Richard McElreath
January 04, 2023
1.5k

# Statistical Rethinking 2023 - Lecture 02

January 04, 2023

## Transcript

1. Statistical Rethinking
2. e Garden of Forking Data
2023

2. What
proportion of
the surface is
covered with
water?

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

4. 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

5. 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

6. 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

7. 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?

8. 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

9. 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

10. 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.

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

12. 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.

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

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

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

16. First Possibility
Figure 2.2

17. Second Possibility
Figure 2.2

18. ird Possibility
Figure 2.2

19. Figure 2.2
First Observation

20. Figure 2.2
Second Observation

21. Figure 2.2
ird Observation

22. Figure 2.2
3 Ways to see
for 25% water

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

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

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

26. (3)

27. (4)

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

29. '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.

30. '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.

31. '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.

32. '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.

33. '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.

34. 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

35. 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

36. 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

37. 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

38. 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

39. 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

40. 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:

41. 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
< > = ×
< > = ×
< > = ×
< > = ×
< > = ×

42. 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
< > = ×
< > = ×
< > = ×
< > = ×
< > = ×

43. 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
< > = ×
< > = ×
< > = ×
< > = ×
< > = ×

44. 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
< > = ×
< > = ×
< > = ×
< > = ×
< > = ×

45. 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
< > = ×
< > = ×
< > = ×
< > = ×
< > = ×

46. 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

47. 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

48. 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

49. 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

50. ESB
0 0.25 0.5 0.75 1
proportion water
0.0 0.
3 DPEF
sample W L p ways prob 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

51. 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

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

53. 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 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)

54. 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 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

55. 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 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 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

56. /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"

57. /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

58. IF YOU TEST
NOTHING
YOU MISS
EVERYTHING

59. Code the estimator
26"-*5: "4463"/\$&
# function to compute posterior distribution
compute_posterior W L ways post bars 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

60. Code the estimator
26"-*5: "4463"/\$&
# function to compute posterior distribution
compute_posterior W L ways post bars 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

61. Code the estimator
26"-*5: "4463"/\$&
# function to compute posterior distribution
compute_posterior W L ways post bars 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

62. Code the estimator
26"-*5: "4463"/\$&
# function to compute posterior distribution
compute_posterior W L ways post bars 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

63. Code the estimator
26"-*5: "4463"/\$&
# function to compute posterior distribution
compute_posterior W L ways post bars 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

64. 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

65. PAUSE

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

67. 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]

68. 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]

69. More possibilities

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 W 5 possibilities

70. More possibilities

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

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 W 5 possibilities

71. More possibilities

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

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 W 5 possibilities

72. 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

73. 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

74. 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

75. Ten tosses of the globe

76. 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

77. 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

78. (1) No minimum sample size

79. (2) Shape embodies sample size

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

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

82. 0.0 1.0 2.0
proportion water
density
0 0.5 1
(4) No one true interval
Intervals
communicate shape
of posterior
50%

83. 0.0 1.0 2.0
proportion water
density
0 0.5 1
(4) No one true interval
Intervals
communicate shape
of posterior
89%

84. 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%

85. 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.

86. 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

87. 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

88. 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 /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

89. Uncertainty Causal model Implications
ꔄ ꔄ

90. 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 pred_post tab_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

91. 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

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

93. 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.

94. 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

95. 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

96. BONUS
ROUND

97. 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
-

98. 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

99. 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

100. 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

101. 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

102. 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

103. Misclassi cation simulation
Obey the work ow! Code a generative model:
.&"463&.&/5 "/% .*4\$-"44*'*\$"5*0/
3 DPEF

sim_globe2 true_sample obs_sample 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

104. Misclassi cation simulation
Obey the work ow! Code a generative model:
.&"463&.&/5 "/% .*4\$-"44*'*\$"5*0/
3 DPEF

sim_globe2 true_sample obs_sample 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

105. Misclassi cation simulation
Obey the work ow! Code a generative model:
.&"463&.&/5 "/% .*4\$-"44*'*\$"5*0/
3 DPEF

sim_globe2 true_sample obs_sample 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

106. Misclassi cation simulation
Obey the work ow! Code a generative model:
.&"463&.&/5 "/% .*4\$-"44*'*\$"5*0/
3 DPEF

sim_globe2 true_sample obs_sample 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

107. 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

108. true samples

109. true samples
observed samples
1-in-3 misclassi ed

110. Observe — How many ways can this happen?

111. 6 ways to observe water, when true sample is water

✓ ✓

112. 1 way to observe water, when true sample is land

113. 3×2 + 1×1 = 7 ways to observe water

✓ ✓

114. 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)

115. 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 Pr(p|W,L,x)
probability of each water probability of each land

116. 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 Pr(p|W,L,x)
probability of each water probability of each land
some unpleasant
normalizing constant

117. 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

118. 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