Richard McElreath
January 09, 2023
3.4k

# Statistical Rethinking 2023 - Lecture 03

January 09, 2023

## Transcript

2. ### Mars: June 2020 until Feb 2021 — Tunç Tezel —

https://vimeo.com/user48630149

8. ### 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

11. ### 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.
12. ### Making Geocentric Models Skill development goals: (1) Language for representing

models (2) Calculate posterior distributions with multiple unknowns (3) Constructing & understanding linear models

14. ### 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
15. ### Linear Regression Drawing the Owl (1) Question/goal/estimand (2) Scientific model

(3) Statistical model(s) (4) Validate model (5) Analyze data
16. ### 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)
17. ### 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)
18. ### 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)
19. ### 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”
20. ### 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
21. ### (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
22. ### 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 JUT  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 8FMM VTF ćF XBZ *MM EP UIJT JT UP TJNVMBUF PVS VOPCTFSWFE 6 WBSJBCMF GSPN UI ćFO XF DBO DPNQVUF B QFSTPOT XFJHIU BT 8 = β) + 6 F DPEF UP EP UIJT n to simulate weights of individuals from height t <- function(H,b,sd) {
23. ### Generative model: H → W ćF XBZ *MM EP UIJT

JT UP TJNVMBUF PVS VOPCTFSWFE 6 WBSJBCMF GSPN UI ćFO XF DBO DPNQVUF B QFSTPOT 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 JUT  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 8FMM VTF (BVTTJBO WBSJBUJPO ćF XBZ *MM EP UIJT JT UP TJNVMBUF PVS VOPCTFSWFE 6 WBSJBCMF GSPN UIF QSFWJPVT TFDUJPO ćFO XF DBO DPNQVUF B QFSTPOT XFJHIU BT 8 = β) + 6 )FSFT 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:
24. ### F )FSFT 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 )
25. ### 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
26. ### Describing models Conventional statistical model notation: (1) List the variables

(2) Define each variable as a deterministic or distributional function of the other variables
27. ### Describing models F 8 = β) + 6 )FSFT 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 XFMM 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
28. ### EBSE XBZ PG EFTDSJCJOH NPEFMT MJLF UIF POF BCPWF VTJOH

TUBUJTUJDBM O IFMQGVM GPS CPUI HFOFSBUJWF NPEFMT MJLF UIF POF BCPWF BOE GPS TUBU XFMM 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 JOEJWJEVBMT 8w 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
29. ### EBSE XBZ PG EFTDSJCJOH NPEFMT MJLF UIF POF BCPWF VTJOH

TUBUJTUJDBM O IFMQGVM GPS CPUI HFOFSBUJWF NPEFMT MJLF UIF POF BCPWF BOE GPS TUBU XFMM 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 JOEJWJEVBMT 8w 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
30. ### EBSE XBZ PG EFTDSJCJOH NPEFMT MJLF UIF POF BCPWF VTJOH

TUBUJTUJDBM O IFMQGVM GPS CPUI HFOFSBUJWF NPEFMT MJLF UIF POF BCPWF BOE GPS TUBU XFMM 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 JOEJWJEVBMT 8w 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
31. ### EBSE XBZ PG EFTDSJCJOH NPEFMT MJLF UIF POF BCPWF VTJOH

TUBUJTUJDBM O IFMQGVM GPS CPUI HFOFSBUJWF NPEFMT MJLF UIF POF BCPWF BOE GPS TUBU XFMM 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 JOEJWJEVBMT 8w 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
32. ### EBSE XBZ PG EFTDSJCJOH NPEFMT MJLF UIF POF BCPWF VTJOH

TUBUJTUJDBM O IFMQGVM GPS CPUI HFOFSBUJWF NPEFMT MJLF UIF POF BCPWF BOE GPS TUBU XFMM 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 JOEJWJEVBMT 8w 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
33. ### EBSE XBZ PG EFTDSJCJOH NPEFMT MJLF UIF POF BCPWF VTJOH

TUBUJTUJDBM O IFMQGVM GPS CPUI HFOFSBUJWF NPEFMT MJLF UIF POF BCPWF BOE GPS TUBU XFMM 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 JOEJWJEVBMT 8w 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
34. ### F E XIFSF β JT UIF QSPQPSUJPOBMJUZ DPOTUBOU *G JUT

 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 8FMM VTF (BVTTJBO WBSJBUJPO ćF XBZ *MM EP UIJT JT UP TJNVMBUF PVS VOPCTFSWFE 6 WBSJBCMF GSPN UIF QSFWJPVT TFDUJPO ćFO XF DBO DPNQVUF B QFSTPOT XFJHIU BT 8 = β) + 6 )FSFT 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 JOEJWJEVBMT 8w 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 σw ćF MBTU MJOF JOEJDBUFT BOPUIFS Q CVU GPS ) ćFTF XFSF VOJGPSN WBMVFT CFUXFFO  BOE 

36. ### 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,]
37. ### 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
38. ### 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 -FUT NBLF JU F σ =  BOE α =  4P UIFO XF KVTU OFFE UP DPOTJEFS EJČFSFOU QPTTJCMF WBMVF
39. ### 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 -FUT NBLF JU F σ =  BOE α =  4P UIFO XF KVTU OFFE UP DPOTJEFS EJČFSFOU QPTTJCMF WBMVF
40. ### 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 -FUT NBLF JU F σ =  BOE α =  4P UIFO XF KVTU OFFE UP DPOTJEFS EJČFSFOU QPTTJCMF WBMVF garden of forking data
41. ### 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 -FUT NBLF JU F σ =  BOE α =  4P UIFO XF KVTU OFFE UP DPOTJEFS EJČFSFOU QPTTJCMF WBMVF garden of forking data
42. ### 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 -FUT NBLF JU F σ =  BOE α =  4P UIFO XF KVTU OFFE UP DPOTJEFS EJČFSFOU QPTTJCMF WBMVF garden of forking data
43. ### 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 -FUT 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 σw *UT 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 JUT FOUJSFMZ B GVODUJ SJBCMFT *U JT UIFSF TP JU JT FBTJFS UP SFBE UIF NPEFM "OE UIJT JT BMNPTU BMX PEFMT BSF EFĕOFE 4P XFMM GPMMPX UIF DPOWFOUJPO W is distributed normally with mean that is a linear function of H
44. ### 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 -FUT QSPKFDU UIF QPTUFSJPS EJTUSJCVUJPO
45. ### 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 -FUT QSPKFDU UIF QPTUFSJPS EJTUSJCVUJPO 130 140 150 160 170 50 60 70 80 90 height (cm) weight (kg) N = 3
46. ### 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

49. ### 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 XFWF 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 )FSFT NZ 8J ∼ /PSNBM(µJ, σ) µJ = α + β)J α ∼ /PSNBM(, ) β ∼ 6OJGPSN(, ) σ ∼ 6OJGPSN(, ) PU NPSF BCPVU UIFTF QSJPST JO B NPNFOU #VU IFSFT 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) )
50. ### 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 XFWF 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 )FSFT NZ 8J ∼ /PSNBM(µJ, σ) µJ = α + β)J α ∼ /PSNBM(, ) β ∼ 6OJGPSN(, ) σ ∼ 6OJGPSN(, ) PU NPSF BCPVU UIFTF QSJPST JO B NPNFOU #VU IFSFT 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) )
51. ### 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 JTOU 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(, ) 8FSF HPJOH UP UIJOL B MPU NPSF BCPVU UIFTF QSJPST JO B NPNFO TUBUJTUJDBM NPEFM
52. ### 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 JTOU 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(, ) 8FSF HPJOH UP UIJOL B MPU NPSF BCPVU UIFTF QSJPST JO B NPNFO TUBUJTUJDBM NPEFM
53. ### 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 TPDBMMFE CZ "OESFX (FMNBO  JT 8IFO ZPV IBWF DPN QVUBUJPOBM QSPCMFNT PęFO UIFSFT B QSPCMFN XJUI ZPVS NPEFM  1SJPS QSFEJDUJWF EJTUSJCVUJPOT #FGPSF UBDLMJOH B SFBM TBNQMF BOE DPNQVUJOH BO FTUJ NBUF MFUT 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 JTOU 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(, ) 8FSF HPJOH UP UIJOL B MPU NPSF BCPVU UIFTF QSJPST JO B NPNFO TUBUJTUJDBM NPEFM
54. ### 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 TPDBMMFE CZ "OESFX (FMNBO  JT 8IFO ZPV IBWF DPN QVUBUJPOBM QSPCMFNT PęFO UIFSFT B QSPCMFN XJUI ZPVS NPEFM  1SJPS QSFEJDUJWF EJTUSJCVUJPOT #FGPSF UBDLMJOH B SFBM TBNQMF BOE DPNQVUJOH BO FTUJ NBUF MFUT 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)
55. ### 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)
56. ### 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,]
57. ### 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
58. ### # 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)
59. ### 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,]
60. ### 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
61. ### 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
62. ### 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
63. ### 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
64. ### 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 )
65. ### 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
66. ### 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 :PVWF 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 JTOU 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?
67. ### 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