Bayesian Inference is Just Counting

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BAYESIAN INFERENCE IS JUST COUNTING Richard McElreath MPI-EVA p(x|y)p(y)/p(x)

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The Golem of Prague go•lem |gōlǝm| noun • (in Jewish legend) a clay figure brought to life by magic.   • an automaton or robot. ORIGIN late 19th cent.: from Yiddish goylem, from Hebrew gōlem ‘shapeless mass.’

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The Golem of Prague “Even the most perfect of Golem, risen to life to protect us, can easily change into a destructive force. Therefore let us treat carefully that which is strong, just as we bow kindly and patiently to that which is weak.” Rabbi Judah Loew ben Bezalel (1512–1609) From Breath of Bones: A Tale of the Golem

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The Golems of Science Golem • Made of clay • Animated by “truth” • Powerful • Blind to creator’s intent • Easy to misuse • Fictional Model • Made of...silicon? • Animated by “truth” • Hopefully powerful • Blind to creator’s intent • Easy to misuse • Not even false

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Bayesian data analysis • Use probability to describe uncertainty • Extends ordinary logic (true/false) to continuous plausibility • Computationally difficult • Markov chain Monte Carlo (MCMC) to the rescue • Used to be controversial • Ronald Fisher: Bayesian analysis “must be wholly rejected.” Pierre-Simon Laplace (1749–1827) Sir Harold Jeffreys (1891–1989) with Bertha Swirles, aka Lady Jeffreys (1903–1999)

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Bayesian data analysis Count all the ways data can happen, according to assumptions. Assumptions with more ways that are consistent with data are more plausible.

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Bayesian data analysis • Contrast with frequentist view • Probability is just limiting frequency • Uncertainty arises from sampling variation • Bayesian probability much more general • Probability is in the golem, not in the world • Coins are not random, but our ignorance makes them so Saturn as Galileo saw it

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Garden of Forking Data • The future: • Full of branching paths • Each choice closes some • The data: • Many possible events • Each observation eliminates some

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Garden of Forking Data (1) (2) (3) (4) (5) Contains 4 marbles ? Possible contents: Observe:

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Conjecture: Data:

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Conjecture: Data:

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Conjecture: Data:

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Conjecture: Data: 3 paths consistent with data

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

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

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3 ways 9 ways 8 ways

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3 ways 9 ways 8 ways

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3 ways 9 ways 8 ways

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Garden of Forking Data OE XIJUF UIFSF BSF QBUIT UIBU TVSWJWF WF DPOTJEFSFE ĕWF EJČFSFOU DPOKFDUVSFT BCPVU UIF DPOUFOUT PG UIF CBH F NBSCMFT UP GPVS CMVF NBSCMFT 'PS FBDI PG UIFTF DPOKFDUVSFT XFWF TFRVFODFT QBUIT UISPVHI UIF HBSEFO PG GPSLJOH EBUB DPVME QPUFOUJBMMZ EBUB $POKFDUVSF 8BZT UP QSPEVDF < > × × = < > × × = < > × × = < > × × = < > × × = S PG XBZT UP QSPEVDF UIF EBUB GPS FBDI DPOKFDUVSF DBO CF DPNQVUFE VNCFS PG QBUIT JO FBDI iSJOHw PG UIF HBSEFO BOE UIFO CZ NVMUJQMZJOH ćJT JT KVTU B DPNQVUBUJPOBM EFWJDF *U UFMMT VT UIF TBNF UIJOH BT 'ĶĴ BWJOH UP ESBX UIF HBSEFO ćF GBDU UIBU OVNCFST BSF NVMUJQMJFE EVSJOH OHF UIF GBDU UIBU UIJT JT TUJMM KVTU DPVOUJOH PG MPHJDBMMZ QPTTJCMF QBUIT

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Updating Another draw from the bag: QBUIT DPNQBUJCMF XJUI UIF EBUB TFRVFODF 0S ZPV DPVME UBLF UIF Q PWFS DPOKFDUVSFT BOE KVTU VQEBUF UIFN JO MJHIU PG UIF OFX PCTFS PVU UIBU UIFTF UXP NFUIPET BSF NBUIFNBUJDBMMZ JEFOUJDBM "T MPOH BT UIF OFX JT MPHJDBMMZ JOEFQFOEFOU PG UIF QSFWJPVT PCTFSWBUJPOT SFT IPX UP EP JU 'JSTU XF DPVOU UIF OVNCFST PG XBZT FBDI DPOKFDUVSF DPVME Q X PCTFSWBUJPO ćFO XF NVMUJQMZ FBDI PG UIFTF OFX DPVOUT CZ UIF QSFWJPVT OV GPS FBDI DPOKFDUVSF *O UBCMF GPSN $POKFDUVSF 8BZT UP QSPEVDF 1SFWJPVT DPVOUT /FX DPVOU < > × = < > × = < > × = < > × = < > × = X DPVOUT JO UIF SJHIUIBOE DPMVNO BCPWF TVNNBSJ[F BMM UIF FWJEFODF GPS FBDI T OFX EBUB BSSJWF BOE QSPWJEFE UIPTF EBUB BSF JOEFQFOEFOU PG QSFWJPVT PCTFSW F OVNCFS PG MPHJDBMMZ QPTTJCMF XBZT GPS B DPOKFDUVSF UP QSPEVDF BMM UIF EBUB VQ BO CF DPNQVUFE KVTU CZ NVMUJQMZJOH UIF OFX DPVOU CZ UIF PME DPVOU T VQEBUJOH BQQSPBDI BNPVOUT UP OPUIJOH NPSF UIBO BTTFSUJOH UIBU XIFO X VT JOGPSNBUJPO TVHHFTUJOH UIFSF BSF 8QSJPS XBZT GPS B DPOKFDUVSF UP QSPEVDF B Q 4

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Using other information marbles rare, but every bag contains at least one. Factory says: IJT FYBNQMF UIF QSJPS EBUB BOE OFX EBUB BSF PG UIF TBNF UZQF NBSCMFT ESBX #VU JO HFOFSBM UIF QSJPS EBUB BOE OFX EBUB DBO CF PG EJČFSFOU UZQFT 4VQ UIBU TPNFPOF GSPN UIF NBSCMF GBDUPSZ UFMMT ZPV UIBU CMVF NBSCMFT BSF SBSF H DPOUBJOJOH < > UIFZ NBEF CBHT DPOUBJOJOH < > BOE CBHT > ćFZ BMTP FOTVSFE UIBU FWFSZ CBH DPOUBJOFE BU MFBTU POF CMVF BOE PO 8F DBO VQEBUF PVS DPVOUT BHBJO $POKFDUVSF 1SJPS XBZT 'BDUPSZ DPVOU /FX DPVOU < > × = < > × = < > × = < > × = < > × = DPOKFDUVSF < > JT NPTU QMBVTJCMF CVU CBSFMZ CFUUFS UIBO < > * E EJČFSFODF JO UIFTF DPVOUT BU XIJDI XF DBO TBGFMZ EFDJEF UIBU POF PG UIF DPO SSFDU POF :PVMM TQFOE UIF OFYU DIBQUFS FYQMPSJOH UIBU RVFTUJPO OH 0SJHJOBM JHOPSBODF 8IJDI BTTVNQUJPO TIPVME XF VTF XIFO UIFSF JT OP QSF UIF QSJPS EBUB BOE OFX EBUB BSF PG UIF TBNF UZQF NBSCMFT ESBXO GSPN FSBM UIF QSJPS EBUB BOE OFX EBUB DBO CF PG EJČFSFOU UZQFT 4VQQPTF GPS POF GSPN UIF NBSCMF GBDUPSZ UFMMT ZPV UIBU CMVF NBSCMFT BSF SBSF 4P GPS H < > UIFZ NBEF CBHT DPOUBJOJOH < > BOE CBHT DPOUBJO Z BMTP FOTVSFE UIBU FWFSZ CBH DPOUBJOFE BU MFBTU POF CMVF BOE POF XIJUF EBUF PVS DPVOUT BHBJO $POKFDUVSF 1SJPS XBZT 'BDUPSZ DPVOU /FX DPVOU > × = > × = > × = > × = > × = < > JT NPTU QMBVTJCMF CVU CBSFMZ CFUUFS UIBO < > *T UIFSF B F JO UIFTF DPVOUT BU XIJDI XF DBO TBGFMZ EFDJEF UIBU POF PG UIF DPOKFDUVSFT :PVMM TQFOE UIF OFYU DIBQUFS FYQMPSJOH UIBU RVFTUJPO BM JHOPSBODF 8IJDI BTTVNQUJPO TIPVME XF VTF XIFO UIFSF JT OP QSFWJPVT JO

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Using other information marbles rare. Factory says: IJT FYBNQMF UIF QSJPS EBUB BOE OFX EBUB BSF PG UIF TBNF UZQF NBSCMFT ESBX #VU JO HFOFSBM UIF QSJPS EBUB BOE OFX EBUB DBO CF PG EJČFSFOU UZQFT 4VQ UIBU TPNFPOF GSPN UIF NBSCMF GBDUPSZ UFMMT ZPV UIBU CMVF NBSCMFT BSF SBSF H DPOUBJOJOH < > UIFZ NBEF CBHT DPOUBJOJOH < > BOE CBHT > ćFZ BMTP FOTVSFE UIBU FWFSZ CBH DPOUBJOFE BU MFBTU POF CMVF BOE PO 8F DBO VQEBUF PVS DPVOUT BHBJO $POKFDUVSF 1SJPS XBZT 'BDUPSZ DPVOU /FX DPVOU < > × = < > × = < > × = < > × = < > × = DPOKFDUVSF < > JT NPTU QMBVTJCMF CVU CBSFMZ CFUUFS UIBO < > * E EJČFSFODF JO UIFTF DPVOUT BU XIJDI XF DBO TBGFMZ EFDJEF UIBU POF PG UIF DPO SSFDU POF :PVMM TQFOE UIF OFYU DIBQUFS FYQMPSJOH UIBU RVFTUJPO OH 0SJHJOBM JHOPSBODF 8IJDI BTTVNQUJPO TIPVME XF VTF XIFO UIFSF JT OP QSF

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Counts to plausibility Unglamorous basis of applied probability: Things that can happen more ways are more plausible. J[F JT UP BEE VQ BMM PG UIF QSPEVDUT POF GPS FBDI WBMVF Q DBO UBLF XBZT Q DBO QSPEVDF %OFX × QSJPS QMBVTJCJMJUZ Q UIFO EJWJEF FBDI QSPEVDU CZ UIF TVN PG QSPEVDUT QMBVTJCJMJUZ PG Q BęFS %OFX = XBZT Q DBO QSPEVDF %OFX × QSJPS QMBVTJCJMJUZ Q TVN PG QSPEVDUT FT OPUIJOH TQFDJBM SFBMMZ BCPVU TUBOEBSEJ[JOH UP POF "OZ WBMVF XJMM EP #VU VTJOH CFS FOET VQ NBLJOH UIF NBUIFNBUJDT NPSF DPOWFOJFOU $POTJEFS BHBJO UIF UBCMF GSPN CFGPSF OPX VQEBUFE VTJOH PVS EFĕOJUJPOT PG Q BOE iQ UZw 1PTTJCMF DPNQPTJUJPO Q XBZT UP QSPEVDF EBUB QMBVTJCJMJUZ < > < > . < > . < > . < > DBO RVJDLMZ DPNQVUF UIFTF QMBVTJCJMJUJFT JO 3 ʄǤ ǭ ƾ ǐ ǃ ǐ Ǆ Ǯ

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Counts to plausibility TVN PG QSPEVDUT FT OPUIJOH TQFDJBM SFBMMZ BCPVU TUBOEBSEJ[JOH UP POF "OZ WBMVF XJMM EP #VU VTJOH CFS FOET VQ NBLJOH UIF NBUIFNBUJDT NPSF DPOWFOJFOU $POTJEFS BHBJO UIF UBCMF GSPN CFGPSF OPX VQEBUFE VTJOH PVS EFĕOJUJPOT PG Q BOE iQ UZw 1PTTJCMF DPNQPTJUJPO Q XBZT UP QSPEVDF EBUB QMBVTJCJMJUZ < > < > . < > . < > . < > DBO RVJDLMZ DPNQVUF UIFTF QMBVTJCJMJUJFT JO 3 ʄǤ ǭ ƾ ǐ ǃ ǐ Ǆ Ǯ ǳ.0(ǭ24.Ǯ ƻǏƼǀ ƻǏƿƻ ƻǏƿǀ ćFTF QMBVTJCJMJUJFT BSF BMTP QSPCBCJMJUJFTUIFZ BSF OPOOFHBUJWF [FSP PS QPTJUJWF CFST UIBU TVN UP POF "OE BMM PG UIF NBUIFNBUJDBM UIJOHT ZPV DBO EP XJUI QSPCBCJ EBSEJ[F JT UP BEE VQ BMM PG UIF QSPEVDUT POF GPS FBDI WBMVF Q DBO UBLF XBZT Q DBO QSPEVDF %OFX × QSJPS QMBVTJCJMJUZ Q "OE UIFO EJWJEF FBDI QSPEVDU CZ UIF TVN PG QSPEVDUT QMBVTJCJMJUZ PG Q BęFS %OFX = XBZT Q DBO QSPEVDF %OFX × QSJPS QMBVTJCJMJUZ Q TVN PG QSPEVDUT ćFSFT OPUIJOH TQFDJBM SFBMMZ BCPVU TUBOEBSEJ[JOH UP POF "OZ WBMVF XJMM EP #VU VTJOH UIF OVNCFS FOET VQ NBLJOH UIF NBUIFNBUJDT NPSF DPOWFOJFOU $POTJEFS BHBJO UIF UBCMF GSPN CFGPSF OPX VQEBUFE VTJOH PVS EFĕOJUJPOT PG Q BOE iQMBV TJCJMJUZw 1PTTJCMF DPNQPTJUJPO Q XBZT UP QSPEVDF EBUB QMBVTJCJMJUZ < > < > . < > . < > . < > :PV DBO RVJDLMZ DPNQVUF UIFTF QMBVTJCJMJUJFT JO 3 3 DPEF 24. ʄǤ ǭ ƾ ǐ ǃ ǐ Ǆ Ǯ 24.ǳ.0(ǭ24.Ǯ ǯƼǰ ƻǏƼǀ ƻǏƿƻ ƻǏƿǀ ćFTF QMBVTJCJMJUJFT BSF BMTP QSPCBCJMJUJFTUIFZ BSF OPOOFHBUJWF [FSP PS QPTJUJWF SFBM OVNCFST UIBU TVN UP POF "OE BMM PG UIF NBUIFNBUJDBM UIJOHT ZPV DBO EP XJUI QSPCBCJMJUJFT ZPV DBO BMTP EP XJUI UIFTF WBMVFT 4QFDJĕDBMMZ FBDI QJFDF PG UIF DBMDVMBUJPO IBT B EJSFDU QBSUOFS JO BQQMJFE QSPCBCJMJUZ UIFPSZ ćFTF QBSUOFST IBWF TUFSFPUZQFE OBNFT TP JUT XPSUI

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Building a model • How to use probability to do typical statistical modeling? 1. Design the model (data story) 2. Condition on the data (update) 3. Evaluate the model (critique)

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Nine tosses of the globe: W L W W W L W L W

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Design > Condition > Evaluate • Data story motivates the model • How do the data arise? • For W L W W W L W L W: • Some true proportion of water, p • Toss globe, probability p of observing W, 1–p of L • Each toss therefore independent of other tosses • Translate data story into probability statements

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Design > Condition > Evaluate • Bayesian updating defines optimal learning in small world, converts prior into posterior • Give your golem an information state, before the data: Here, an initial confidence in each possible value of p between zero and one • Condition on data to update information state: New confidence in each value of p, conditional on data

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probability of water 0 0.5 1 n = 1 W L W W W L W L W confidence probability of water 0 0.5 1 n = 2 W L W W W L W L W W n = 4 W L W W W L W L W confidence n = 5 W L W W W L W L W W prior p, proportion W plausibility

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probability of water 0 0.5 1 n = 1 W L W W W L W L W confidence probability of water 0 0.5 1 n = 2 W L W W W L W L W W n = 4 W L W W W L W L W confidence n = 5 W L W W W L W L W W prior posterior p, proportion W plausibility

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probability of water 0 0.5 1 n = 1 W L W W W L W L W confidence probability of water 0 0.5 1 n = 2 W L W W W L W L W probability of water 0 0.5 1 n = 3 W L W W W L W L W probability of water 0 0.5 1 n = 4 W L W W W L W L W confidence probability of water 0 0.5 1 n = 5 W L W W W L W L W probability of water 0 0.5 1 n = 6 W L W W W L W L W probability of water 0 0.5 1 n = 7 W L W W W L W L W confidence probability of water 0 0.5 1 n = 8 W L W W W L W L W probability of water 0 0.5 1 n = 9 W L W W W L W L W proportion water 0 0.5 1 plausibility n = 0 proportion water 0 0.5 1 plausibility n = 0 W L W W W L W L W proportion water 0 0.5 1 plausibility n = 0 W L W W W L W L W proportion water 0 0.5 1 plausibility n = 0 W L W W W L W L W proportion water 0 0.5 1 plausibility n = 0 W L W W W L W L W proportion water 0 0.5 1 plausibility n = 0 W L W W W L W L W

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Design > Condition > Evaluate • Data order irrelevant, because golem assumes order irrelevant • All-at-once, one-at-a-time, shuﬄed order all give same posterior • Every posterior is a prior for next observation • Every prior is posterior of some other inference • Sample size automatically embodied in posterior 4."-- 803-%4 "/% -"3(& 803-%4 probability of water 0 0.5 1 n = 1 W L W W W L W L W confidence probability of water 0 0.5 1 n = 2 W L W W W L W L W probability of water 0 0.5 1 n = 3 W L W W W L W L W probability of water 0 0.5 1 n = 4 W L W W W L W L W confidence probability of water 0 0.5 1 n = 5 W L W W W L W L W probability of water 0 0.5 1 n = 6 W L W W W L W L W probability of water 0 0.5 1 n = 7 W L W W W L W L W confidence probability of water 0 0.5 1 n = 8 W L W W W L W L W probability of water 0 0.5 1 n = 9 W L W W W L W L W proportion water 0 0.5 1 plausibility n = 0 W L W W W L W L W proportion water 0 0.5 1 plausibility n = 0 W L W W W L W L W proportion water 0 0.5 1 plausibility n = 0 W L W W W L W L W proportion water 0 0.5 1 plausibility n = 0 W L W W W L W L W proportion water 0 0.5 1 plausibility n = 0 W L W W W L W L W proportion water 0 0.5 1 plausibility n = 0 W L W W W L W L W 'ĶĴłĿĲ Ɗƍ )PX B #BZFTJBO NPEFM MFBSOT &BDI UPTT PG UIF HMPCF QSPEVDFT BO PCTFSWBUJPO PG XBUFS 8 PS MBOE - ćF NPEFMT FTUJNBUF PG UIF QSP QPSUJPO PG XBUFS PO UIF HMPCF JT B QMBVTJCJMJUZ GPS FWFSZ QPTTJCMF WBMVF ćF MJOFT BOE DVSWFT JO UIJT ĕHVSF BSF UIFTF DPMMFDUJPOT PG QMBVTJCJMJUJFT *O FBDI QMPU B QSFWJPVT QMBVTJCJMJUJFT EBTIFE DVSWF BSF VQEBUFE JO MJHIU PG UIF MBUFTU

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Design > Condition > Evaluate • Bayesian inference: Logical answer to a question in the form of a model    “How plausible is each proportion of water, given these data?”  • Golem must be supervised • Did the golem malfunction? • Does the golem’s answer make sense? • Does the question make sense? • Check sensitivity of answer to changes in assumptions

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Construction perspective • Build joint model: (1) List variables (2) Define generative relations (3) ??? (4) Profit • Input: Joint prior • Deduce: Joint posterior

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The Joint Model σ α ρσα σβ ρσα σβ σ β 8 ∼ #JOPNJBM(/, Q) Q ∼ 6OJGPSN(, ) • Bayesian models are generative • Can be run forward to generate predictions or simulate date • Can be run in reverse to infer process from data

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The Joint Model σ α ρσα σβ ρσα σβ σ β 8 ∼ #JOPNJBM(/, Q) Q ∼ 6OJGPSN(, ) • Run forward:

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Run in Reverse:   Computing the posterior 1. Analytical approach (often impossible) 2. Grid approximation (very intensive) 3. Quadratic approximation (limited) 4. Markov chain Monte Carlo (intensive)

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Predictive checks • Something like a significance test, but not • No universally best way to evaluate adequacy of model-based predictions • No way to justify always using a threshold like 5% • Good predictive checks always depend upon purpose and imagination “It would be very nice to have a formal apparatus that gives us some ‘optimal’ way of recognizing unusual phenomena and inventing new classes of hypotheses [...]; but this remains an art for the creative human mind.”   —E.T. Jaynes (1922–1998)

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Triumph of Geocentrism • Claudius Ptolemy (90–168) • Egyptian mathematician • Accurate model of planetary motion • Epicycles: orbits on orbits • Fourier series -*/&" Earth equant planet epicycle deferent

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Geocentrism • Descriptively accurate • Mechanistically wrong • General method of approximation • Known to be wrong Regression • Descriptively accurate • Mechanistically wrong • General method of approximation • Taken too seriously

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Linear regression • Simple statistical golems • Model of mean and variance of normally (Gaussian) distributed measure • Mean as additive combination of weighted variables • Constant variance

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1809 Bayesian argument for normal error and least-squares estimation

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Why normal? • Why are normal (Gaussian) distributions so common in statistics? 1. Easy to calculate with 2. Common in nature 3. Very conservative assumption 0.0 0.1 0.2 0.3 0.4 x density −4σ −2σ 0 2σ 4σ 95%

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-6 -3 0 3 6 0.00 0.10 0.20 position Density 16 steps -6 -3 0 3 6 0.0 0.1 0.2 0.3 position Density 4 steps -6 -3 0 3 6 0.00 0.10 0.20 position Density 8 steps 0 4 8 12 16 -6 -3 0 3 6 step number position 'ĶĴłĿĲ ƌƊ 3BOEPN XBMLT PO UIF TPDDFS ĕFME DPOWFSHF UP B OPSNBM EJT USJCVUJPO ćF NPSF TUFQT BSF UBLFO UIF DMPTFS UIF NBUDI CFUXFFO UIF SFBM Figure 4.2

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Why normal? • Processes that produce normal distributions • Addition • Products of small deviations • Logarithms of products Francis Galton’s 1894 “bean machine” for simulating normal distributions

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Why normal? • Ontological perspective • Processes which add fluctuations result in dampening • Damped fluctuations end up Gaussian • No information left, except mean and variance • Can’t infer process from distribution! • Epistemological perspective • Know only mean and variance • Then least surprising and most conservative (maximum entropy) distribution is Gaussian • Nature likes maximum entropy distributions

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Linear models • Models of normally distributed data common • “General Linear Model”: t-test, single regression, multiple regression, ANOVA, ANCOVA, MANOVA, MANCOVA, yadda yadda yadda • All the same thing • Learn strategy, not procedure Willard Boepple

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30 35 40 45 50 55 60 140 150 160 170 180 weight height N = 10 30 35 40 45 50 55 60 140 150 160 170 180 weight height N = 20 30 35 40 45 50 55 60 140 150 160 170 180 weight height N = 50 30 35 40 45 50 55 60 140 150 160 170 180 weight height N = 100 30 35 40 45 50 55 60 140 150 160 170 180 weight height N = 200 30 35 40 45 50 55 60 140 150 160 170 180 weight height N = 350

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Regression as a wicked oracle • Regression automatically focuses on the most informative cases • Cases that don’t help are automatically ignored • But not kind — ask carefully

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Why not just add everything? • Could just add all available predictors to model • “We controlled for...” • Almost always a bad idea • Adding variables creates confounds • Residual confounding • Overfitting

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AGE HEIGHT MATH ? MATH independent of HEIGHT, conditional on AGE

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M 5 6 7 8 9 10 50 55 60 65 70 75 80 5 6 7 8 9 10 A 50 55 60 65 70 75 80 110 120 130 110 120 130 H

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MATH independent of HEIGHT, conditional on AGE 118 119 120 121 122 55 60 65 70 75 H M A = 7 123 124 125 126 127 60 65 70 75 H M A = 8 128 129 130 131 132 60 65 70 75 80 H M A = 9 133 134 135 136 137 65 70 75 80 H M A = 10

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LIGHT POWER SWITCH SWITCH dependent on POWER, conditional on LIGHT SWITCH independent of POWER

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LIGHT POWER SWITCH ON ON ? OFF ON ? SWITCH dependent on POWER, conditional on LIGHT This eﬀect known as “collider bias”

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MARRIED AGE HAPPY HAPPY dependent on AGE, conditional on MARRIED HAPPY independent of AGE

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Why not just add everything? • Matters for experiments as well • Conditioning on post-treatment variables can be very bad • Conditioning on pre-treatment can also be bad (colliders) • Good news! • Causal inference possible in observational settings • But requires good theory

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Texts in Statistical Science Richard McElreath Statistical Rethinking A Bayesian Course with Examples in R and Stan SECOND EDITION ond on JUST COUNTING IMPLICATIONS OF ASSUMPTIONS