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Statistical Rethinking 2. e Garden of Forking Data 2023

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What proportion of the surface is covered with water?

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How should we use the sample? How to produce a summary? How to represent uncertainty?

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Work ow (1) De ne generative model of the sample (2) De ne a speci c estimand (3) Design a statistical way to produce estimate (4) Test (3) using (1) (5) Analyze sample, summarize

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Generative model of the globe Begin conceptually: How do the variables in uence one another? OVNCFS PG HMPCF UPTTFT / ćJT JT DIPTFO CZ UIF FYQFSJNFOUFS OVNCFS PG XBUFS QPJOUT PCTFSWFE 8 OVNCFS PG MBOE QPJOUT PCTFSWFE - EJBHSBN UIBU TIPXT UIFTF GPVS WBSJBCMFT BOE DPOOFDUT TPNF PG UIFN DBVTBM JOĘVFODF -FUT TUBSU XJUI B CMBOL EJBHSBN BOE BEE UIF BSSP Q / 8 - BEE TPNF BSSPXT "SSPXT JO UIFTF EJBHSBNT JOEJDBUF DBVTBM JOĘ proportion of water number of tosses water observations land observations

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BEE TPNF BSSPXT "SSPXT JO UIFTF EJBHSBNT JOEJDBUF DBVTBM JO BCPVU XIBU iDBVTBM JOĘVFODFw NFBOT IFSF JT UP JNBHJOF DIBOHJOH IJDI PUIFS WBSJBCMFT BMTP DIBOHF BT B DPOTFRVFODF 'PS FYBNQMF PG HMPCF UPTTFT / UIFO CPUI 8 BOE - NJHIU DIBOHF #VU Q XPV BOE - CVU OPU Q 8F ESBX UIBU MJLF UIJT Q / 8 - Generative model of the globe Begin conceptually: How do the variables in uence one another? N in uences W and L in uence in uence

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 4."-- 803-%4 "/% -"3(& 803-%4 TU DIBOHJOH 8 BOE - EJSFDUMZ‰GPS FYBNQMF CZ NBOJQVMBUJOH UIF PO Q PS / #VU DIBOHJOH Q CZ GPS FYBNQMF FSBTJOH B SBOEPN DPO JOĘVFODF 8 BOE - BU MFBTU PO BWFSBHF 4P XF OFFE UXP NPSF BSSP Q / 8 - S DBVTBM EJBHSBN PG UIF HMPCF UPTTJOH FYQFSJNFOU ćFSF BSF TPN Generative model of the globe Begin conceptually: How do the variables in uence one another?

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Generative model of the globe Generative assumptions: What do the arrows mean exactly? W,L = f(p, N)  4."-- 803-%4 "/% -"3(& 803-%4 TU DIBOHJOH 8 BOE - EJSFDUMZ‰GPS FYBNQMF CZ NBOJQVMBUJOH UIF PO Q PS / #VU DIBOHJOH Q CZ GPS FYBNQMF FSBTJOH B SBOEPN DPO JOĘVFODF 8 BOE - BU MFBTU PO BWFSBHF 4P XF OFFE UXP NPSF BSSP Q / 8 - S DBVTBM EJBHSBN PG UIF HMPCF UPTTJOH FYQFSJNFOU ćFSF BSF TPN

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Work ow (1) De ne generative model of the sample (2) De ne a speci c estimand (3) Design a statistical way to produce estimate (4) Test (3) using (1) (5) Analyze sample, summarize

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Bayesian data analysis For each possible explanation of the sample, Count all the ways the sample could happen. Explanations with more ways to produce the sample are more plausible.

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e Garden of Forking Data El jardín de los datos que se bifurcan

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For each possible proportion of water on the globe, Count all the ways the sample of tosses could happen. Proportions with more ways to produce the sample are more plausible.

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A Four-sided Globe 1 1 2 3 1 4 covered 25% by water

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Garden of Forking Data Observe: (1) (2) (3) (4) (5) Possible d4 globes:

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Garden of Forking Data Observe: (1) (2) (3) (4) (5) Possible d4 globes: 25%

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First Possibility Figure 2.2

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Second Possibility Figure 2.2

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ird Possibility Figure 2.2

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Figure 2.2 First Observation

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Figure 2.2 Second Observation

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Figure 2.2 ird Observation

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Figure 2.2 3 Ways to see for 25% water

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Garden of Forking Data (1) (2) (3) (4) (5) Possible globes: Ways to produce ? 3 ? ? ?

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Garden of Forking Data (1) (2) (3) (4) (5) Possible globes: Ways to produce 0 3 ? ? ?

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Garden of Forking Data (1) (2) (3) (4) (5) Possible globes: Ways to produce 0 3 ? ? 0

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(3)

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(4)

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Garden of Forking Data (1) (2) (3) (4) (5) Possible globes: Ways to produce 0 3 8 9 0

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'ĶĴłĿIJ ƊƋ "ęFS FMJNJOBUJOH QBUIT JODPOTJTUFOU XJUI UIF TFRVFODF 8-8 POMZ  PG UIF  QBUIT SFNBJO VQ IPX NBOZ TFRVFODFT QBUIT UISPVHI UIF HBSEFO PG GPSLJOH EBUB DPVME QPUF UIF ĕSTU UISFF PCTFSWFE TBNQMFT  1PTTJCJMJUZ 8BZT UP QSPEVDF < >   ×  ×  =  < >   ×  ×  =  < >   ×  ×  =  < >   ×  ×  =  < >   ×  ×  =  IBU UIF OVNCFS PG XBZT UP QSPEVDF UIF EBUB GPS FBDI QPTTJCJJMUZ DBO CF DPN Counts to plausibility Unglamorous basis of applied probability: ings that can happen more ways are more plausible.

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'ĶĴłĿIJ ƊƋ "ęFS FMJNJOBUJOH QBUIT JODPOTJTUFOU XJUI UIF TFRVFODF 8-8 POMZ  PG UIF  QBUIT SFNBJO VQ IPX NBOZ TFRVFODFT QBUIT UISPVHI UIF HBSEFO PG GPSLJOH EBUB DPVME QPUF UIF ĕSTU UISFF PCTFSWFE TBNQMFT  1PTTJCJMJUZ 8BZT UP QSPEVDF < >   ×  ×  =  < >   ×  ×  =  < >   ×  ×  =  < >   ×  ×  =  < >   ×  ×  =  IBU UIF OVNCFS PG XBZT UP QSPEVDF UIF EBUB GPS FBDI QPTTJCJJMUZ DBO CF DPN Counts to plausibility Unglamorous basis of applied probability: ings that can happen more ways are more plausible.

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'ĶĴłĿIJ ƊƋ "ęFS FMJNJOBUJOH QBUIT JODPOTJTUFOU XJUI UIF TFRVFODF 8-8 POMZ  PG UIF  QBUIT SFNBJO VQ IPX NBOZ TFRVFODFT QBUIT UISPVHI UIF HBSEFO PG GPSLJOH EBUB DPVME QPUF UIF ĕSTU UISFF PCTFSWFE TBNQMFT  1PTTJCJMJUZ 8BZT UP QSPEVDF < >   ×  ×  =  < >   ×  ×  =  < >   ×  ×  =  < >   ×  ×  =  < >   ×  ×  =  IBU UIF OVNCFS PG XBZT UP QSPEVDF UIF EBUB GPS FBDI QPTTJCJJMUZ DBO CF DPN Counts to plausibility Unglamorous basis of applied probability: ings that can happen more ways are more plausible.

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'ĶĴłĿIJ ƊƋ "ęFS FMJNJOBUJOH QBUIT JODPOTJTUFOU XJUI UIF TFRVFODF 8-8 POMZ  PG UIF  QBUIT SFNBJO VQ IPX NBOZ TFRVFODFT QBUIT UISPVHI UIF HBSEFO PG GPSLJOH EBUB DPVME QPUF UIF ĕSTU UISFF PCTFSWFE TBNQMFT  1PTTJCJMJUZ 8BZT UP QSPEVDF < >   ×  ×  =  < >   ×  ×  =  < >   ×  ×  =  < >   ×  ×  =  < >   ×  ×  =  IBU UIF OVNCFS PG XBZT UP QSPEVDF UIF EBUB GPS FBDI QPTTJCJJMUZ DBO CF DPN Counts to plausibility Unglamorous basis of applied probability: ings that can happen more ways are more plausible.

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'ĶĴłĿIJ ƊƋ "ęFS FMJNJOBUJOH QBUIT JODPOTJTUFOU XJUI UIF TFRVFODF 8-8 POMZ  PG UIF  QBUIT SFNBJO VQ IPX NBOZ TFRVFODFT QBUIT UISPVHI UIF HBSEFO PG GPSLJOH EBUB DPVME QPUF UIF ĕSTU UISFF PCTFSWFE TBNQMFT  1PTTJCJMJUZ 8BZT UP QSPEVDF < >   ×  ×  =  < >   ×  ×  =  < >   ×  ×  =  < >   ×  ×  =  < >   ×  ×  =  IBU UIF OVNCFS PG XBZT UP QSPEVDF UIF EBUB GPS FBDI QPTTJCJJMUZ DBO CF DPN Counts to plausibility Unglamorous basis of applied probability: ings that can happen more ways are more plausible.

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TFF XIBU IBQQFOT )FSFT UIF TBNQMF BHBJO BT B SFNJOEFS 8 - 8 8 8 - 8 - 8 ćF GPVSUI PCTFSWBUJPO JT 8  5P VQEBUF PVS QSFWJPVT DPVOUT GPS FBDI QPTTJCJMJUZ XF KVTU OFFE UP NVMUJQMZ CZ UIF BQQSPQSJBUF OVNCFS PG XBZT UP TFF UIJT TJOHMF 8 'PS  UIBUT  'PS  UIBUT  'PS  UIBUT  6QEBUJOH PVS UBCMF 1PTTJCJMJUZ 8BZT UP QSPEVDF 8BZT UP QSPEVDF 8BZT UP QSPEVDF < >  ×  ×  =    ×  =  < >  ×  ×  =    ×  =  < >  ×  ×  =    ×  =  < >  ×  ×  =    ×  =  < >  ×  ×  =    ×  =  Updating Another draw from the bag: 4

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TFF XIBU IBQQFOT )FSFT UIF TBNQMF BHBJO BT B SFNJOEFS 8 - 8 8 8 - 8 - 8 ćF GPVSUI PCTFSWBUJPO JT 8  5P VQEBUF PVS QSFWJPVT DPVOUT GPS FBDI QPTTJCJMJUZ XF KVTU OFFE UP NVMUJQMZ CZ UIF BQQSPQSJBUF OVNCFS PG XBZT UP TFF UIJT TJOHMF 8 'PS  UIBUT  'PS  UIBUT  'PS  UIBUT  6QEBUJOH PVS UBCMF 1PTTJCJMJUZ 8BZT UP QSPEVDF 8BZT UP QSPEVDF 8BZT UP QSPEVDF < >  ×  ×  =    ×  =  < >  ×  ×  =    ×  =  < >  ×  ×  =    ×  =  < >  ×  ×  =    ×  =  < >  ×  ×  =    ×  =  Updating Another draw from the bag: 4

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TFF XIBU IBQQFOT )FSFT UIF TBNQMF BHBJO BT B SFNJOEFS 8 - 8 8 8 - 8 - 8 ćF GPVSUI PCTFSWBUJPO JT 8  5P VQEBUF PVS QSFWJPVT DPVOUT GPS FBDI QPTTJCJMJUZ XF KVTU OFFE UP NVMUJQMZ CZ UIF BQQSPQSJBUF OVNCFS PG XBZT UP TFF UIJT TJOHMF 8 'PS  UIBUT  'PS  UIBUT  'PS  UIBUT  6QEBUJOH PVS UBCMF 1PTTJCJMJUZ 8BZT UP QSPEVDF 8BZT UP QSPEVDF 8BZT UP QSPEVDF < >  ×  ×  =    ×  =  < >  ×  ×  =    ×  =  < >  ×  ×  =    ×  =  < >  ×  ×  =    ×  =  < >  ×  ×  =    ×  =  Updating Another draw from the bag: 4

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TFF XIBU IBQQFOT )FSFT UIF TBNQMF BHBJO BT B SFNJOEFS 8 - 8 8 8 - 8 - 8 ćF GPVSUI PCTFSWBUJPO JT 8  5P VQEBUF PVS QSFWJPVT DPVOUT GPS FBDI QPTTJCJMJUZ XF KVTU OFFE UP NVMUJQMZ CZ UIF BQQSPQSJBUF OVNCFS PG XBZT UP TFF UIJT TJOHMF 8 'PS  UIBUT  'PS  UIBUT  'PS  UIBUT  6QEBUJOH PVS UBCMF 1PTTJCJMJUZ 8BZT UP QSPEVDF 8BZT UP QSPEVDF 8BZT UP QSPEVDF < >  ×  ×  =    ×  =  < >  ×  ×  =    ×  =  < >  ×  ×  =    ×  =  < >  ×  ×  =    ×  =  < >  ×  ×  =    ×  =  Updating Another draw from the bag: 4

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TFF XIBU IBQQFOT )FSFT UIF TBNQMF BHBJO BT B SFNJOEFS 8 - 8 8 8 - 8 - 8 ćF GPVSUI PCTFSWBUJPO JT 8  5P VQEBUF PVS QSFWJPVT DPVOUT GPS FBDI QPTTJCJMJUZ XF KVTU OFFE UP NVMUJQMZ CZ UIF BQQSPQSJBUF OVNCFS PG XBZT UP TFF UIJT TJOHMF 8 'PS  UIBUT  'PS  UIBUT  'PS  UIBUT  6QEBUJOH PVS UBCMF 1PTTJCJMJUZ 8BZT UP QSPEVDF 8BZT UP QSPEVDF 8BZT UP QSPEVDF < >  ×  ×  =    ×  =  < >  ×  ×  =    ×  =  < >  ×  ×  =    ×  =  < >  ×  ×  =    ×  =  < >  ×  ×  =    ×  =  Updating Another draw from the bag: 4

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TFF XIBU IBQQFOT )FSFT UIF TBNQMF BHBJO BT B SFNJOEFS 8 - 8 8 8 - 8 - 8 ćF GPVSUI PCTFSWBUJPO JT 8  5P VQEBUF PVS QSFWJPVT DPVOUT GPS FBDI QPTTJCJMJUZ XF KVTU OFFE UP NVMUJQMZ CZ UIF BQQSPQSJBUF OVNCFS PG XBZT UP TFF UIJT TJOHMF 8 'PS  UIBUT  'PS  UIBUT  'PS  UIBUT  6QEBUJOH PVS UBCMF 1PTTJCJMJUZ 8BZT UP QSPEVDF 8BZT UP QSPEVDF 8BZT UP QSPEVDF < >  ×  ×  =    ×  =  < >  ×  ×  =    ×  =  < >  ×  ×  =    ×  =  < >  ×  ×  =    ×  =  < >  ×  ×  =    ×  =  Updating Another draw from the bag: 4

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TFF XIBU IBQQFOT )FSFT UIF TBNQMF BHBJO BT B SFNJOEFS 8 - 8 8 8 - 8 - 8 ćF GPVSUI PCTFSWBUJPO JT 8  5P VQEBUF PVS QSFWJPVT DPVOUT GPS FBDI QPTTJCJMJUZ XF KVTU OFFE UP NVMUJQMZ CZ UIF BQQSPQSJBUF OVNCFS PG XBZT UP TFF UIJT TJOHMF 8 'PS  UIBUT  'PS  UIBUT  'PS  UIBUT  6QEBUJOH PVS UBCMF 1PTTJCJMJUZ 8BZT UP QSPEVDF 8BZT UP QSPEVDF 8BZT UP QSPEVDF < >  ×  ×  =    ×  =  < >  ×  ×  =    ×  =  < >  ×  ×  =    ×  =  < >  ×  ×  =    ×  =  < >  ×  ×  =    ×  =  Updating Another draw from the bag:

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e whole sample 8F DBO LFFQ BQQMZJOH UIJT SVMF BHBJO BOE BHBJO UP VQEBUF GPS FBDI OFX PCTFSWBUJPO 'PS BMM OJOF PCTFSWBUJPOT UIF DPNQMFUF UBCMF JT CFMPX 'PS FBDI QPTTJCJMJUZ UIF UPUBM DPVOU JT KVTU UIF QSPEVDU PG UIF OVNCFS PG XBZT UP TFF 8 UP UIF QPXFS PG UIF OVNCFS PG UJNFT 8 XBT TBNQMFE BOE UIF OVNCFS PG XBZT UP TFF - UP UIF QPXFS PG UIF OVNCFS PG UJNFT - XBT TBNQMFE ćBU JT BO BXGVM UIJOH UP XSJUF EPXO CVU DPODFQUVBMMZ XF KVTU NVMUJQMZ UIF DPVOU FBDI UJNF CZ UIF OVNCFS PG XBZT UP TFF UIF NPTU SFDFOU PCTFSWBUJPO 1PTTJCJMJUZ 0CTFSWBUJPOT < >          =  ×  < >          =  ×  < >          =  ×  < >          =  ×  < >          =  × 

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e whole sample 8F DBO LFFQ BQQMZJOH UIJT SVMF BHBJO BOE BHBJO UP VQEBUF GPS FBDI OFX PCTFSWBUJPO 'PS BMM OJOF PCTFSWBUJPOT UIF DPNQMFUF UBCMF JT CFMPX 'PS FBDI QPTTJCJMJUZ UIF UPUBM DPVOU JT KVTU UIF QSPEVDU PG UIF OVNCFS PG XBZT UP TFF 8 UP UIF QPXFS PG UIF OVNCFS PG UJNFT 8 XBT TBNQMFE BOE UIF OVNCFS PG XBZT UP TFF - UP UIF QPXFS PG UIF OVNCFS PG UJNFT - XBT TBNQMFE ćBU JT BO BXGVM UIJOH UP XSJUF EPXO CVU DPODFQUVBMMZ XF KVTU NVMUJQMZ UIF DPVOU FBDI UJNF CZ UIF OVNCFS PG XBZT UP TFF UIF NPTU SFDFOU PCTFSWBUJPO 1PTTJCJMJUZ 0CTFSWBUJPOT < >          =  ×  < >          =  ×  < >          =  ×  < >          =  ×  < >          =  × 

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e whole sample 8F DBO LFFQ BQQMZJOH UIJT SVMF BHBJO BOE BHBJO UP VQEBUF GPS FBDI OFX PCTFSWBUJPO 'PS BMM OJOF PCTFSWBUJPOT UIF DPNQMFUF UBCMF JT CFMPX 'PS FBDI QPTTJCJMJUZ UIF UPUBM DPVOU JT KVTU UIF QSPEVDU PG UIF OVNCFS PG XBZT UP TFF 8 UP UIF QPXFS PG UIF OVNCFS PG UJNFT 8 XBT TBNQMFE BOE UIF OVNCFS PG XBZT UP TFF - UP UIF QPXFS PG UIF OVNCFS PG UJNFT - XBT TBNQMFE ćBU JT BO BXGVM UIJOH UP XSJUF EPXO CVU DPODFQUVBMMZ XF KVTU NVMUJQMZ UIF DPVOU FBDI UJNF CZ UIF OVNCFS PG XBZT UP TFF UIF NPTU SFDFOU PCTFSWBUJPO 1PTTJCJMJUZ 0CTFSWBUJPOT < >          =  ×  < >          =  ×  < >          =  ×  < >          =  ×  < >          =  × 

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e whole sample 8F DBO LFFQ BQQMZJOH UIJT SVMF BHBJO BOE BHBJO UP VQEBUF GPS FBDI OFX PCTFSWBUJPO 'PS BMM OJOF PCTFSWBUJPOT UIF DPNQMFUF UBCMF JT CFMPX 'PS FBDI QPTTJCJMJUZ UIF UPUBM DPVOU JT KVTU UIF QSPEVDU PG UIF OVNCFS PG XBZT UP TFF 8 UP UIF QPXFS PG UIF OVNCFS PG UJNFT 8 XBT TBNQMFE BOE UIF OVNCFS PG XBZT UP TFF - UP UIF QPXFS PG UIF OVNCFS PG UJNFT - XBT TBNQMFE ćBU JT BO BXGVM UIJOH UP XSJUF EPXO CVU DPODFQUVBMMZ XF KVTU NVMUJQMZ UIF DPVOU FBDI UJNF CZ UIF OVNCFS PG XBZT UP TFF UIF NPTU SFDFOU PCTFSWBUJPO 1PTTJCJMJUZ 0CTFSWBUJPOT < >          =  ×  < >          =  ×  < >          =  ×  < >          =  ×  < >          =  × 

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e whole sample 8F DBO LFFQ BQQMZJOH UIJT SVMF BHBJO BOE BHBJO UP VQEBUF GPS FBDI OFX PCTFSWBUJPO 'PS BMM OJOF PCTFSWBUJPOT UIF DPNQMFUF UBCMF JT CFMPX 'PS FBDI QPTTJCJMJUZ UIF UPUBM DPVOU JT KVTU UIF QSPEVDU PG UIF OVNCFS PG XBZT UP TFF 8 UP UIF QPXFS PG UIF OVNCFS PG UJNFT 8 XBT TBNQMFE BOE UIF OVNCFS PG XBZT UP TFF - UP UIF QPXFS PG UIF OVNCFS PG UJNFT - XBT TBNQMFE ćBU JT BO BXGVM UIJOH UP XSJUF EPXO CVU DPODFQUVBMMZ XF KVTU NVMUJQMZ UIF DPVOU FBDI UJNF CZ UIF OVNCFS PG XBZT UP TFF UIF NPTU SFDFOU PCTFSWBUJPO 1PTTJCJMJUZ 0CTFSWBUJPOT < >          =  ×  < >          =  ×  < >          =  ×  < >          =  ×  < >          =  × 

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e whole sample 8F DBO LFFQ BQQMZJOH UIJT SVMF BHBJO BOE BHBJO UP VQEBUF GPS FBDI OFX PCTFSWBUJPO 'PS BMM OJOF PCTFSWBUJPOT UIF DPNQMFUF UBCMF JT CFMPX 'PS FBDI QPTTJCJMJUZ UIF UPUBM DPVOU JT KVTU UIF QSPEVDU PG UIF OVNCFS PG XBZT UP TFF 8 UP UIF QPXFS PG UIF OVNCFS PG UJNFT 8 XBT TBNQMFE BOE UIF OVNCFS PG XBZT UP TFF - UP UIF QPXFS PG UIF OVNCFS PG UJNFT - XBT TBNQMFE ćBU JT BO BXGVM UIJOH UP XSJUF EPXO CVU DPODFQUVBMMZ XF KVTU NVMUJQMZ UIF DPVOU FBDI UJNF CZ UIF OVNCFS PG XBZT UP TFF UIF NPTU SFDFOU PCTFSWBUJPO 1PTTJCJMJUZ 0CTFSWBUJPOT < >          =  ×  < >          =  ×  < >          =  ×  < >          =  ×  < >          =  ×  Ways for p to produce W,L = (4p)W × (4–4p)L

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Probability Probability: Non-negative values that sum to one Suppose W=20, L=10. en p=0.5 has ways to produce sample. Better to convert to probability. 2W × 2L = 1,073,741,824

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Probability ĻıĮĿıĶŇIJ UIF DPVOUT ćJT KVTU NFBOT UP EJWJEF FBDI DPVOU CZ UIFJS UPUBM FX WBMVFT TVN UP POF 8F DBO TBGFMZ EP UIJT CFDBVTF EJWJEJOH UIF DPVOUT CFS UIFJS UPUBM EPFTOU EJTDBSE BOZ JOGPSNBUJPO *U DBO CF SFWFSTFE *U KVTU ST FBTJFS UP DPNQBSF J[JOH UIF DPVOUT JT XIFSF ĽĿļįĮįĶĹĶŁņ DPNFT JO 'PS UIF PSJHJOBM TBNQMF -888-8-8 TUBOEBSEJ[JOH HJWFT VT UIFTF QSPCBCJMJUJFT 1PTTJCMF 8BZT UP 1SPCBCJMJUZ PG QSPQPSUJPO QSPEVDF TBNQMF QSPQPSUJPO    .   .   .      EF UP DBMDVMBUF UIF XBZT BOE UIF QSPCBCJMJUJFT GSPN UIF TBNQMF ę   4."-- 803-% 0 0.25 0.5 0.75 1 proportion water probability 0.0 0.1 0.2 0.3 0.4 0.5

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Probability ĻıĮĿıĶŇIJ UIF DPVOUT ćJT KVTU NFBOT UP EJWJEF FBDI DPVOU CZ UIFJS UPUBM FX WBMVFT TVN UP POF 8F DBO TBGFMZ EP UIJT CFDBVTF EJWJEJOH UIF DPVOUT CFS UIFJS UPUBM EPFTOU EJTDBSE BOZ JOGPSNBUJPO *U DBO CF SFWFSTFE *U KVTU ST FBTJFS UP DPNQBSF J[JOH UIF DPVOUT JT XIFSF ĽĿļįĮįĶĹĶŁņ DPNFT JO 'PS UIF PSJHJOBM TBNQMF -888-8-8 TUBOEBSEJ[JOH HJWFT VT UIFTF QSPCBCJMJUJFT 1PTTJCMF 8BZT UP 1SPCBCJMJUZ PG QSPQPSUJPO QSPEVDF TBNQMF QSPQPSUJPO    .   .   .      EF UP DBMDVMBUF UIF XBZT BOE UIF QSPCBCJMJUJFT GSPN UIF TBNQMF ę   4."-- 803-% 0 0.25 0.5 0.75 1 proportion water probability 0.0 0.1 0.2 0.3 0.4 0.5 Posterior distribution

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ESB 0 0.25 0.5 0.75 1 proportion water 0.0 0. 3 DPEF  sample <- c("W","L","W","W","W","L","W","L","W") W <- sum(sample=="W") # number of W observed L <- sum(sample=="L") # number of L observed p <- c(0,0.25,0.5,0.75,1) # proportions W ways <- sapply( p , function(q) (q*4)^W * ((1-q)*4)^L ) prob <- ways/sum(ways) cbind( p , ways , prob ) p ways prob [1,] 0.00 0 0.00000000 [2,] 0.25 27 0.02129338 [3,] 0.50 512 0.40378549 [4,] 0.75 729 0.57492114 [5,] 1.00 0 0.00000000 ćFTF QSPCBCJMJUJFT BSF SFMBUJWF QMBVTJCJMJUJFT GPS UIF EJČFSFOU QSPQPSUJPOT PG XBUFS ćFZ BSF DPNQVUFE BęFS VQEBUJOH GPS BMM UIF PCTFSWBUJPOT BOE UIF TFU PG UIFTF QSPCBCJMJUJFT JT VTVBMMZ Probability ę   4."-- 803-% 0 0.25 0.5 0.75 1 proportion water probability 0.0 0.1 0.2 0.3 0.4 0.5

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Work ow (1) De ne generative model of the sample (2) De ne a speci c estimand (3) Design a statistical way to produce estimate (4) Test (3) using (1) (5) Analyze sample, summarize

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Test Before You Est(imate) (1) Code a generative simulation (2) Code an estimator (3) Test (2) with (1) Extremely powerful, fun

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Generative simulation F E 0VS BQQSPBDI JO UIJT CPPL XJMM CF UP XSJUF DPEF UIBU TIBEPXT FBDI TUFQ FTUJNBOE → FTUJNBUPS → FTUJNBUF "OE XF DBO UFTU FBDI TUFQ BT XF HP *O GBDU BT UIF TUBUJTUJDBM NPEFMT HFU NPSF DPNQMFY XFMM IBWF B MBEEFS PG UFTUT UP NBLF UIF DPOTUSVDUJPO PG UIF NPEFM FBTJFS BOE TBGFS  4ZOUIFUJD TBNQMF ćF ĕSTU UIJOH UP EP JT UP TJNVMBUF B TBNQMF GSPN B HFOFSBUJWF NPEFM XIJDI JT VTFE UP EFĕOF UIF FTUJNBOE ćJT QSPEVDFT POF PS NBOZ ŀņĻŁĵIJŁĶİ TBN QMFT ćFO XF DBO GFFE UIF TZOUIFUJD TBNQMFT JOUP UIF TUBUJTUJDBM QSPDFEVSF BOE TFF UIBU JU CFIBWFT BT XF IPQF 'PS UIF HMPCF UPTTJOH QSPCMFN XF XBOU UP TJNVMBUF TBNQMJOH GSPN UIF HMPCF * BN HPJOH UP XSJUF B GVODUJPO UIBU TJNVMBUFT TBNQMJOH GSPN UIF HMPCF *G ZPV BSF OPU GBNJMJBS XJUI GVODUJPOT ZPV DBO UIJOL PG UIFN BT OBNFT GPS QJFDFT PG DPEF UIBU ZPV XBOU UP SFVTF *O BEEJUJPO UP NBLJOH JU FBTJFS UP SFQFBU UIF DPEF B GVODUJPO DBO BMTP NBLF UFTUJOH FBTJFS )FSFT B WFSZ TJNQMF GVODUJPO UIBU TJNVMBUFT TBNQMJOH OJOF UJNFT GSPN B HMPCF XJUI B  XBUFS 3 DPEF  # function to toss a globe covered p by water N times sim_globe <- function( p=0.7 , N=9 ) { sample(c("W","L"),size=N,prob=c(p,1-p),replace=TRUE) } /PUIJOH IBQQFOT VOUJM XF DBMM UIF GVODUJPO CZ JUT OBNF #VU DIBOHJOH Q CZ GPS FYBNQMF FSBTJOH B SBOEPN DPOUJOFOU PO UI BOE - BU MFBTU PO BWFSBHF 4P XF OFFE UXP NPSF BSSPXT Q / 8 - SBN PG UIF HMPCF UPTTJOH FYQFSJNFOU ćFSF BSF TPNF HFOFSBM BOE OH UIFTF EJBHSBNT #VU XF EPOU OFFE UIFN SJHIU OPX 4P JOTUFBE NBLFT B MPU PG TUSPOH BTTVNQUJPOT CFDBVTF PG WBSJBCMFT BOE BSSPX NQMF UIF TBNQMF TJ[F / JT JOEFQFOEFOU PG Q BOE UIF SFTVMUT 8 BOE W,L = f(p, N)

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 BEEJUJPO UP NBLJOH JU FBTJFS UP SFQFBU UIF DPEF B GVODUJPO DBO BMTP NBLF UFTUJOH FBTJFS )FSFT B WFSZ TJNQMF GVODUJPO UIBU TJNVMBUFT TBNQMJOH OJOF UJNFT GSPN B HMPCF XJUI B  XBUFS 3 DPEF  # function to toss a globe covered p by water N times sim_globe <- function( p=0.7 , N=9 ) { sample(c("W","L"),size=N,prob=c(p,1-p),replace=TRUE) } /PUIJOH IBQQFOT VOUJM XF DBMM UIF GVODUJPO CZ JUT OBNF 3 DPEF  sim_globe() [1] "L" "W" "W" "W" "L" "L" "L" "W" "L" 3FQFBU DBMMJOH UIF GVODUJPO UP TFF UIBU JU TJNVMBUFT B EJČFSFOU TBNQMF FBDI UJNF "OE CZ OBNJOH UIF QSPQPSUJPO PG XBUFS p BOE OVNCFS PG UPTTFT N JO UIF GVODUJPO EFĕOJUJPO XF DBO FBTJMZ DIBOHF UIFTF WBMVFT XIFO XF DBMM UIF GVODUJPO Possible observations Number of tosses Probability of each possible observation

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 BEEJUJPO UP NBLJOH JU FBTJFS UP SFQFBU UIF DPEF B GVODUJPO DBO BMTP NBLF UFTUJOH FBTJFS )FSFT B WFSZ TJNQMF GVODUJPO UIBU TJNVMBUFT TBNQMJOH OJOF UJNFT GSPN B HMPCF XJUI B  XBUFS 3 DPEF  # function to toss a globe covered p by water N times sim_globe <- function( p=0.7 , N=9 ) { sample(c("W","L"),size=N,prob=c(p,1-p),replace=TRUE) } /PUIJOH IBQQFOT VOUJM XF DBMM UIF GVODUJPO CZ JUT OBNF 3 DPEF  sim_globe() [1] "L" "W" "W" "W" "L" "L" "L" "W" "L" 3FQFBU DBMMJOH UIF GVODUJPO UP TFF UIBU JU TJNVMBUFT B EJČFSFOU TBNQMF FBDI UJNF "OE CZ OBNJOH UIF QSPQPSUJPO PG XBUFS p BOE OVNCFS PG UPTTFT N JO UIF GVODUJPO EFĕOJUJPO XF DBO FBTJMZ DIBOHF UIFTF WBMVFT XIFO XF DBMM UIF GVODUJPO  BEEJUJPO UP NBLJOH JU FBTJFS UP SFQFBU UIF DPEF B GVODUJPO DBO BMTP NBLF UFTUJOH FBTJFS )FSFT B WFSZ TJNQMF GVODUJPO UIBU TJNVMBUFT TBNQMJOH OJOF UJNFT GSPN B HMPCF XJUI B  XBUFS 3 DPEF  # function to toss a globe covered p by water N times sim_globe <- function( p=0.7 , N=9 ) { sample(c("W","L"),size=N,prob=c(p,1-p),replace=TRUE) } /PUIJOH IBQQFOT VOUJM XF DBMM UIF GVODUJPO CZ JUT OBNF 3 DPEF  sim_globe() [1] "L" "W" "W" "W" "L" "L" "L" "W" "L" 3FQFBU DBMMJOH UIF GVODUJPO UP TFF UIBU JU TJNVMBUFT B EJČFSFOU TBNQMF FBDI UJNF "OE CZ OBNJOH UIF QSPQPSUJPO PG XBUFS p BOE OVNCFS PG UPTTFT N JO UIF GVODUJPO EFĕOJUJPO XF DBO FBTJMZ DIBOHF UIFTF WBMVFT XIFO XF DBMM UIF GVODUJPO

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/PUIJOH IBQQFOT VOUJM XF DBMM UIF GVODUJPO CZ JUT OBNF 3 DPEF  sim_globe() [1] "L" "W" "W" "W" "L" "L" "L" "W" "L" 3FQFBU DBMMJOH UIF GVODUJPO UP TFF UIBU JU TJNVMBUFT B EJČFSFOU TBNQMF FBDI UJNF "OE CZ OBNJOH UIF QSPQPSUJPO PG XBUFS p BOE OVNCFS PG UPTTFT N JO UIF GVODUJPO EFĕOJUJPO XF DBO FBTJMZ DIBOHF UIFTF WBMVFT XIFO XF DBMM UIF GVODUJPO replicate(sim_globe(p=0.5,N=9),n=10) [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [1,] "W" "L" "L" "W" "W" "L" "L" "W" "W" "L" [2,] "W" "L" "W" "L" "W" "L" "L" "W" "L" "L" [3,] "W" "L" "L" "L" "L" "W" "L" "W" "W" "W" [4,] "W" "W" "L" "W" "L" "W" "W" "W" "W" "W" [5,] "L" "W" "W" "W" "W" "W" "L" "W" "L" "L" [6,] "L" "W" "L" "L" "W" "L" "W" "W" "W" "W" [7,] "W" "W" "W" "L" "W" "W" "W" "L" "L" "L" [8,] "L" "W" "L" "L" "L" "W" "L" "W" "W" "W" [9,] "W" "L" "L" "W" "L" "W" "W" "W" "L" "L"

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/PUIJOH IBQQFOT VOUJM XF DBMM UIF GVODUJPO CZ JUT OBNF 3 DPEF  sim_globe() [1] "L" "W" "W" "W" "L" "L" "L" "W" "L" 3FQFBU DBMMJOH UIF GVODUJPO UP TFF UIBU JU TJNVMBUFT B EJČFSFOU TBNQMF FBDI UJNF "OE CZ OBNJOH UIF QSPQPSUJPO PG XBUFS p BOE OVNCFS PG UPTTFT N JO UIF GVODUJPO EFĕOJUJPO XF DBO FBTJMZ DIBOHF UIFTF WBMVFT XIFO XF DBMM UIF GVODUJPO   4."-- 803-%4 "/% -"3(& 803-%4 3 DPEF  sim_globe( p=1 , N=11 ) [1] "W" "W" "W" "W" "W" "W" "W" "W" "W" "W" "W" /PX XF IBWF B TBNQMF PG  UPTTFT GSPN B HMPCF DPWFSFE FOUJSFMZ JO XBUFS ćFZ TIPVME BMM CF 8 ćJT JT B UFTU PG PVS TJNVMBUJPO ćF ĕSTU UIJOH UP EP FBDI UJNF ZPV XSJUF B TZOUIFUJD EBUB TJNVMBUJPO JT UP UFTU JU GPS JOQVUT GPS XIJDI ZPV BMSFBEZ LOPX IPX JU TIPVME CFIBWF ćFTF JOQVUT XJMM VTVBMMZ CF FYUSFNF WBMVFT 4P HP BIFBE BOE USZ p=0 UPP :PV TIPVME POMZ HFU - -FUT USZ B USJDLJFS UFTU "T UIF TBNQMF TJ[F JODSFBTFT UIF QSPQPSUJPO PG 8 JO UIF TBNQMF TIPVME HFU DMPTF UP p 4P MFUT USZ 3 DPEF Test the simulation on extreme settings ę   4."-- 803-%4 "/% -"3(& 803-%4 3 DPEF  sim_globe( p=1 , N=11 ) [1] "W" "W" "W" "W" "W" "W" "W" "W" "W" "W" "W" /PX XF IBWF B TBNQMF PG  UPTTFT GSPN B HMPCF DPWFSFE FOUJSFMZ JO XBUFS ćFZ TIPVME BMM CF 8 ćJT JT B UFTU PG PVS TJNVMBUJPO ćF ĕSTU UIJOH UP EP FBDI UJNF ZPV XSJUF B TZOUIFUJD EBUB TJNVMBUJPO JT UP UFTU JU GPS JOQVUT GPS XIJDI ZPV BMSFBEZ LOPX IPX JU TIPVME CFIBWF ćFTF JOQVUT XJMM VTVBMMZ CF FYUSFNF WBMVFT 4P HP BIFBE BOE USZ p=0 UPP :PV TIPVME POMZ HFU - -FUT USZ B USJDLJFS UFTU "T UIF TBNQMF TJ[F JODSFBTFT UIF QSPQPSUJPO PG 8 JO UIF TBNQMF TIPVME HFU DMPTF UP p 4P MFUT USZ 3 DPEF  sum( sim_globe( p=0.5 , N=1e4 ) == "W" ) / 1e4 [1] 0.505 5SZ TPNF PUIFS WBMVFT GPS p UP NBLF TVSF UIF TJNVMBUJPO JT GVODUJPOJOH DPSSFDUMZ /PUJDF UIBU ZPV BSF BMNPTU OFWFS HPJOH UP HFU FYBDUMZ p CBDL ćJT JT OPSNBM 4BNQMFT BSF ĕOJUF BOE

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IF YOU TEST NOTHING YOU MISS EVERYTHING

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Code the estimator  26"-*5: "4463"/$&  # function to compute posterior distribution compute_posterior <- function( the_sample , poss=c(0,0.25,0.5,0.75,1) ) { W <- sum(the_sample=="W") # number of W observed L <- sum(the_sample=="L") # number of L observed ways <- sapply( poss , function(q) (q*4)^W * ((1-q)*4)^L ) post <- ways/sum(ways) bars <- sapply( post, function(q) make_bar(q) ) data.frame( poss , ways , post=round(post,3) , bars ) } 5P VTF UIJT GVODUJPO ZPV OFFE UP HJWF JU B TBNQMF "OE XF DBO KVTU FNCFE UIF QSFWJPVT TJNVMBUJPO GVODUJPO JOTJEF JU Ways for p to produce W,L = (4p)W × (4–4p)L

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Code the estimator  26"-*5: "4463"/$&  # function to compute posterior distribution compute_posterior <- function( the_sample , poss=c(0,0.25,0.5,0.75,1) ) { W <- sum(the_sample=="W") # number of W observed L <- sum(the_sample=="L") # number of L observed ways <- sapply( poss , function(q) (q*4)^W * ((1-q)*4)^L ) post <- ways/sum(ways) bars <- sapply( post, function(q) make_bar(q) ) data.frame( poss , ways , post=round(post,3) , bars ) } 5P VTF UIJT GVODUJPO ZPV OFFE UP HJWF JU B TBNQMF "OE XF DBO KVTU FNCFE UIF QSFWJPVT TJNVMBUJPO GVODUJPO JOTJEF JU Ways for p to produce W,L = (4p)W × (4–4p)L

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Code the estimator  26"-*5: "4463"/$&  # function to compute posterior distribution compute_posterior <- function( the_sample , poss=c(0,0.25,0.5,0.75,1) ) { W <- sum(the_sample=="W") # number of W observed L <- sum(the_sample=="L") # number of L observed ways <- sapply( poss , function(q) (q*4)^W * ((1-q)*4)^L ) post <- ways/sum(ways) bars <- sapply( post, function(q) make_bar(q) ) data.frame( poss , ways , post=round(post,3) , bars ) } 5P VTF UIJT GVODUJPO ZPV OFFE UP HJWF JU B TBNQMF "OE XF DBO KVTU FNCFE UIF QSFWJPVT TJNVMBUJPO GVODUJPO JOTJEF JU Ways for p to produce W,L = (4p)W × (4–4p)L

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Code the estimator  26"-*5: "4463"/$&  # function to compute posterior distribution compute_posterior <- function( the_sample , poss=c(0,0.25,0.5,0.75,1) ) { W <- sum(the_sample=="W") # number of W observed L <- sum(the_sample=="L") # number of L observed ways <- sapply( poss , function(q) (q*4)^W * ((1-q)*4)^L ) post <- ways/sum(ways) bars <- sapply( post, function(q) make_bar(q) ) data.frame( poss , ways , post=round(post,3) , bars ) } 5P VTF UIJT GVODUJPO ZPV OFFE UP HJWF JU B TBNQMF "OE XF DBO KVTU FNCFE UIF QSFWJPVT TJNVMBUJPO GVODUJPO JOTJEF JU Ways for p to produce W,L = (4p)W × (4–4p)L

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Code the estimator  26"-*5: "4463"/$&  # function to compute posterior distribution compute_posterior <- function( the_sample , poss=c(0,0.25,0.5,0.75,1) ) { W <- sum(the_sample=="W") # number of W observed L <- sum(the_sample=="L") # number of L observed ways <- sapply( poss , function(q) (q*4)^W * ((1-q)*4)^L ) post <- ways/sum(ways) bars <- sapply( post, function(q) make_bar(q) ) data.frame( poss , ways , post=round(post,3) , bars ) } 5P VTF UIJT GVODUJPO ZPV OFFE UP HJWF JU B TBNQMF "OE XF DBO KVTU FNCFE UIF QSFWJPVT TJNVMBUJPO GVODUJPO JOTJEF JU Ways for p to produce W,L = (4p)W × (4–4p)L

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ESB TJNVMBUJPO GVODUJPO JOTJEF JU 3 DPEF  compute_posterior( sim_globe() ) poss ways post bars 1 0.00 0 0.000 2 0.25 243 0.291 ###### 3 0.50 512 0.612 ############ 4 0.75 81 0.097 ## 5 1.00 0 0.000 3FQFBU UIJT GVODUJPO DBMM B GFX UJNFT UP TIPX UIBU BT UIF TBNQMF WBSJFT TP UPP EPFT UIF QPTUF SJPS EJTUSJCVUJPO )PX EP XF UFTU PVS FTUJNBUPS "HBJO UIF ĕSTU UIJOH UP USZ BSF TPNF FYUSFNF TBNQMFT XJUI LOPXO QSPQFSUJFT ćFO XF DBO USZ JODSFBTJOH UIF TBNQMF TJ[F BOE FOTVSJOH UIBU UIF QPTUFSJPS EJTUSJCVUJPO CFIBWFT DPSSFDUMZ ćF ĕSTU FYUSFNF UFTU JT B TBNQMF XJUI POMZ 8 3 DPEF  compute_posterior( rep("W",times=9) ) poss ways post bars 1 0.00 0 0.000 2 0.25 1 0.000 (1) Test the estimator where the answer is known (2) Explore di erent sampling designs (3) Develop intuition for sampling and estimation

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PAUSE

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More possibilities 4-sided globe [0 0.25 0.5 0.75 1]

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More possibilities 4-sided globe 10-sided globe [0 0.25 0.5 0.75 1] [0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1]

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More possibilities 4-sided globe 10-sided globe [0 0.25 0.5 0.75 1] [0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1] 20-sided globe [0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1]

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More possibilities Bę   4."-- 803-%4 "/% -"3(& 803-%4 0 0.25 0.5 0.75 1 proportion water probability 0.0 0.1 0.2 0.3 0.4 0.5 'ĶĴłĿIJ Ɗƍ ćF QPTUFSJPS QSPCBCJMJUZ EJTUSJ CVUJPO GPS UIF TBNQMF 8-888-8-8 GPS UIF QSPQPSUJPOT     BOE  EF  sample <- c("W","L","W","W","W","L","W","L","W") W <- sum(sample=="W") # number of W observed 5 possibilities

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More possibilities Bę   4."-- 803-%4 "/% -"3(& 803-%4 0 0.2 0.4 0.6 0.8 1 proportion water posterior probability 0.00 0.05 0.10 0.15 0.20 0.25 0.30 11 possibilities 0 0.1 0.25 0.4 0.55 0.7 0.85 1 proportion water posterior probability 0.00 0.05 0.10 0.15 0.20 0.25 0.30 21 possibilities 'ĶĴłĿIJ ƊƎ ćF QPTUFSJPS EJTUSJCVUJPO GPS UIF HMPCF TBNQMF DPNQVUFE XJUI JODSFBTJOH OVNCFST PG QPTTJCMF QSPQPSUJPOT PG XBUFS -Fę  QPTTJCJMJUJFT 3JHIU  QPTTJCJMJUJFT Bę   4."-- 803-%4 "/% -"3(& 803-%4 0 0.25 0.5 0.75 1 proportion water probability 0.0 0.1 0.2 0.3 0.4 0.5 'ĶĴłĿIJ Ɗƍ ćF QPTUFSJPS QSPCBCJMJUZ EJTUSJ CVUJPO GPS UIF TBNQMF 8-888-8-8 GPS UIF QSPQPSUJPOT     BOE  EF  sample <- c("W","L","W","W","W","L","W","L","W") W <- sum(sample=="W") # number of W observed 5 possibilities

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More possibilities Bę   4."-- 803-%4 "/% -"3(& 803-%4 0 0.2 0.4 0.6 0.8 1 proportion water posterior probability 0.00 0.05 0.10 0.15 0.20 0.25 0.30 11 possibilities 0 0.1 0.25 0.4 0.55 0.7 0.85 1 proportion water posterior probability 0.00 0.05 0.10 0.15 0.20 0.25 0.30 21 possibilities 'ĶĴłĿIJ ƊƎ ćF QPTUFSJPS EJTUSJCVUJPO GPS UIF HMPCF TBNQMF DPNQVUFE XJUI JODSFBTJOH OVNCFST PG QPTTJCMF QSPQPSUJPOT PG XBUFS -Fę  QPTTJCJMJUJFT 3JHIU  QPTTJCJMJUJFT Bę   4."-- 803-%4 "/% -"3(& 803-%4 0 0.25 0.5 0.75 1 proportion water probability 0.0 0.1 0.2 0.3 0.4 0.5 'ĶĴłĿIJ Ɗƍ ćF QPTUFSJPS QSPCBCJMJUZ EJTUSJ CVUJPO GPS UIF TBNQMF 8-888-8-8 GPS UIF QSPQPSUJPOT     BOE  EF  sample <- c("W","L","W","W","W","L","W","L","W") W <- sum(sample=="W") # number of W observed 5 possibilities

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In nite possibilities e globe is a polyhedron with an in nite number of sides e posterior probability of any “side” p is proportional to: Only trick is normalizing to probability. A er a little calculus:  #":&4*"/ 61%"5*/( " DPOUJOVPVT TPMVUJPO *O UIJT FYBNQMF JUT OPU IBSE UP EFSJWF UIF QPTUFSJPS EJTUSJ BO FYBDU DPOUJOVPVT EJTUSJCVUJPO GVODUJPO *U UVSOT PVU UIBU UIF QPTUFSJPS QSPCBCJ PTTJCMF QSPQPSUJPO PG XBUFS Q CFUXFFO [FSP BOE POF JT QSPQPSUJPOBM UP Q8 ( − Q)- 8 JT UIF OVNCFS PG XBUFS PCTFSWFE BOE - JT UIF OVNCFS PG MBOE PCTFSWFE 8I UIJT JT UIF FYBDU FYQSFTTJPO UIBU XF BMSFBEZ VTFE UP DBMDVMBUF UIF SFMBUJWF OVNCFS Z WBMVF Q DPVME QSPEVDF B TBNQMF XJUI 8 XBUFS BOE - MBOE *U JT B MPHJDBM JNQMJDBUJ BSEFO PG GPSLJOH QBUIT 8IFO XF VTFE UIJT FYQSFTTJPO PSJHJOBMMZ XF NVMUJQMJFE Q IF TVN PG BMM OVNFSBUPST GPS FWFSZ QPTTJCMF Q 'PS B ĕOJUF OVNCFS PG E TVN ; = Q Q8(−Q)- XIFSF UIF Q OPUBUJPO NFBOT UP FWBMVBUF UIFO BEE UIFN 'PS BO JOĕOJUF OVNCFS PG QPTTJCMF Q WBMVFT GSPN [FSP PVT QBSUOFS PG BO JOUFHSBM ; = Q8 ( − Q)-EQ iGBNPVTw ZPV NFBO iLOPXOw :PV DBO MPPL JU VQ POMJOF *O UIF DBTF PG  ; = 8! -! (8 + - + )! CBCJMJUZ PG BOZ TQFDJĕD Q JT FYBDUMZ = #FUB(8 + , - + ) = 8!-! (8 + - + )! Q8 ( − Q)- F TIBQF DPNFT FOUJSFMZ GSPN UIF Q8( − Q)- UFSN BOE UIF SFTU JT KVTU

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In nite possibilities e globe is a polyhedron with an in nite number of sides e posterior probability of any “side” p is proportional to: Only trick is normalizing to probability. A er a little calculus:  #":&4*"/ 61%"5*/( " DPOUJOVPVT TPMVUJPO *O UIJT FYBNQMF JUT OPU IBSE UP EFSJWF UIF QPTUFSJPS EJTUSJ BO FYBDU DPOUJOVPVT EJTUSJCVUJPO GVODUJPO *U UVSOT PVU UIBU UIF QPTUFSJPS QSPCBCJ PTTJCMF QSPQPSUJPO PG XBUFS Q CFUXFFO [FSP BOE POF JT QSPQPSUJPOBM UP Q8 ( − Q)- 8 JT UIF OVNCFS PG XBUFS PCTFSWFE BOE - JT UIF OVNCFS PG MBOE PCTFSWFE 8I UIJT JT UIF FYBDU FYQSFTTJPO UIBU XF BMSFBEZ VTFE UP DBMDVMBUF UIF SFMBUJWF OVNCFS Z WBMVF Q DPVME QSPEVDF B TBNQMF XJUI 8 XBUFS BOE - MBOE *U JT B MPHJDBM JNQMJDBUJ BSEFO PG GPSLJOH QBUIT 8IFO XF VTFE UIJT FYQSFTTJPO PSJHJOBMMZ XF NVMUJQMJFE Q Posterior probability of p = OPNJOBUPS ; JT UIF TVN PG BMM OVNFSBUPST GPS FWFSZ QPTTJCMF Q 'PS B ĕOJUF OVNCF ; JT KVTU B TUBOEBSE TVN ; = Q Q8(−Q)- XIFSF UIF Q OPUBUJPO NFBOT UP FWBMV O GPS FWFSZ Q BOE UIFO BEE UIFN 'PS BO JOĕOJUF OVNCFS PG QPTTJCMF Q WBMVFT GSPN [ TU VTF UIF DPOUJOVPVT QBSUOFS PG BO JOUFHSBM ; = Q8 ( − Q)-EQ PVT JOUFHSBM JG CZ iGBNPVTw ZPV NFBO iLOPXOw :PV DBO MPPL JU VQ POMJOF *O UIF DBT OE - JU JT HJWFO CZ ; = 8! -! (8 + - + )! UIF QPTUFSJPS QSPCBCJMJUZ PG BOZ TQFDJĕD Q JT FYBDUMZ 1S(Q|8, -) = #FUB(8 + , - + ) = (8 + - + )! 8!-! Q8 ( − Q)- NQMJDBUFE CVU UIF TIBQF DPNFT FOUJSFMZ GSPN UIF Q8( − Q)- UFSN BOE UIF SFTU JT

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In nite possibilities Posterior probability of p = Normalizing constant relative number of ways to observe sample e “Beta” distribution ; = Q ( − Q) EQ PVT JOUFHSBM JG CZ iGBNPVTw ZPV NFBO iLOPXOw :PV DBO MPPL JU VQ POMJOF *O UIF DBT OE - JU JT HJWFO CZ ; = 8! -! (8 + - + )! UIF QPTUFSJPS QSPCBCJMJUZ PG BOZ TQFDJĕD Q JT FYBDUMZ 1S(Q|8, -) = #FUB(8 + , - + ) = (8 + - + )! 8!-! Q8 ( − Q)- NQMJDBUFE CVU UIF TIBQF DPNFT FOUJSFMZ GSPN UIF Q8( − Q)- UFSN BOE UIF SFTU JT UIF BSFB VOEFS UIF DVSWF TVNT UP  TP UIBU JU JT B QSPQFS QSPCBCJMJUZ EJTUSJCVUJPO " )- UFSN JT KVTU BO JNQMJDBUJPO PG UIF HBSEFO PG GPSLJOH QBUIT ćFSFT OPUIJOH FMTF HP #FUB EJTUSJCVUJPO JT HJWFO CZ UIF GVODUJPO dbeta() JO 3

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Ten tosses of the globe

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Bę posterior probability 0 0.5 1 0 W 0 0.5 1 0 W L 0 0.5 1 0 W L W posterior probability 0 0.5 1 0 W L W W 0 0.5 1 0 W L W W W 0 0.5 1 0 W L W W W L

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ESB posterior probability 0 0.5 1 0 W L W W 0 0.5 1 0 W L W W W 0 0.5 1 0 W L W W W L proportion water (p) posterior probability 0 0.5 1 0 W L W W W L W proportion water (p) 0 0.5 1 0 W L W W W L W L proportion water (p) 0 0.5 1 0 W L W W W L W L W

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(1) No minimum sample size

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(2) Shape embodies sample size

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(3) No point estimate mean mode e distribution is the estimate Always use the entire distribution

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(4) No one true interval Intervals communicate shape of posterior 0.0 1.0 2.0 proportion water density 0 0.5 1

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0.0 1.0 2.0 proportion water density 0 0.5 1 (4) No one true interval Intervals communicate shape of posterior 50%

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0.0 1.0 2.0 proportion water density 0 0.5 1 (4) No one true interval Intervals communicate shape of posterior 89%

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0.0 1.0 2.0 proportion water density 0 0.5 1 (4) No one true interval Intervals communicate shape of posterior 95% is obvious superstition. Nothing magical happens at the boundary. 99%

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Letters From My Reviewers “ e author uses these cute 89% intervals, but we need to see the 95% intervals so we can tell whether any of the e ects are robust.” at an arbitrary interval contains an arbitrary value is not meaningful. Use the whole distribution.

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Work ow (1) De ne generative model of the sample (2) De ne a speci c estimand (3) Design a statistical way to produce estimate (4) Test (3) using (1) (5) Analyze sample, summarize

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From Posterior to Prediction Implications of model depend upon entire posterior Must average any inference over entire posterior is usually requires integral calculus OR we can just take samples from the posterior

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Sampling the posterior XF XJMM VTF TUBUJTUJDBM QSPDFEVSFT UIBU FTUJNBUF UIF QPTUFSJPS EJTUSJCVUJPO XJUI TBNQMFT ćFSF XJMM CF OP PUIFS SFQSFTFOUBUJPO PG JU 4P JG ZPV HFU VTFE UP XPSLJOH XJUI QPTUFSJPS TBNQMFT OPX ZPV XPOU IBWF UP SFMFBSO BOZUIJOH MBUFS *O UIJT DBTF XF DBO ESBX TBNQMFT GSPN UIF QPTUFSJPS XJUI 3 DPEF  post_samples <- rbeta( 1e3 , 6+1 , 3+1 ) /PX post_samples DPOUBJOT  QSPQPSUJPOT PG XBUFS  +VTU TIPX UIF QPTUFSJPS ćF CFTU TVNNBSZ PG UIF QPTUFSJPS EJTUSJCVUJPO JT UIF QPTUF SJPS EJTUSJCVUJPO +VTU ESBX JU *O NPSF DPNQMJDBUFE NPEFMT XIBU XFMM ESBX JT B QPTUFSJPS F ES proportion water ćF SFE DVSWF JT BO FTUJNBUF PG UIF EJTUSJCVUJPO CBTFE PO UIF TBNQMFT GSPN JU ćF EBTIFE DVSWF JT UIF BOBMZUJDBM FYBDU QPTUFSJPS EJTUSJCVUJPO ćF TIBQF PG UIF SFE DVSWF EFQFOET VQPO IPX ZPV FTUJNBUF JU GSPN UIF TBNQMFT‰JUT B TUBUJTUJDBM FTUJNBUF JUTFMG 4P EPOU TUBSU QFFSJOH BU UIF XJHHMFT BOE USZJOH UP NBLF TFOTF PG UIFN ćFZ BSF KVTU TBNQMJOH WBSJBUJPO "OE JG XF DIBOHF IPX UP DVSWF JT FTUJNBUFE XFMM HFU NPSF PS GFXFS XJHHMFT 8JUI 3T EFOTJUZ FTUJNBUPS NBLJOH adj TNBMMFS QSPEVDFT NPSF MPDBM FTUJNBUJPO 3 DPEF  dens( post_samples , lwd=4 , col=2 , xlab="proportion water" , adj=0.1 ) curve( dbeta(x,6+1,3+1) , add=TRUE , lty=2 , lwd=3 ) 0.2 0.4 0.6 0.8 0 1 2 3 proportion water Density beta distribution samples

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Uncertainty Causal model Implications ꔄ ꔄ

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plot( table(pred_64) , xlim=c(0,10) , xlab="number of W" , ylab="count" , lwd=10 , col=1 ) # now simulate posterior predictive distribution post_samples <- rbeta(1e4,6+1,3+1) pred_post <- sapply( post_samples , function(p) sum(sim_globe(p,10)=="W") ) tab_post <- table(pred_post) for ( i in 0:10 ) lines(c(i,i),c(0,tab_post[i+1]),lwd=4,col=4)  46.."3*;*/( 1045&3*03 %*453*#65*0/4  0 500 1500 2500 number of W count 0 1 2 3 4 5 6 7 8 9 10 ćF CMBDL IJTUPHSBN TIPXT UIF QSFEJDUJWF EJTUSJCVUJPO GPS Q = . UIF QPTUFSJPS NFBO ćF p = 0.64 entire posterior

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Sampling is Fun & Easy Sample from posterior, compute desired quantity for each sample, pro t Much easier than doing integrals Turn a calculus problem into a data summary problem MCMC produces only samples anyway

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Sampling is Handsome & Handy ings we’ll compute with sampling: Model-based forecasts Causal e ects Counterfactuals Prior predictions

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Bayesian data analysis For each possible explanation of the data, Count all the ways data can happen. Explanations with more ways to produce the data are more plausible.

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Bayesian modesty No guarantees except logical Probability theory is a method of logically deducing implications of data under assumptions that you must choose Any framework selling you more is hiding assumptions

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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 / 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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BONUS ROUND

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Misclassi cation  .FBTVSFNFOU BOE NJTDMBTTJĕDBUJPO UFS IBT GPDVTFE PO B TJNQMF EFTDSJQUJWF FTUJNBOE UIF QSPQPSUJPO OUFE JU JO UIF DPOUFYU PG B TJNQMF DBVTBM EJBHSBN Q / 8 -

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Misclassi cation JT B CJU SFEVOEBOU CFDBVTF JG XF LOPX / BOE 8 XF DBO KVTU DBMD P MFUT SFESBX UIF EJBHSBN XJUI UIBU JO NJOE "OE *MM BEE TPNF PS UIJOH FMTF BT XFMM Q / 8 N 8 BOE / BSF ļįŀIJĿŃIJı‰XF LOPX UIFJS WBMVFT ćF WBSJBCM FBE JU JT PVS FTUJNBOE 0OF DPOWFOUJPO GPS TIPXJOH XIJDI WBSJBCM XIJDI IBWF OPU JT UP ESBX DJSDMFT BSPVOE VOPCTFSWFE WBSJBCMFT unobserved

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Misclassi cation JT B CJU SFEVOEBOU CFDBVTF JG XF LOPX / BOE 8 XF DBO KVTU DBMD P MFUT SFESBX UIF EJBHSBN XJUI UIBU JO NJOE "OE *MM BEE TPNF PS UIJOH FMTF BT XFMM Q / 8 N 8 BOE / BSF ļįŀIJĿŃIJı‰XF LOPX UIFJS WBMVFT ćF WBSJBCM FBE JU JT PVS FTUJNBOE 0OF DPOWFOUJPO GPS TIPXJOH XIJDI WBSJBCM XIJDI IBWF OPU JT UP ESBX DJSDMFT BSPVOE VOPCTFSWFE WBSJBCMFT population size unobserved

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Misclassi cation ĹĮŀŀĶijĶİĮŁĶļĻ FSSPS "HBJO UIJOL BCPVU HMPCF UPTTJOH #VU PVOUJOH 8 BOE - NBLFT NJTUBLFT  PG UIF UJNF UIFZ XSJU PO TXJUDIJOH 8 GPS - BOE - GPS 8 ćJT JT QBSU PG IPX UIF TBN IF TBNQMF "OE XF TIPVME CF BCMF BOE SFBEZ UP JODMVEF JU JO UIF Q / 8 8 PX XF EP OPU PCTFSWF UIF USVF DPVOU 8 *OTUFBE XF PCTFSWF U OE 8 JT DBVTFE CZ UXP WBSJBCMFT UIF USVF DPVOU 8 BOE UIF NF true samples unobserved

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Misclassi cation ijĶİĮŁĶļĻ FSSPS "HBJO UIJOL BCPVU HMPCF UPTTJOH #VU OPX H 8 BOE - NBLFT NJTUBLFT  PG UIF UJNF UIFZ XSJUF EPX JUDIJOH 8 GPS - BOE - GPS 8 ćJT JT QBSU PG IPX UIF TBNQMF BSJ QMF "OE XF TIPVME CF BCMF BOE SFBEZ UP JODMVEF JU JO UIF DBVTB Q / 8 8 . EP OPU PCTFSWF UIF USVF DPVOU 8 *OTUFBE XF PCTFSWF UIF NJ JT DBVTFE CZ UXP WBSJBCMFT UIF USVF DPVOU 8 BOE UIF NFBTVSFN misclassi ed samples

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Misclassi cation ĶļĻ FSSPS "HBJO UIJOL BCPVU HMPCF UPTTJOH #VU OPX PVS BT OE - NBLFT NJTUBLFT  PG UIF UJNF UIFZ XSJUF EPXO UIF H 8 GPS - BOE - GPS 8 ćJT JT QBSU PG IPX UIF TBNQMF BSJTFT TP OE XF TIPVME CF BCMF BOE SFBEZ UP JODMVEF JU JO UIF DBVTBM EJBH Q / 8 8 . PU PCTFSWF UIF USVF DPVOU 8 *OTUFBE XF PCTFSWF UIF NJTDMBTT VTFE CZ UXP WBSJBCMFT UIF USVF DPVOU 8 BOE UIF NFBTVSFNFOU Q measurement process

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Misclassi cation simulation Obey the work ow! Code a generative model:  .&"463&.&/5 "/% .*4$-"44*'*$"5*0/  3 DPEF  sim_globe2 <- function( p=0.7 , N=9 , x=0.1 ) { true_sample <- sample(c("W","L"),size=N,prob=c(p,1-p),replace=TRUE) obs_sample <- ifelse( runif(N) < x , ifelse( true_sample=="W" , "L" , "W" ) , # error true_sample ) # no error return(obs_sample) } 5P VOEFSTUBOE UIF QSPCMFN NJTDMBTTJĕDBUJPO DBVTFT GPS PVS QSFWJPVT FTUJNBUPS DPOTJEFS BO FYUSFNF DBTF MJLF Q =  /PX XJUIPVU FSSPS XFE OFWFS PCTFSWF XBUFS #VU XJUI FSSPS XFMM PCTFSWF XBUFS  PG UIF UJNF 4JNJMBSMZ PO UIF PUIFS FYUSFNF FOE Q =  #VU OPX XF TIPVME OFWFS PCTFSWF MBOE CVU XF PCTFSWF JU JOTUFBE  PG UIF UJNF (P BIFBE BOE UFTU UIF TJNVMBUJPO DPEF BCPWF UP NBLF TVSF JU XPSLT BT FYQFDUFE

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Misclassi cation simulation Obey the work ow! Code a generative model:  .&"463&.&/5 "/% .*4$-"44*'*$"5*0/  3 DPEF  sim_globe2 <- function( p=0.7 , N=9 , x=0.1 ) { true_sample <- sample(c("W","L"),size=N,prob=c(p,1-p),replace=TRUE) obs_sample <- ifelse( runif(N) < x , ifelse( true_sample=="W" , "L" , "W" ) , # error true_sample ) # no error return(obs_sample) } 5P VOEFSTUBOE UIF QSPCMFN NJTDMBTTJĕDBUJPO DBVTFT GPS PVS QSFWJPVT FTUJNBUPS DPOTJEFS BO FYUSFNF DBTF MJLF Q =  /PX XJUIPVU FSSPS XFE OFWFS PCTFSWF XBUFS #VU XJUI FSSPS XFMM PCTFSWF XBUFS  PG UIF UJNF 4JNJMBSMZ PO UIF PUIFS FYUSFNF FOE Q =  #VU OPX XF TIPVME OFWFS PCTFSWF MBOE CVU XF PCTFSWF JU JOTUFBE  PG UIF UJNF (P BIFBE BOE UFTU UIF TJNVMBUJPO DPEF BCPWF UP NBLF TVSF JU XPSLT BT FYQFDUFE

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Misclassi cation simulation Obey the work ow! Code a generative model:  .&"463&.&/5 "/% .*4$-"44*'*$"5*0/  3 DPEF  sim_globe2 <- function( p=0.7 , N=9 , x=0.1 ) { true_sample <- sample(c("W","L"),size=N,prob=c(p,1-p),replace=TRUE) obs_sample <- ifelse( runif(N) < x , ifelse( true_sample=="W" , "L" , "W" ) , # error true_sample ) # no error return(obs_sample) } 5P VOEFSTUBOE UIF QSPCMFN NJTDMBTTJĕDBUJPO DBVTFT GPS PVS QSFWJPVT FTUJNBUPS DPOTJEFS BO FYUSFNF DBTF MJLF Q =  /PX XJUIPVU FSSPS XFE OFWFS PCTFSWF XBUFS #VU XJUI FSSPS XFMM PCTFSWF XBUFS  PG UIF UJNF 4JNJMBSMZ PO UIF PUIFS FYUSFNF FOE Q =  #VU OPX XF TIPVME OFWFS PCTFSWF MBOE CVU XF PCTFSWF JU JOTUFBE  PG UIF UJNF (P BIFBE BOE UFTU UIF TJNVMBUJPO DPEF BCPWF UP NBLF TVSF JU XPSLT BT FYQFDUFE

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Misclassi cation simulation Obey the work ow! Code a generative model:  .&"463&.&/5 "/% .*4$-"44*'*$"5*0/  3 DPEF  sim_globe2 <- function( p=0.7 , N=9 , x=0.1 ) { true_sample <- sample(c("W","L"),size=N,prob=c(p,1-p),replace=TRUE) obs_sample <- ifelse( runif(N) < x , ifelse( true_sample=="W" , "L" , "W" ) , # error true_sample ) # no error return(obs_sample) } 5P VOEFSTUBOE UIF QSPCMFN NJTDMBTTJĕDBUJPO DBVTFT GPS PVS QSFWJPVT FTUJNBUPS DPOTJEFS BO FYUSFNF DBTF MJLF Q =  /PX XJUIPVU FSSPS XFE OFWFS PCTFSWF XBUFS #VU XJUI FSSPS XFMM PCTFSWF XBUFS  PG UIF UJNF 4JNJMBSMZ PO UIF PUIFS FYUSFNF FOE Q =  #VU OPX XF TIPVME OFWFS PCTFSWF MBOE CVU XF PCTFSWF JU JOTUFBE  PG UIF UJNF (P BIFBE BOE UFTU UIF TJNVMBUJPO DPEF BCPWF UP NBLF TVSF JU XPSLT BT FYQFDUFE

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Misclassi cation estimator Use the intuition from the generative model to draw out the Garden of Forking Data, build a Bayesian estimator. Two stages: (1) true samples, (2) misclassi cation

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true samples

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true samples observed samples 1-in-3 misclassi ed

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Observe — How many ways can this happen?

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6 ways to observe water, when true sample is water ✓ ✓ ✓ ✓ ✓ ✓

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1 way to observe water, when true sample is land ✓

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3×2 + 1×1 = 7 ways to observe water ✓ ✓ ✓ ✓ ✓ ✓ ✓

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Misclassi cation estimator Posterior distribution for p given W,L,x: UPUBM 4P XF FOE VQ XJUI  QBUIT PVU PG  ćJT JT UIF QSPCBCJMJUZ UIBU XF EJOH NJTDMBTTJĕDBUJPO JO UIF QSPDFTT PWF DPVOUJOH UP QSJNF PVU JOUVJUJPO XF DBO XSJUF B QSPCBCJMJUZ FYQSFTTJPO Z PG PCTFSWJOH 8 PO BOZ HJWFO UPTT PG UIF HMPCF *U JT 1S(XBUFS|Q, Y) = Q( − Y) + ( − Q)Y PQPSUJPO PG XBUFS PO UIF HMPCF BOE Y UIF DIBODF PG NJTDMBTTJĕDBUJPO ćJT F TBNF TUSVDUVSF BT UIF UPUBM XBZT FYQSFTTJPO  ×  +  ×  =  "OE NF SFTVMU . ×   + . ×   = / 4JNJMBSMZ GPS UIF QSPCBCJMJUZ PG 1S(MBOE|Q, Y) = ( − Q)( − Y) + QY VTU EFSJWFE IFSF JO BO JOGPSNBM XBZ JT UIF SVMF PG QSPCBCJMJUZ UIFPSZ UIBU PU IBQQFO UPHFUIFS BMUFSOBUJWFT BSF BEEFE XIFSF FWFOUT UIBU IBQQFO UP MJFE ćF USVF TUBUF DBO CF 8 PS - *U DBOOPU CF CPUI BU UIF TBNF UJNF T DBO ĕOE UIF EPPS PVU PO UIFJS PXO  4P XF FOE VQ BEEJOH UIF XBZT UP TFF BMMZ 8 UP UIF XBZT UP TFF 8 XIFO JU JT BDUVBMMZ - 8F EJEOU XPSSZ BCPVU QPSUJPO PG XBUFS PO UIF HMPCF BOE Y UIF DIBODF PG NJTDMBTTJĕDBUJPO ćJT F TBNF TUSVDUVSF BT UIF UPUBM XBZT FYQSFTTJPO  ×  +  ×  =  "OE NF SFTVMU . ×   + . ×   = / 4JNJMBSMZ GPS UIF QSPCBCJMJUZ PG 1S(MBOE|Q, Y) = ( − Q)( − Y) + QY TU EFSJWFE IFSF JO BO JOGPSNBM XBZ JT UIF SVMF PG QSPCBCJMJUZ UIFPSZ UIBU U IBQQFO UPHFUIFS BMUFSOBUJWFT BSF BEEFE XIFSF FWFOUT UIBU IBQQFO UP JFE ćF USVF TUBUF DBO CF 8 PS - *U DBOOPU CF CPUI BU UIF TBNF UJNF DBO ĕOE UIF EPPS PVU PO UIFJS PXO  4P XF FOE VQ BEEJOH UIF XBZT UP TFF BMMZ 8 UP UIF XBZT UP TFF 8 XIFO JU JT BDUVBMMZ - 8F EJEOU XPSSZ BCPVU SF XIFO XF PSJHJOBMMZ TUBSUJOH DPVOUJOH HBSEFO QBUIT CFDBVTF UIFSF XFSF 8F XFSF OFWFS XPOEFSJOH XIBU IBQQFOFE JO UIF TBNQMJOH 8F LOFX XIBU F IBWF BO PCTFSWBUJPO UIBU JT DPOTJTUFOU XJUI EJČFSFOU USVF FWFOUT 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1 proportion of water DPVOUT GPS NJTDMBTTJĕDBUJPO ćF CMBDL DVSW JT PVS QSFWJPVT QPTUFSJPS XIJDI JHOPSFT NJT DMBTTJĕDBUJPO 4P XIBU JT PVS #BZFTJBO FTUJNBUPS OPX 'PS PVS PSJHJOBM TBNQMF XJUI 8 =  BOE - =  VNJOH NJTDMBTTJĕDBUJPO BU B SBUF PG Y UIF OFX QPTUFSJPS EJTUSJCVUJPO MPPLT MJLF UIJT 1S(Q|8, -) = [Q( − Y) + ( − Q)Y]8 × [( − Q)( − Y) + QY]- ; FSF BT BMXBZT UIF EFOPNJOBUPS ; JT KVTU UIF TVN PG FWFSZ OVNFSBUPS GPS FWFSZ WBMVF P *U KVTU OPSNBMJ[FT UIF DPVOUT TP UIFZ TVN UP POF BOE BSF QSPQFS QSPCBCJMJUJFT #VU UI Pr(p|W,L,x)

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4P XIBU JT PVS #BZFTJBO FTUJNBUPS OPX 'PS PVS PSJHJOBM TBNQMF XJUI 8 =  BOE - =  VNJOH NJTDMBTTJĕDBUJPO BU B SBUF PG Y UIF OFX QPTUFSJPS EJTUSJCVUJPO MPPLT MJLF UIJT 1S(Q|8, -) = [Q( − Y) + ( − Q)Y]8 × [( − Q)( − Y) + QY]- ; FSF BT BMXBZT UIF EFOPNJOBUPS ; JT KVTU UIF TVN PG FWFSZ OVNFSBUPS GPS FWFSZ WBMVF P *U KVTU OPSNBMJ[FT UIF DPVOUT TP UIFZ TVN UP POF BOE BSF QSPQFS QSPCBCJMJUJFT #VU UI ODFQUVBM IFBSU JT UIF OVNFSBUPS "OE JU KVTU DPVOUJOH BMM UIF XBZT UP TFF B TBNQMF XJUI 8 UFS BOE - MBOE BTTVNJOH NJTDMBTTJĕDBUJPO QSPCBCJMJUZ Y ćF OPSNBMJ[JOH DPOTUBOU ; JT B XBZT KVTU B OVJTBODF CVU JG ZPV BSF DVSJPVT TFF UIF 0WFSUIJOLJOH CPY GVSUIFS EPXO -FUT QMPU PVS OFX QPTUFSJPS EJTUSJCVUJPO BOE DPNQBSF JU UP UIF QSFWJPVT POF code for the normalizing constant eta <- function(x,a,b) exp( pbeta(x,a,b,log.p=TRUE) + lbeta(a,b) ) Pr(p|W,L,x) probability of each water probability of each land

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4P XIBU JT PVS #BZFTJBO FTUJNBUPS OPX 'PS PVS PSJHJOBM TBNQMF XJUI 8 =  BOE - =  VNJOH NJTDMBTTJĕDBUJPO BU B SBUF PG Y UIF OFX QPTUFSJPS EJTUSJCVUJPO MPPLT MJLF UIJT 1S(Q|8, -) = [Q( − Y) + ( − Q)Y]8 × [( − Q)( − Y) + QY]- ; FSF BT BMXBZT UIF EFOPNJOBUPS ; JT KVTU UIF TVN PG FWFSZ OVNFSBUPS GPS FWFSZ WBMVF P *U KVTU OPSNBMJ[FT UIF DPVOUT TP UIFZ TVN UP POF BOE BSF QSPQFS QSPCBCJMJUJFT #VU UI ODFQUVBM IFBSU JT UIF OVNFSBUPS "OE JU KVTU DPVOUJOH BMM UIF XBZT UP TFF B TBNQMF XJUI 8 UFS BOE - MBOE BTTVNJOH NJTDMBTTJĕDBUJPO QSPCBCJMJUZ Y ćF OPSNBMJ[JOH DPOTUBOU ; JT B XBZT KVTU B OVJTBODF CVU JG ZPV BSF DVSJPVT TFF UIF 0WFSUIJOLJOH CPY GVSUIFS EPXO -FUT QMPU PVS OFX QPTUFSJPS EJTUSJCVUJPO BOE DPNQBSF JU UP UIF QSFWJPVT POF code for the normalizing constant eta <- function(x,a,b) exp( pbeta(x,a,b,log.p=TRUE) + lbeta(a,b) ) Pr(p|W,L,x) probability of each water probability of each land some unpleasant normalizing constant

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Misclassi cation posterior ę   4."-- 803-%4 "/% -"3(& 803-%4 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 proportion of water posterior probability 'ĶĴłĿIJ ƊƐ ćF QPTUFSJPS HMPCF UPTTJOH FYQFSJNFOU UJPO ćF SFE DVSWF JT UI DPVOUT GPS NJTDMBTTJĕDBUJP JT PVS QSFWJPVT QPTUFSJPS DMBTTJĕDBUJPO previous posterior misclassi cation posterior

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Measurement matters When there is measurement error, better to model it than to ignore it Same goes for: missing data, compliance, inclusion, etc Good news: Samples do not need to be representative of population in order to provide good estimates of population What matters is why the sample di ers

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