Statistical Rethinking 7. Fitting Over & Under 2023 

Mikołaj Kopernik (1473–1543) 

In nite causes, nite data Estimator might exist, but not be useful Struggle against causation: How to use causal assumptions to design estimators, contrast alternative models Struggle against data: How to make the estimators work X Y Z B A C 

Problems of Prediction What function describes these points? ( tting, compression) What function explains these points? (causal inference) What would happen if we changed a point’s mass? (intervention) What is the next observation from the same process? (prediction) 35 40 45 50 55 60 600 800 1000 1200 mass (kg) brain volume (cc) 

Leave-one-out cross-validation (1) Drop one point (2) Fit line to remaining (3) Predict dropped point (4) Repeat (1) with next point (5) Score is error on dropped 

Leave-one-out cross-validation (1) Drop one point (2) Fit line to remaining (3) Predict dropped point (4) Repeat (1) with next point (5) Score is error on dropped 

Leave-one-out cross-validation (1) Drop one point (2) Fit line to remaining (3) Predict dropped point (4) Repeat (1) with next point (5) Score is error on dropped 

Leave-one-out cross-validation (1) Drop one point (2) Fit line to remaining (3) Predict dropped point (4) Repeat (1) with next point (5) Score is error on dropped 

Leave-one-out cross-validation (1) Drop one point (2) Fit line to remaining (3) Predict dropped point (4) Repeat (1) with next point (5) Score is error on dropped In: 318 Out: 619 

Bayesian Cross-Validation We use the entire posterior, not just a point prediction Cross-validation score is: MJNJU UIFPSFN UP QSPWJEF B NFBTVSF PG UIF TUBOEBSE FSSPS TĽŀĶŀ = / WBS(QTJT J) NCFS PG PCTFSWBUJPOT BOE QTJT J JT UIF 14*4 FTUJNBUF GPS PCTFSWBUJPO J *G UIJT F TFOTF CF TVSF UP MPPL BU UIF DPEF CPY BU UIF FOE PG UIJT TFDUJPO QBHF   FUPTNPPUIFE DSPTTWBMJEBUJPO $SPTTWBMJEBUJPO FTUJNBUFT UIF PVUPGTBNQMF ĿIJıĶİŁĶŃIJıIJĻŀĶŁņ MQQE QBHF   *G ZPV IBWF / PCTFSWBUJPOT BOE ĕU UIF QQJOH B TJOHMF PCTFSWBUJPO ZJ FBDI UJNF UIFO UIF PVUPGTBNQMF MQQE JT UIF TVN BDZ GPS FBDI PNJUUFE ZJ  MQQE $7 = / J=  4 4 T= MPH 1S(ZJ|θ−J,T) NQMFT GSPN B .BSLPW DIBJO BOE θ−J,T JT UIF TUI TBNQMF GSPN UIF QPTUFSJPS EJTUSJ S PCTFSWBUJPOT PNJUUJOH ZJ  NQMJOH SFQMBDFT UIF DPNQVUBUJPO PG / QPTUFSJPS EJTUSJCVUJPOT CZ VTJOH BO FTUJNBUF -5 -4 -3 -2 -1 0 0.0 0.5 1.0 1.5 log posterior prob of observation Density 

Bayesian Cross-Validation FSF / JT UIF OVNCFS PG PCTFSWBUJPOT BOE QTJT J JT UIF 14*4 FTUJNBUF GPS PCTFSWBUJPO J *G TOU RVJUF NBLF TFOTF CF TVSF UP MPPL BU UIF DPEF CPY BU UIF FOE PG UIJT TFDUJPO QBHF  FSUIJOLJOH 1BSFUPTNPPUIFE DSPTTWBMJEBUJPO $SPTTWBMJEBUJPO FTUJNBUFT UIF PVUPGTBN ĴĽļĶĻŁńĶŀIJĽĿIJıĶİŁĶŃIJıIJĻŀĶŁņ MQQE QBHF   *G ZPV IBWF / PCTFSWBUJPOT BOE ĕU EFM / UJNFT ESPQQJOH B TJOHMF PCTFSWBUJPO ZJ FBDI UJNF UIFO UIF PVUPGTBNQMF MQQE JT UIF T IF BWFSBHF BDDVSBDZ GPS FBDI PNJUUFE ZJ  MQQE $7 = / J=  4 4 T= MPH 1S(ZJ|θ−J,T) SF T JOEFYFT TBNQMFT GSPN B .BSLPW DIBJO BOE θ−J,T JT UIF TUI TBNQMF GSPN UIF QPTUFSJPS EJ PO DPNQVUFE GPS PCTFSWBUJPOT PNJUUJOH ZJ  *NQPSUBODF TBNQMJOH SFQMBDFT UIF DPNQVUBUJPO PG / QPTUFSJPS EJTUSJCVUJPOT CZ VTJOH BO FTUJN IF JNQPSUBODF PG FBDI J UP UIF QPTUFSJPS EJTUSJCVUJPO 8F ESBX TBNQMFT GSPN UIF GVMM QPTUFSJPS VUJPO Q(θ|Z) CVU XF XBOU TBNQMFT GSPN UIF SFEVDFE MFBWFPOFPVU QPTUFSJPS EJTUSJCVUJPO Q(θ|Z XF SFXFJHIU FBDI TBNQMF T CZ UIF JOWFSTF PG UIF QSPCBCJMJUZ PG UIF PNJUUFE PCTFSWBUJPO  log pointwise predictive density Pages 210 and 218 N data points S samples from posterior log probability of each point i, computed with posterior that omits point i average log probability for point i 

In: 318 Out: 619 In: 289 Out: 865 [1] [2] 

318 619 289 865 In: 201 Out: 12,538 [3] [1] [2] 

201 12,538 120 25,530 7 293,840 318 619 289 865 [1] [2] [3] [4] [5] 

Cross-validation For simple models, more parameters improves t to sample But may reduce accuracy of predictions out of sample Most accurate model trades o exibility with over tting 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 polynomial terms relative error error in sample error out of sample 

1st degree polynomial 6th degree polynomial 

2nd degree polynomial error in sample error out of sample 1 2 3 4 5 6 200 600 1000 polynomial terms prediction error 

2nd degree polynomial error in sample error out of sample 1 2 3 4 5 150 200 250 300 polynomial terms prediction error 

Regularization Over tting depends upon the priors Skeptical priors have tighter variance, reduce exibility Regularization: Function nds regular features of process Good priors are o en tighter than you think! 

1 2 3 4 5 150 200 250 300 polynomial terms prediction error in out β j ∼ Normal(0,10) μ i = α + m ∑ j=1 β j xj i 

1 2 3 4 5 150 200 250 300 polynomial terms prediction error in out β j ∼ Normal(0,10) β j ∼ Normal(0,10) β j ∼ Normal(0,1) β j ∼ Normal(0,1) 

1 2 3 4 5 150 200 250 300 polynomial terms prediction error in out β j ∼ Normal(0,10) β j ∼ Normal(0,10) β j ∼ Normal(0,1) β j ∼ Normal(0,1) β j ∼ Normal(0,0.5) β j ∼ Normal(0,0.5) 

in out 1 2 3 4 5 150 200 250 300 350 400 polynomial terms prediction error β j ∼ Normal(0,0.1) 

Regularizing priors How to choose width of prior? For causal inference, use science For pure prediction, can tune the prior using cross-validation Many tasks are a mix of inference and prediction No need to be perfect, just better 

PAUSE 

Prediction penalty 1 2 3 4 5 150 200 250 300 polynomial terms prediction error in out 

Prediction penalty 1 2 3 4 5 150 200 250 300 polynomial terms prediction error 1 2 3 4 5 0 50 100 150 200 polynomial terms out-of-sample penalty in out 

Penalty prediction For N points, cross-validation requires tting N models What if you could estimate the penalty from a single model t? Good news! You can: Importance sampling (PSIS) Information criteria (WAIC) 1 2 3 4 5 0 50 100 150 200 polynomial terms out-of-sample penalty 

1 2 3 4 5 150 200 250 300 polynomial terms prediction error 1 2 3 4 5 30 35 40 45 50 55 60 polynomial terms log pointwise predictive density WAIC PSIS lppd leave-one-out cross-validation 

Over t Under t WAIC,PSIS,CV measure over tting Regularization manages over tting None directly address causal inference Important for understanding statistical inference 

Model Mis-selection Do not use predictive criteria (WAIC, PSIS, CV) to choose a causal estimate Predictive criteria actually prefer confounds & colliders Example: Plant growth experiment H0 H1 T F 

H 1 ∼ Normal(μ i , σ) μ i = H 0 × p i p i = α + β T T i + β F F i H 1 ∼ Normal(μ i , σ) μ i = H 0 × p i p i = α + β T T i Wrong adjustment set for total causal e ect of treatment (blocks mediating path) Correct adjustment set for total causal e ect of treatment H0 H1 T F 

H 1 ∼ Normal(μ i , σ) μ i = H 0 × p i p i = α + β T T i + β F F i H 1 ∼ Normal(μ i , σ) μ i = H 0 × p i p i = α + β T T i -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0 4 8 12 effect of treatment (posterior) Density correct biased 

H 1 ∼ Normal(μ i , σ) μ i = H 0 × p i p i = α + β T T i + β F F i H 1 ∼ Normal(μ i , σ) μ i = H 0 × p i p i = α + β T T i m6.8 m6.7 350 360 370 380 390 400 410 deviance PSIS H1 ~ H0 + T + F H1 ~ H0 + T Wrong model wins at prediction 

m6.8 m6.7 350 360 370 380 390 400 410 deviance PSIS H1 ~ H0 + T + F H1 ~ H0 + T Wrong model wins at prediction Score in sample Score out of sample Standard error of score PSIS contrast and standard error 

1.2 1.4 1.6 1.8 2.0 growth no fungus yo fungus treatment control H0 H1 T F Why does the wrong model win at prediction? 

1.2 1.4 1.6 1.8 2.0 growth no fungus yo fungus 1.2 1.4 1.6 1.8 2.0 growth control treatment treatment control fungus no fungus H0 H1 T F Why does the wrong model win at prediction? Fungus is in fact a better predictor than treatment 

Model Mis-selection Do not use predictive criteria (WAIC, PSIS, CV) to choose a causal estimate However, many analyses are mixes of inferential and predictive chores Still need help nding good functional descriptions while avoiding over tting H0 H1 T F 

Outliers & Robust Regression Some points are more in uential than others “Outliers”: Observations in the tails of predictive distribution Outliers indicate predictions are possibly overcon dent, unreliable e model doesn’t expect enough variation -4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4 

Outliers & Robust Regression Dropping outliers is bad: Just ignores the problem; predictions are still bad! It’s the model that’s wrong, not the data First, quantify in uence of each point Second, use a mixture model (robust regression) -4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4 

Outliers & Robust Regression Divorce rate example Maine and Idaho both unusual Maine: high divorce for trend Idaho: low divorce for trend -2 -1 0 1 2 3 -2 -1 0 1 2 Age at marriage (std) Divorce rate (std) Idaho Maine 

Outliers & Robust Regression Quantify in uence: PSIS k statistic WAIC penalty term (“e ective number of parameters”) 0.0 0.5 1.0 0.0 0.5 1.0 1.5 2.0 PSIS Pareto k WAIC penalty Idaho Maine 

Outliers & Robust Regression 0.0 0.5 1.0 0.0 0.5 1.0 1.5 2.0 PSIS Pareto k WAIC penalty Idaho Maine -2 -1 0 1 2 3 -2 -1 0 1 2 Age at marriage (std) Divorce rate (std) Idaho Maine 

Mixing Gaussians -6 -4 -2 0 2 4 6 0.0 0.4 0.8 value density 

Mixing Gaussians -6 -4 -2 0 2 4 6 0.0 0.2 0.4 value density Student-t Gaussian 

m5.3 <- quap( alist( D ~ dnorm( mu , sigma ) , mu <- a + bM*M + bA*A , a ~ dnorm( 0 , 0.2 ) , bM ~ dnorm( 0 , 0.5 ) , bA ~ dnorm( 0 , 0.5 ) , sigma ~ dexp( 1 ) ) , data = dat ) m5.3t <- quap( alist( D ~ dstudent( 2 , mu , sigma ) , mu <- a + bM*M + bA*A , a ~ dnorm( 0 , 0.2 ) , bM ~ dnorm( 0 , 0.5 ) , bA ~ dnorm( 0 , 0.5 ) , sigma ~ dexp( 1 ) ) , data = dat ) 

m5.3 <- quap( alist( D ~ dnorm( mu , sigma ) , mu <- a + bM*M + bA*A , a ~ dnorm( 0 , 0.2 ) , bM ~ dnorm( 0 , 0.5 ) , bA ~ dnorm( 0 , 0.5 ) , sigma ~ dexp( 1 ) ) , data = dat ) m5.3t <- quap( alist( D ~ dstudent( 2 , mu , sigma ) , mu <- a + bM*M + bA*A , a ~ dnorm( 0 , 0.2 ) , bM ~ dnorm( 0 , 0.5 ) , bA ~ dnorm( 0 , 0.5 ) , sigma ~ dexp( 1 ) ) , data = dat ) -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 bA (effect of age of marriage) Density Student-t model Gaussian model 

Robust Regressions Unobserved heterogeneity => mixture of Gaussians ick tails means model is less surprised by extreme values Usually impossible to estimate distribution of extreme values Student-t regression as default? -2 -1 0 1 2 3 -2 -1 0 1 2 Age at marriage (std) Divorce rate (std) Idaho Maine 

https://www.vox.com/2015/5/21/8635369/pinker-taleb 

Problems of Prediction What is the next observation from the same process? (prediction) Possible to make very good predictions without knowing causes Optimizing prediction does not reliably reveal causes Powerful tools (PSIS, regularization) for measuring and managing accuracy 1 2 3 4 5 150 200 250 300 polynomial terms prediction error 

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 Over tting / MCMC Chapters 7, 8, 9 Week 5 Generalized Linear Models Chapters 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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