT BO FYBNQMF ćFO XF DBO TJNVMBUF BOE QMPU PVS TZOUIFUJD QFPQMF 3 DPEF  H <- runif( 200 , min=130 , max=170 ) W <- sim_weight( H , b=0.5 , sd=5 ) plot( W ~ H , col=2 , lwd=3 )  W <- b*H + U return(W) } ćF BSHVNFOU sd JT UIF TUBOEBSE EFWJBUJPO PG 6 *U EFUFSNJOFT IPX NVDI WBSJBUJPO UIFSF JT BSPVOE UIF FYQFDUFE WBMVF PG 8 GPS FBDI ) WBMVF 5P NBLF UIJT TJNVMBUJPO XPSL XF BMTP OFFE UP TJNVMBUF IFJHIU *MM KVTU VTF B VOJGPSN EJTUSJCVUJPO PG BEVMU IFJHIU GSPN DN UP DN BT BO FYBNQMF ćFO XF DBO TJNVMBUF BOE QMPU PVS TZOUIFUJD QFPQMF 3 DPEF  H <- runif( 200 , min=130 , max=170 ) W <- sim_weight( H , b=0.5 , sd=5 ) plot( W ~ H , col=2 , lwd=3 ) 

QMPU PVS TZOUIFUJD QFPQMF 3 DPEF  H <- runif( 200 , min=130 , max=170 ) W <- sim_weight( H , b=0.5 , sd=5 ) plot( W ~ H , col=2 , lwd=3 ) ę  8)"5 *4 5)& */'-6&/$& 0' )&*()5 0/ 8&*()5  130 140 150 160 170 50 60 70 80 90 H W ćF TJNVMBUJPO DBO QSPEVDF NBOZ EJČFSFOU SFMBUJPOTIJQT &YQFSJNFOU XJUI UIF b QBSBNFUFS 

Describing models Conventional statistical model notation: (1) List the variables (2) Define each variable as a deterministic or distributional function of the other variables 

Describing models F 8 = β) + 6 )FSFT TPNF DPEF UP EP UIJT 3 DPEF  # function to simulate weights of individuals from height sim_weight <- function(H,b,sd) { U <- rnorm( length(H) , 0 , sd ) W <- b*H + U return(W) } ćF BSHVNFOU sd JT UIF TUBOEBSE EFWJBUJPO PG 6 *U EFUFSNJOFT IPX NVDI WBSJBUJPO UIFSF JT BSPVOE UIF FYQFDUFE WBMVF PG 8 GPS FBDI ) WBMVF 5P NBLF UIJT TJNVMBUJPO XPSL XF BMTP OFFE UP TJNVMBUF IFJHIU *MM KVTU VTF B VOJGPSN EJTUSJCVUJPO PG BEVMU IFJHIU GSPN DN UP DN BT BO FYBNQMF ćFO XF DBO TJNVMBUF BOE QMPU PVS TZOUIFUJD QFPQMF 3 DPEF  H <- runif( 200 , min=130 , max=170 ) W <- sim_weight( H , b=0.5 , sd=5 ) QBSBNFUFS DSJCJOH NPEFMT #FGPSF NPWJOH PO UIF EFWFMPQJOH UIF FTUJNBUPS * EBSE XBZ PG EFTDSJCJOH NPEFMT MJLF UIF POF BCPWF VTJOH TUBUJTUJDBM O IFMQGVM GPS CPUI HFOFSBUJWF NPEFMT MJLF UIF POF BCPWF BOE GPS TUBU XFMM EFWFMPQ "OE JU JT WFSZ DPNNPO JO UIF TDJFODFT FJHIU TJNVMBUJPO BCPWF JT EFĕOFE BT 8J = β)J + 6J 6J ∼ /PSNBM(, σ) )J ∼ 6OJGPSN(, ) OF JT UIF FRVBUJPO GPS 8 ćF MJUUMF J PO 8 ) BOE 6 JOEJDBUFT BO J 

EBSE XBZ PG EFTDSJCJOH NPEFMT MJLF UIF POF BCPWF VTJOH TUBUJTUJDBM O IFMQGVM GPS CPUI HFOFSBUJWF NPEFMT MJLF UIF POF BCPWF BOE GPS TUBU XFMM EFWFMPQ "OE JU JT WFSZ DPNNPO JO UIF TDJFODFT FJHIU TJNVMBUJPO BCPWF JT EFĕOFE BT 8J = β)J + 6J 6J ∼ /PSNBM(, σ) )J ∼ 6OJGPSN(, ) OF JT UIF FRVBUJPO GPS 8 ćF MJUUMF J PO 8 ) BOE 6 JOEJDBUFT BO J OVNCFS :PV DBO SFBE JU MJLF iFBDI JOEJWJEVBMT 8w ćF TFDPOE MJOF E G UIPTF VOPCTFSWFE 6 WBMVFT POF GPS FBDI JOEJWJEVBM ćFTF XFSF TJ JTUSJCVUJPO XJUI TUBOEBSE EFWJBUJPO σ ćBU ∼ TZNCPM JOEJDBUFT B variables definitions 

EBSE XBZ PG EFTDSJCJOH NPEFMT MJLF UIF POF BCPWF VTJOH TUBUJTUJDBM O IFMQGVM GPS CPUI HFOFSBUJWF NPEFMT MJLF UIF POF BCPWF BOE GPS TUBU XFMM EFWFMPQ "OE JU JT WFSZ DPNNPO JO UIF TDJFODFT FJHIU TJNVMBUJPO BCPWF JT EFĕOFE BT 8J = β)J + 6J 6J ∼ /PSNBM(, σ) )J ∼ 6OJGPSN(, ) OF JT UIF FRVBUJPO GPS 8 ćF MJUUMF J PO 8 ) BOE 6 JOEJDBUFT BO J OVNCFS :PV DBO SFBE JU MJLF iFBDI JOEJWJEVBMT 8w ćF TFDPOE MJOF E G UIPTF VOPCTFSWFE 6 WBMVFT POF GPS FBDI JOEJWJEVBM ćFTF XFSF TJ JTUSJCVUJPO XJUI TUBOEBSE EFWJBUJPO σ ćBU ∼ TZNCPM JOEJDBUFT B individuals 

EBSE XBZ PG EFTDSJCJOH NPEFMT MJLF UIF POF BCPWF VTJOH TUBUJTUJDBM O IFMQGVM GPS CPUI HFOFSBUJWF NPEFMT MJLF UIF POF BCPWF BOE GPS TUBU XFMM EFWFMPQ "OE JU JT WFSZ DPNNPO JO UIF TDJFODFT FJHIU TJNVMBUJPO BCPWF JT EFĕOFE BT 8J = β)J + 6J 6J ∼ /PSNBM(, σ) )J ∼ 6OJGPSN(, ) OF JT UIF FRVBUJPO GPS 8 ćF MJUUMF J PO 8 ) BOE 6 JOEJDBUFT BO J OVNCFS :PV DBO SFBE JU MJLF iFBDI JOEJWJEVBMT 8w ćF TFDPOE MJOF E G UIPTF VOPCTFSWFE 6 WBMVFT POF GPS FBDI JOEJWJEVBM ćFTF XFSF TJ JTUSJCVUJPO XJUI TUBOEBSE EFWJBUJPO σ ćBU ∼ TZNCPM JOEJDBUFT B deterministic distributed as 

EBSE XBZ PG EFTDSJCJOH NPEFMT MJLF UIF POF BCPWF VTJOH TUBUJTUJDBM O IFMQGVM GPS CPUI HFOFSBUJWF NPEFMT MJLF UIF POF BCPWF BOE GPS TUBU XFMM EFWFMPQ "OE JU JT WFSZ DPNNPO JO UIF TDJFODFT FJHIU TJNVMBUJPO BCPWF JT EFĕOFE BT 8J = β)J + 6J 6J ∼ /PSNBM(, σ) )J ∼ 6OJGPSN(, ) OF JT UIF FRVBUJPO GPS 8 ćF MJUUMF J PO 8 ) BOE 6 JOEJDBUFT BO J OVNCFS :PV DBO SFBE JU MJLF iFBDI JOEJWJEVBMT 8w ćF TFDPOE MJOF E G UIPTF VOPCTFSWFE 6 WBMVFT POF GPS FBDI JOEJWJEVBM ćFTF XFSF TJ JTUSJCVUJPO XJUI TUBOEBSE EFWJBUJPO σ ćBU ∼ TZNCPM JOEJDBUFT B Equation for expected weight 

EBSE XBZ PG EFTDSJCJOH NPEFMT MJLF UIF POF BCPWF VTJOH TUBUJTUJDBM O IFMQGVM GPS CPUI HFOFSBUJWF NPEFMT MJLF UIF POF BCPWF BOE GPS TUBU XFMM EFWFMPQ "OE JU JT WFSZ DPNNPO JO UIF TDJFODFT FJHIU TJNVMBUJPO BCPWF JT EFĕOFE BT 8J = β)J + 6J 6J ∼ /PSNBM(, σ) )J ∼ 6OJGPSN(, ) OF JT UIF FRVBUJPO GPS 8 ćF MJUUMF J PO 8 ) BOE 6 JOEJDBUFT BO J OVNCFS :PV DBO SFBE JU MJLF iFBDI JOEJWJEVBMT 8w ćF TFDPOE MJOF E G UIPTF VOPCTFSWFE 6 WBMVFT POF GPS FBDI JOEJWJEVBM ćFTF XFSF TJ JTUSJCVUJPO XJUI TUBOEBSE EFWJBUJPO σ ćBU ∼ TZNCPM JOEJDBUFT B Gaussian error with standard deviation sigma 

EBSE XBZ PG EFTDSJCJOH NPEFMT MJLF UIF POF BCPWF VTJOH TUBUJTUJDBM O IFMQGVM GPS CPUI HFOFSBUJWF NPEFMT MJLF UIF POF BCPWF BOE GPS TUBU XFMM EFWFMPQ "OE JU JT WFSZ DPNNPO JO UIF TDJFODFT FJHIU TJNVMBUJPO BCPWF JT EFĕOFE BT 8J = β)J + 6J 6J ∼ /PSNBM(, σ) )J ∼ 6OJGPSN(, ) OF JT UIF FRVBUJPO GPS 8 ćF MJUUMF J PO 8 ) BOE 6 JOEJDBUFT BO J OVNCFS :PV DBO SFBE JU MJLF iFBDI JOEJWJEVBMT 8w ćF TFDPOE MJOF E G UIPTF VOPCTFSWFE 6 WBMVFT POF GPS FBDI JOEJWJEVBM ćFTF XFSF TJ JTUSJCVUJPO XJUI TUBOEBSE EFWJBUJPO σ ćBU ∼ TZNCPM JOEJDBUFT B Height uniformly distributed from 130cm to 170cm 

F E XIFSF β JT UIF QSPQPSUJPOBMJUZ DPOTUBOU *G JUT  GPS FYBNQMF UIFO B QFSTPO XIP JT DN UBMM XFJHIT LH 0CWJPVTMZ OPU FWFSZPOF XJUI UIF TBNF IFJHIU IBT FYBDUMZ UIF TBNF XFJHIU "OE UIF TJNVMBUJPO TIPVME SFĘFDU UIJT 4P XF OFFE UP JOUSPEVDF TPNF WBSJBUJPO 8FMM VTF (BVTTJBO WBSJBUJPO ćF XBZ *MM EP UIJT JT UP TJNVMBUF PVS VOPCTFSWFE 6 WBSJBCMF GSPN UIF QSFWJPVT TFDUJPO ćFO XF DBO DPNQVUF B QFSTPOT XFJHIU BT 8 = β) + 6 )FSFT TPNF DPEF UP EP UIJT 3 DPEF  # function to simulate weights of individuals from height sim_weight <- function(H,b,sd) { U <- rnorm( length(H) , 0 , sd ) W <- b*H + U return(W) } ćF BSHVNFOU sd JT UIF TUBOEBSE EFWJBUJPO PG 6 *U EFUFSNJOFT IPX NVDI WBSJBUJPO UIFSF JT BSPVOE UIF FYQFDUFE WBMVF PG 8 GPS FBDI ) WBMVF FJHIU TJNVMBUJPO BCPWF JT EFĕOFE BT 8J = β)J + 6J 6J ∼ /PSNBM(, σ) )J ∼ 6OJGPSN(, ) OF JT UIF FRVBUJPO GPS 8 ćF MJUUMF J PO 8 ) BOE 6 JOEJDBUFT BO J OVNCFS :PV DBO SFBE JU MJLF iFBDI JOEJWJEVBMT 8w ćF TFDPOE MJOF E G UIPTF VOPCTFSWFE 6 WBMVFT POF GPS FBDI JOEJWJEVBM ćFTF XFSF TJ JTUSJCVUJPO XJUI TUBOEBSE EFWJBUJPO σ ćBU ∼ TZNCPM JOEJDBUFT B Q :PV DBO SFBE JU BT iUIF 6 WBMVFT BSF EJTUSJCVUFE BDDPSEJOH UP B OP NFBO [FSP BOE TUBOEBSE EFWJBUJPO σw ćF MBTU MJOF JOEJDBUFT BOPUIFS Q CVU GPS ) ćFTF XFSF VOJGPSN WBMVFT CFUXFFO  BOE  

PAUSE 

Linear Regression Drawing the Owl (1) Question/goal/estimand (2) Scientific model (3) Statistical model(s) (4) Validate model (5) Analyze data 140 150 160 170 180 30 35 40 45 50 55 60 height (cm) weight (kg) data(Howell1) d <- Howell1[Howell1$age>=18,] 

Estimator We want to estimate how the average weight changes with height. F UIF PCTFSWFE CPEZ XFJHIUT #VU UIF HBSEFO EFQFOET VQPO NPSF UIBO POF VO ODF B OPSNBM EJTUSJCVUJPO EFQFOET VQPO CPUI B NFBO BOE WBSJBODF 4P UIFSF BS MJUJFT UP DPOTJEFS -VDLJMZ QSPCBCJMJUZ UIFPSZ DBO IBOEMF UIFN BMM FBTJMZ *MM U VDUJPO TMPX BOE UIFO UIFSF XJMM CF FYBNQMFT BHJOF B MJOF EFTDSJCJOH UIF SFMBUJPOTIJQ CFUXFFO BOZ QBSUJDVMBS WBMVF )J BOE 8J GPS B TQFDJĕD )J XIJDI JT PęFO XSJUUFO &(8J|)J) " MJOF GPS UIJT FYQFDUB CZ &(8J|)J) = α + β)J SJBCMF α IFSF JT UIF ĶĻŁIJĿİIJĽŁ 8IFO )J =  UIFO &(8J|)J) = α "OE β PG UIF MJOF F MBTU UIJOH XF OFFE CFGPSF XF DBO SFBMMZ CVJME UIF FTUJNBUPS UIF QPTUFSJPS EJTUSJ OTJEFS 6 UIF WBSJBUJPO BSPVOE UIF FYQFDUBUJPO ćF MBSHFS σ JT UIF NPSF WB NQMJFT B OPSNBM EJTUSJCVUJPO XJUI NFBO &(8J|)J) BOE TUBOEBSE EFWJBUJPO σ Average weight conditional on height intercept slope 

Posterior distribution CJMJUJFT .BZCF ĕY σ =  BOE UIFO DPOTJEFS POMZ α = , β =  BOE α = QQPTF XF BMSFBEZ IBWF B QSFMJNJOBSZ QPTUFSJPS EJTUSJCVUJPO 1S(α, β, σ) GP JUJFT /PX XF HFU UIF EBUB GPS B OFX JOEJWJEVBM )J BOE 8J  8F XBOU UP VQ PS TP JU JODMVEFT UIJT JOEJWJEVBM 'PS BOZ DPNCJOBUJPO PG α BOE β BOE σ UIF Q MJUZ JT 1S(α, β, σ|)J, 8J) = 1S(8J|)J, α, β, σ) 1S(α, β, σ) ; ; JT PVS GSJFOEMZ OPSNBMJ[JOH DPOTUBOU ćF BDUJPO BT BMXBZT JT PO UPQ *O # XF UBLF UIF QSFWJPVT QPTUFSJPS QSPCBCJMJUZ UIF QSJPS QSPCBCJMJUZ PG UIJT DPNC , σ) BOE NVMUJQMZ JU CZ UIF SFMBUJWF OVNCFS PG XBZT QSPCBCJMJUZ UIBU UIJT DPNC FT DPVME QSPEVDF UIF PCTFSWBUJPO 8J XIJDI JT 1S(8J|)J, α, β, σ) T UBLF XIBU XF IBWF TP GBS BOE BDUVBMMZ EP TPNF DBMDVMBUJPOT -FUT NBLF JU F σ =  BOE α =  4P UIFO XF KVTU OFFE UP DPOTJEFS EJČFSFOU QPTTJCMF WBMVF 

Posterior distribution posterior probability of specific line CJMJUJFT .BZCF ĕY σ =  BOE UIFO DPOTJEFS POMZ α = , β =  BOE α = QQPTF XF BMSFBEZ IBWF B QSFMJNJOBSZ QPTUFSJPS EJTUSJCVUJPO 1S(α, β, σ) GP JUJFT /PX XF HFU UIF EBUB GPS B OFX JOEJWJEVBM )J BOE 8J  8F XBOU UP VQ PS TP JU JODMVEFT UIJT JOEJWJEVBM 'PS BOZ DPNCJOBUJPO PG α BOE β BOE σ UIF Q MJUZ JT 1S(α, β, σ|)J, 8J) = 1S(8J|)J, α, β, σ) 1S(α, β, σ) ; ; JT PVS GSJFOEMZ OPSNBMJ[JOH DPOTUBOU ćF BDUJPO BT BMXBZT JT PO UPQ *O # XF UBLF UIF QSFWJPVT QPTUFSJPS QSPCBCJMJUZ UIF QSJPS QSPCBCJMJUZ PG UIJT DPNC , σ) BOE NVMUJQMZ JU CZ UIF SFMBUJWF OVNCFS PG XBZT QSPCBCJMJUZ UIBU UIJT DPNC FT DPVME QSPEVDF UIF PCTFSWBUJPO 8J XIJDI JT 1S(8J|)J, α, β, σ) T UBLF XIBU XF IBWF TP GBS BOE BDUVBMMZ EP TPNF DBMDVMBUJPOT -FUT NBLF JU F σ =  BOE α =  4P UIFO XF KVTU OFFE UP DPOTJEFS EJČFSFOU QPTTJCMF WBMVF 

Posterior distribution posterior probability of specific line CJMJUJFT .BZCF ĕY σ =  BOE UIFO DPOTJEFS POMZ α = , β =  BOE α = QQPTF XF BMSFBEZ IBWF B QSFMJNJOBSZ QPTUFSJPS EJTUSJCVUJPO 1S(α, β, σ) GP JUJFT /PX XF HFU UIF EBUB GPS B OFX JOEJWJEVBM )J BOE 8J  8F XBOU UP VQ PS TP JU JODMVEFT UIJT JOEJWJEVBM 'PS BOZ DPNCJOBUJPO PG α BOE β BOE σ UIF Q MJUZ JT 1S(α, β, σ|)J, 8J) = 1S(8J|)J, α, β, σ) 1S(α, β, σ) ; ; JT PVS GSJFOEMZ OPSNBMJ[JOH DPOTUBOU ćF BDUJPO BT BMXBZT JT PO UPQ *O # XF UBLF UIF QSFWJPVT QPTUFSJPS QSPCBCJMJUZ UIF QSJPS QSPCBCJMJUZ PG UIJT DPNC , σ) BOE NVMUJQMZ JU CZ UIF SFMBUJWF OVNCFS PG XBZT QSPCBCJMJUZ UIBU UIJT DPNC FT DPVME QSPEVDF UIF PCTFSWBUJPO 8J XIJDI JT 1S(8J|)J, α, β, σ) T UBLF XIBU XF IBWF TP GBS BOE BDUVBMMZ EP TPNF DBMDVMBUJPOT -FUT NBLF JU F σ =  BOE α =  4P UIFO XF KVTU OFFE UP DPOTJEFS EJČFSFOU QPTTJCMF WBMVF garden of forking data 

Posterior distribution posterior probability of specific line prior CJMJUJFT .BZCF ĕY σ =  BOE UIFO DPOTJEFS POMZ α = , β =  BOE α = QQPTF XF BMSFBEZ IBWF B QSFMJNJOBSZ QPTUFSJPS EJTUSJCVUJPO 1S(α, β, σ) GP JUJFT /PX XF HFU UIF EBUB GPS B OFX JOEJWJEVBM )J BOE 8J  8F XBOU UP VQ PS TP JU JODMVEFT UIJT JOEJWJEVBM 'PS BOZ DPNCJOBUJPO PG α BOE β BOE σ UIF Q MJUZ JT 1S(α, β, σ|)J, 8J) = 1S(8J|)J, α, β, σ) 1S(α, β, σ) ; ; JT PVS GSJFOEMZ OPSNBMJ[JOH DPOTUBOU ćF BDUJPO BT BMXBZT JT PO UPQ *O # XF UBLF UIF QSFWJPVT QPTUFSJPS QSPCBCJMJUZ UIF QSJPS QSPCBCJMJUZ PG UIJT DPNC , σ) BOE NVMUJQMZ JU CZ UIF SFMBUJWF OVNCFS PG XBZT QSPCBCJMJUZ UIBU UIJT DPNC FT DPVME QSPEVDF UIF PCTFSWBUJPO 8J XIJDI JT 1S(8J|)J, α, β, σ) T UBLF XIBU XF IBWF TP GBS BOE BDUVBMMZ EP TPNF DBMDVMBUJPOT -FUT NBLF JU F σ =  BOE α =  4P UIFO XF KVTU OFFE UP DPOTJEFS EJČFSFOU QPTTJCMF WBMVF garden of forking data 

Posterior distribution posterior probability of specific line normalizing constant prior CJMJUJFT .BZCF ĕY σ =  BOE UIFO DPOTJEFS POMZ α = , β =  BOE α = QQPTF XF BMSFBEZ IBWF B QSFMJNJOBSZ QPTUFSJPS EJTUSJCVUJPO 1S(α, β, σ) GP JUJFT /PX XF HFU UIF EBUB GPS B OFX JOEJWJEVBM )J BOE 8J  8F XBOU UP VQ PS TP JU JODMVEFT UIJT JOEJWJEVBM 'PS BOZ DPNCJOBUJPO PG α BOE β BOE σ UIF Q MJUZ JT 1S(α, β, σ|)J, 8J) = 1S(8J|)J, α, β, σ) 1S(α, β, σ) ; ; JT PVS GSJFOEMZ OPSNBMJ[JOH DPOTUBOU ćF BDUJPO BT BMXBZT JT PO UPQ *O # XF UBLF UIF QSFWJPVT QPTUFSJPS QSPCBCJMJUZ UIF QSJPS QSPCBCJMJUZ PG UIJT DPNC , σ) BOE NVMUJQMZ JU CZ UIF SFMBUJWF OVNCFS PG XBZT QSPCBCJMJUZ UIBU UIJT DPNC FT DPVME QSPEVDF UIF PCTFSWBUJPO 8J XIJDI JT 1S(8J|)J, α, β, σ) T UBLF XIBU XF IBWF TP GBS BOE BDUVBMMZ EP TPNF DBMDVMBUJPOT -FUT NBLF JU F σ =  BOE α =  4P UIFO XF KVTU OFFE UP DPOTJEFS EJČFSFOU QPTTJCMF WBMVF garden of forking data 

PS TP JU JODMVEFT UIJT JOEJWJEVBM 'PS BOZ DPNCJOBUJPO PG α BOE β BOE σ UIF Q MJUZ JT 1S(α, β, σ|)J, 8J) = 1S(8J|)J, α, β, σ) 1S(α, β, σ) ; ; JT PVS GSJFOEMZ OPSNBMJ[JOH DPOTUBOU ćF BDUJPO BT BMXBZT JT PO UPQ *O # XF UBLF UIF QSFWJPVT QPTUFSJPS QSPCBCJMJUZ UIF QSJPS QSPCBCJMJUZ PG UIJT DPNC , σ) BOE NVMUJQMZ JU CZ UIF SFMBUJWF OVNCFS PG XBZT QSPCBCJMJUZ UIBU UIJT DPNC FT DPVME QSPEVDF UIF PCTFSWBUJPO 8J XIJDI JT 1S(8J|)J, α, β, σ) T UBLF XIBU XF IBWF TP GBS BOE BDUVBMMZ EP TPNF DBMDVMBUJPOT -FUT NBLF JU F σ =  BOE α =  4P UIFO XF KVTU OFFE UP DPOTJEFS EJČFSFOU QPTTJCMF WBMVF F β DPVME UBLF BOZ WBMVF CFUXFFO [FSP BOE POF 8F FYDMVEF OFHBUJWF WBMVFT X UIBU BWFSBHF XFJHIU JODSFBTFT XJUI IFJHIU "OE XF FYDMVEF WBMVFT HSFBUFS UI QFPQMF BSF KVTU OPU UIBU EFOTF‰B DN QFSTPO EPFT OPU XFJHIU PO BWFSBH EFWJBUJPO σ *O TUBUJTUJDBM NPEFM OPUBUJPO UIJT NFBOT 8J ∼ /PSNBM(α + β)J, σ) EJTUSJCVUFE BDDPSEJOH UP B OPSNBM EJTUSJCVUJPO XJUI NFBO α + β)J BOE O σw *UT DVTUPNBSZ UP XSJUF EFĕOJUJPOT MJLF UIJT XJUI NPSF UIBO POF MJOF TP SFBE 8J ∼ /PSNBM(µJ, σ) µJ = α + β)J BCMF µJ JT KVTU UIF FYQFDUFE WBMVF GPS JOEJWJEVBM J "OE JUT FOUJSFMZ B GVODUJ SJBCMFT *U JT UIFSF TP JU JT FBTJFS UP SFBE UIF NPEFM "OE UIJT JT BMNPTU BMX PEFMT BSF EFĕOFE 4P XFMM GPMMPX UIF DPOWFOUJPO W is distributed normally with mean that is a linear function of H 

Grid approximate posterior  (&0$&/53*$ .0%&-4 0 0.2 0.4 0.6 0.8 1 beta posterior probability 0.0 0.1 0.2 0.3 0.4 0.5 FT UIJT JNQMZ BCPVU UIF MJOF UIPVHI -FUT QSPKFDU UIF QPTUFSJPS EJTUSJCVUJPO 

Grid approximate posterior  (&0$&/53*$ .0%&-4 0 0.2 0.4 0.6 0.8 1 beta posterior probability 0.0 0.1 0.2 0.3 0.4 0.5 FT UIJT JNQMZ BCPVU UIF MJOF UIPVHI -FUT QSPKFDU UIF QPTUFSJPS EJTUSJCVUJPO 130 140 150 160 170 50 60 70 80 90 height (cm) weight (kg) N = 3 

130 140 150 160 170 50 60 70 80 90 height (cm) weight (kg) N = 1 130 140 150 160 170 50 60 70 80 90 height (cm) weight (kg) N = 2 130 140 150 160 170 50 60 70 80 90 height (cm) weight (kg) N = 3 130 140 150 160 170 50 60 70 80 90 weight (kg) N = 10 130 140 150 160 170 50 60 70 80 90 weight (kg) N = 20 130 140 150 160 170 50 60 70 80 90 weight (kg) N = 89 

Updating the posterior 

Updating the posterior 

Enough grid approximation We’ll use quadratic approximation for the rest of the first half of the course. BMSFBEZ 8IBU XF BMTP OFFE UP EFĕOF JT UIF QSJPS *O UIF DPEF XFWF QVU UIF QSFWJPVT QPTUFSJPS PS VTF B QSJPS UIBU BTTJHOFE UIF TBNF JOJUJBM TJCJMJUZ 8IFO UIFSF BSF JOĕOJUF QPTTJCJMJUJFT HJWFO UIFN BMM UIF TBNF E JEFB $POTJEFS GPS FYBNQMF UIF TMPQF β 8F LOPX JU JT QPTJUJWF BOE POF 8F VTFE UIBU LOPXMFEHF XIFO XF CVJMU PVS MJTU PG QPTTJCJMJUJFT FYQSFTT TVDI LOPXMFEHF JO UIF NPEFM EFĕOJUJPO BT XFMM )FSFT NZ 8J ∼ /PSNBM(µJ, σ) µJ = α + β)J α ∼ /PSNBM(, ) β ∼ 6OJGPSN(, ) σ ∼ 6OJGPSN(, ) PU NPSF BCPVU UIFTF QSJPST JO B NPNFOU #VU IFSFT UIF DPEF GPS UIJT m3.1 <- quap( alist( W ~ dnorm(mu,sigma), mu <- a + b*H, a ~ dnorm(0,10), b ~ dunif(0,1), sigma ~ dunif(0,10) ) , data=list(W=W,H=H) ) 

Enough grid approximation We’ll use quadratic approximation for the rest of the first half of the course. BMSFBEZ 8IBU XF BMTP OFFE UP EFĕOF JT UIF QSJPS *O UIF DPEF XFWF QVU UIF QSFWJPVT QPTUFSJPS PS VTF B QSJPS UIBU BTTJHOFE UIF TBNF JOJUJBM TJCJMJUZ 8IFO UIFSF BSF JOĕOJUF QPTTJCJMJUJFT HJWFO UIFN BMM UIF TBNF E JEFB $POTJEFS GPS FYBNQMF UIF TMPQF β 8F LOPX JU JT QPTJUJWF BOE POF 8F VTFE UIBU LOPXMFEHF XIFO XF CVJMU PVS MJTU PG QPTTJCJMJUJFT FYQSFTT TVDI LOPXMFEHF JO UIF NPEFM EFĕOJUJPO BT XFMM )FSFT NZ 8J ∼ /PSNBM(µJ, σ) µJ = α + β)J α ∼ /PSNBM(, ) β ∼ 6OJGPSN(, ) σ ∼ 6OJGPSN(, ) PU NPSF BCPVU UIFTF QSJPST JO B NPNFOU #VU IFSFT UIF DPEF GPS UIJT m3.1 <- quap( alist( W ~ dnorm(mu,sigma), mu <- a + b*H, a ~ dnorm(0,10), b ~ dunif(0,1), sigma ~ dunif(0,10) ) , data=list(W=W,H=H) ) 

Prior predictive distribution Priors should express scientific knowledge, but softly When H = 0, W = 0 Weight increases (on avg) with height Weight (kg) is less than height (cm) sigma must be positive F VTFE TP GBS XF DPVME JOQVU UIF QSFWJPVT QPTUFSJPS PS VTF B QSJPS QSPCBCJMJUZ UP FWFSZ QPTTJCJMJUZ 8IFO UIFSF BSF JOĕOJUF QPTTJCJM QSJPS QSPCBCJMJUZ JT B CBE JEFB $POTJEFS GPS FYBNQMF UIF TMPQF UIBU JU JTOU HSFBUFS UIBO POF 8F VTFE UIBU LOPXMFEHF XIFO XF 8F XBOU UP CF BCMF UP FYQSFTT TVDI LOPXMFEHF JO UIF NPEFM E TUBSUJOH QSPQPTBM 8J ∼ /PSNBM(µJ, σ) µJ = α + β)J α ∼ /PSNBM(, ) β ∼ 6OJGPSN(, ) σ ∼ 6OJGPSN(, ) 8FSF HPJOH UP UIJOL B MPU NPSF BCPVU UIFTF QSJPST JO B NPNFO TUBUJTUJDBM NPEFM 

Prior predictive distribution Understand the implications of priors through simulation What do the observable variables look like with these priors? F VTFE TP GBS XF DPVME JOQVU UIF QSFWJPVT QPTUFSJPS PS VTF B QSJPS QSPCBCJMJUZ UP FWFSZ QPTTJCJMJUZ 8IFO UIFSF BSF JOĕOJUF QPTTJCJM QSJPS QSPCBCJMJUZ JT B CBE JEFB $POTJEFS GPS FYBNQMF UIF TMPQF UIBU JU JTOU HSFBUFS UIBO POF 8F VTFE UIBU LOPXMFEHF XIFO XF 8F XBOU UP CF BCMF UP FYQSFTT TVDI LOPXMFEHF JO UIF NPEFM E TUBSUJOH QSPQPTBM 8J ∼ /PSNBM(µJ, σ) µJ = α + β)J α ∼ /PSNBM(, ) β ∼ 6OJGPSN(, ) σ ∼ 6OJGPSN(, ) 8FSF HPJOH UP UIJOL B MPU NPSF BCPVU UIFTF QSJPST JO B NPNFO TUBUJTUJDBM NPEFM 

F ES NFBOT UIF TFBSDI GBJMFE UP ĕOE UIF DPNCJOBUJPO PG VOLOPXOT UIBU NBYJNJ[FT QPTUFSJPS QSPCBCJMJUZ 4UBSUJOH UIF TFBSDI GSPN BOPUIFS QPTTJCMZ SBOEPN MPDBUJPO PS JODSFBTJOH IPX MPOH UIF TFBSDI JT BM MPXFE UP DPOUJOVF DBO PęFO SFTPMWF UIJT #VU NPSF PęFO JU JOEJDBUFT UIBU UIF NPEFM JT NJTTQFDJĕFE ćF 'PML ćFPSFN PG 4UBUJTUJDBM $PNQVUJOH TPDBMMFE CZ "OESFX (FMNBO  JT 8IFO ZPV IBWF DPN QVUBUJPOBM QSPCMFNT PęFO UIFSFT B QSPCMFN XJUI ZPVS NPEFM  1SJPS QSFEJDUJWF EJTUSJCVUJPOT #FGPSF UBDLMJOH B SFBM TBNQMF BOE DPNQVUJOH BO FTUJ NBUF MFUT UIJOL NPSF BCPVU UIF QSJPS EJTUSJCVUJPO JO PVU MJUUMF HPMFN 3 DPEF  n <- 1e3 a <- rnorm(n,0,10) b <- runif(n,0,1) plot( NULL , xlim=c(130,170) , ylim=c(50,90) , xlab="height (cm)" , ylab="weight (kg)" ) for ( j in 1:50 ) abline( a=a[j] , b=b[j] , lwd=2 , col=2 )  ćF EBUB ćF EBUB DPOUBJOFE JO data(Howell1) BSF QBSUJBM DFOTVT EBUB GPS UIF %PCF BSFB ,VOH 4BO DPNQJMFE GSPN JOUFSWJFXT DPOEVDUFE CZ /BODZ )PXFMM JO UIF MBUF T ćF ,VOH 4BO BSF UIF NPTU GBNPVT GPSBHJOH QPQVMBUJPO PG UIF UXFOUJFUI DFOUVSZ MBSHFMZ CFDBVTF PG EFUBJMFE RVBOUJUBUJWF TUVEJFT CZ QFPQMF MJLF )PXFMM -PBE UIF EBUB BOE QMBDF UIFN JOUP B DPOWFOJFOU PCKFDU XJUI 3 DPEF  library(rethinking) data(Howell1) F VTFE TP GBS XF DPVME JOQVU UIF QSFWJPVT QPTUFSJPS PS VTF B QSJPS QSPCBCJMJUZ UP FWFSZ QPTTJCJMJUZ 8IFO UIFSF BSF JOĕOJUF QPTTJCJM QSJPS QSPCBCJMJUZ JT B CBE JEFB $POTJEFS GPS FYBNQMF UIF TMPQF UIBU JU JTOU HSFBUFS UIBO POF 8F VTFE UIBU LOPXMFEHF XIFO XF 8F XBOU UP CF BCMF UP FYQSFTT TVDI LOPXMFEHF JO UIF NPEFM E TUBSUJOH QSPQPTBM 8J ∼ /PSNBM(µJ, σ) µJ = α + β)J α ∼ /PSNBM(, ) β ∼ 6OJGPSN(, ) σ ∼ 6OJGPSN(, ) 8FSF HPJOH UP UIJOL B MPU NPSF BCPVU UIFTF QSJPST JO B NPNFO TUBUJTUJDBM NPEFM 

F ES NFBOT UIF TFBSDI GBJMFE UP ĕOE UIF DPNCJOBUJPO PG VOLOPXOT UIBU NBYJNJ[FT QPTUFSJPS QSPCBCJMJUZ 4UBSUJOH UIF TFBSDI GSPN BOPUIFS QPTTJCMZ SBOEPN MPDBUJPO PS JODSFBTJOH IPX MPOH UIF TFBSDI JT BM MPXFE UP DPOUJOVF DBO PęFO SFTPMWF UIJT #VU NPSF PęFO JU JOEJDBUFT UIBU UIF NPEFM JT NJTTQFDJĕFE ćF 'PML ćFPSFN PG 4UBUJTUJDBM $PNQVUJOH TPDBMMFE CZ "OESFX (FMNBO  JT 8IFO ZPV IBWF DPN QVUBUJPOBM QSPCMFNT PęFO UIFSFT B QSPCMFN XJUI ZPVS NPEFM  1SJPS QSFEJDUJWF EJTUSJCVUJPOT #FGPSF UBDLMJOH B SFBM TBNQMF BOE DPNQVUJOH BO FTUJ NBUF MFUT UIJOL NPSF BCPVU UIF QSJPS EJTUSJCVUJPO JO PVU MJUUMF HPMFN 3 DPEF  n <- 1e3 a <- rnorm(n,0,10) b <- runif(n,0,1) plot( NULL , xlim=c(130,170) , ylim=c(50,90) , xlab="height (cm)" , ylab="weight (kg)" ) for ( j in 1:50 ) abline( a=a[j] , b=b[j] , lwd=2 , col=2 )  ćF EBUB ćF EBUB DPOUBJOFE JO data(Howell1) BSF QBSUJBM DFOTVT EBUB GPS UIF %PCF BSFB ,VOH 4BO DPNQJMFE GSPN JOUFSWJFXT DPOEVDUFE CZ /BODZ )PXFMM JO UIF MBUF T ćF ,VOH 4BO BSF UIF NPTU GBNPVT GPSBHJOH QPQVMBUJPO PG UIF UXFOUJFUI DFOUVSZ MBSHFMZ CFDBVTF PG EFUBJMFE RVBOUJUBUJWF TUVEJFT CZ QFPQMF MJLF )PXFMM -PBE UIF EBUB BOE QMBDF UIFN JOUP B DPOWFOJFOU PCKFDU XJUI 3 DPEF  library(rethinking) data(Howell1) 130 140 150 160 170 30 40 50 60 70 height (cm) weight (kg) 

Sermon on priors There are no correct priors, only scientifically justifiable priors Justify with information outside the data — like rest of model Priors not so important in simple models Very important/useful in complex models Need to practice now: simulate, understand 130 140 150 160 170 30 40 50 60 70 height (cm) weight (kg) 

Linear Regression Drawing the Owl (1) Question/goal/estimand (2) Scientific model (3) Statistical model(s) (4) Validate model (5) Analyze data 140 150 160 170 180 30 35 40 45 50 55 60 height (cm) weight (kg) data(Howell1) d <- Howell1[Howell1$age>=18,] 

Simulation-Based Validation Bare minimum: Test statistical model with simulated observations from scientific model Golem might be broken Even working golems might not deliver what you hoped Strong test: Simulation-Based Calibration Fahrvergnügen 

# simulate a sample of 10 people set.seed(93) H <- runif(10,130,170) W <- sim_weight(H,b=0.5,sd=5) # run the model library(rethinking) m3.1 <- quap( alist( W ~ dnorm(mu,sigma), mu <- a + b*H, a ~ dnorm(0,10), b ~ dunif(0,1), sigma ~ dunif(0,10) ) , data=list(W=W,H=H) ) # summary precis( m3.1 ) mean sd 5.5% 94.5% a 5.19 9.43 -9.88 20.26 b 0.49 0.07 0.38 0.59 sigma 5.64 1.29 3.57 7.71 Vary slope and make sure posterior mean tracks it Use a large sample to see that it converges to data generating value Same for other unknowns (parameters) 

Linear Regression Drawing the Owl (1) Question/goal/estimand (2) Scientific model (3) Statistical model(s) (4) Validate model (5) Analyze data 140 150 160 170 180 30 35 40 45 50 55 60 height (cm) weight (kg) data(Howell1) d <- Howell1[Howell1$age>=18,] 

Analyze the data dat <- list(W=d2$weight,H=d2$height) m3.2 <- quap( alist( W ~ dnorm(mu,sigma), mu <- a + b*H, a ~ dnorm(0,10), b ~ dunif(0,1), sigma ~ dunif(0,10) ) , data=dat ) precis( m3.2 ) mean sd 5.5% 94.5% a -43.38 4.17 -50.04 -36.71 b 0.57 0.03 0.53 0.61 sigma 4.25 0.16 3.99 4.51 a 0.50 0.55 0.60 0.65 -1 -60 -50 -40 -30 0.11 0.50 0.55 0.60 0.65 b -0.11 -60 -50 -40 -30 3.8 4.0 4.2 4.4 4.6 4.8 3.8 4.2 4.6 sigma 

Obey The Law First Law of Statistical Interpretation: The parameters are not independent of one another and cannot always be independently interpreted Instead: Push out posterior predictions and describe/interpret those 

Posterior predictive distribution 140 150 160 170 180 30 35 40 45 50 55 60 height (cm) weight (kg) post <- extract.samples(m3.2) plot( d2$height , d2$weight , col=2 , lwd=3 , xlab="height (cm)" , ylab="weight (kg)" ) for ( j in 1:20 ) abline( a=post$a[j] , b=post$b[j] , lwd=1 ) The posterior is full of lines 

140 150 160 170 180 30 35 40 45 50 55 60 height (cm) weight (kg) N = 1 140 150 160 170 180 30 35 40 45 50 55 60 height (cm) weight (kg) N = 5 140 150 160 170 180 30 35 40 45 50 55 60 height (cm) weight (kg) N = 25 140 150 160 170 180 30 35 40 45 50 55 60 height (cm) weight (kg) N = 50 140 150 160 170 180 30 35 40 45 50 55 60 height (cm) weight (kg) N = 352 

140 150 160 170 180 30 35 40 45 50 55 60 height (cm) weight (kg) Posterior predictive distribution post <- extract.samples(m3.2) plot( d2$height , d2$weight , col=2 , lwd=3 , xlab="height (cm)" , ylab="weight (kg)" ) for ( j in 1:20 ) abline( a=post$a[j] , b=post$b[j] , lwd=1 ) The posterior is full of lines The posterior is full of people height_seq <- seq(130,190,len=20) W_postpred <- sim( m3.2 , data=list(H=height_seq) ) W_PI <- apply( W_postpred , 2 , PI ) lines( height_seq , W_PI[1,] , lty=2 , lwd=2 ) lines( height_seq , W_PI[2,] , lty=2 , lwd=2 ) 

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Flexible Linear Thermometers Generative model How does height influence weight? W = f(H,U) “Weight is some function of height & unmeasured stuff” H W U 

Flexible Linear Thermometers Generative model How does height influence weight? W = f(H,U) “Weight is some function of height & unmeasured stuff” F E µJ = α + β)J :PVWF TFFO UIJT NVDI BMSFBEZ 8IBU XF BMTP OFFE UP EFĕOF JT UIF VTFE TP GBS XF DPVME JOQVU UIF QSFWJPVT QPTUFSJPS PS VTF B QSJPS UIBU QSPCBCJMJUZ UP FWFSZ QPTTJCJMJUZ 8IFO UIFSF BSF JOĕOJUF QPTTJCJMJUJFT QSJPS QSPCBCJMJUZ JT B CBE JEFB $POTJEFS GPS FYBNQMF UIF TMPQF β 8 UIBU JU JTOU HSFBUFS UIBO POF 8F VTFE UIBU LOPXMFEHF XIFO XF CVJ 8F XBOU UP CF BCMF UP FYQSFTT TVDI LOPXMFEHF JO UIF NPEFM EFĕO TUBSUJOH QSPQPTBM 8J ∼ /PSNBM(µJ, σ) µJ = α + β)J α ∼ /PSNBM(, ) β ∼ 6OJGPSN(, ) σ ∼ 6OJGPSN(, ) H W U Statistical model How does average weight change with height? 

Course Schedule Week 1 Bayesian inference Chapters 1, 2, 3 Week 2 Linear models & Causal Inference Chapter 4 Week 3 Causes, Confounds & Colliders Chapters 5 & 6 Week 4 Overfitting / Interactions Chapters 7 & 8 Week 5 MCMC & Generalized Linear Models Chapters 9, 10, 11 Week 6 Integers & Other Monsters Chapters 11 & 12 Week 7 Multilevel models I Chapter 13 Week 8 Multilevel models II Chapter 14 Week 9 Measurement & Missingness Chapter 15 Week 10 Generalized Linear Madness Chapter 16 https://github.com/rmcelreath/stat_rethinking_2023 

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