Richard McElreath
January 25, 2019

# L10 Statistical Rethinking Winter 2019

Lecture 10 of the Dec 2018 through March 2019 edition of Statistical Rethinking. Covers Chapter 9, Markov chain Monte Carlo.

January 25, 2019

## Transcript

/ Week 5
2. ### 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
3. ### Computing the posterior 1. Analytical approach (often impossible) 2. Grid

approximation (very intensive) 3. Quadratic approximation (limited) 4. Markov chain Monte Carlo (intensive)

6. ### Contract: King Markov must visit each island in proportion to

its population size. Here’s how he does it...
7. ### (1) Flip a coin to choose island on left or

right. Call it the “proposal” island. 1 2 1 2
8. ### (2) Find population of proposal island. (1) Flip a coin

to choose island on left or right. Call it the “proposal” island. proposal
9. ### (2) Find population of proposal island. 1 2 3 4

5 6 7 (1) Flip a coin to choose island on left or right. Call it the “proposal” island. proposal
10. ### (2) Find population of proposal island. 1 2 3 4

5 6 7 p5 (1) Flip a coin to choose island on left or right. Call it the “proposal” island. proposal
11. ### (2) Find population of proposal island. 1 2 3 4

5 6 7 p5 (1) Flip a coin to choose island on left or right. Call it the “proposal” island. (3) Find population of current island. p4 proposal
12. ### (2) Find population of proposal island. 1 2 3 4

5 6 7 p5 (1) Flip a coin to choose island on left or right. Call it the “proposal” island. (3) Find population of current island. p4 (4) Move to proposal, with probability = p5 p4 proposal
13. ### (2) Find population of proposal island. 1 2 3 4

5 6 7 (1) Flip a coin to choose island on left or right. Call it the “proposal” island. (3) Find population of current island. (4) Move to proposal, with probability = p5 p4 (5) Repeat from (1)
14. ### (2) Find population of proposal island. 1 2 3 4

5 6 7 (1) Flip a coin to choose island on left or right. Call it the “proposal” island. (3) Find population of current island. (4) Move to proposal, with probability = p5 p4 (5) Repeat from (1) This procedure ensures visiting each island in proportion to its population, in the long run.
15. ### Markov chain Monte Carlo • Markov chain Monte Carlo (MCMC)

• Understand the approach • Meet different algorithms • Interfaces to MCMC: Stan & ulam • How to sample responsibly • How to recognize and fix problems
16. ### Metropolis algorithm HVBSBOUFFT UIBU UIF LJOH XJMM CF GPVOE PO

FBDI JTMBOE JO QSPQPSUJPO UP JUT QPQVMBUJPO :PV DBO QSPWF UIJT UP ZPVSTFMG CZ TJNVMBUJOH ,JOH .BSLPWT KPVSOFZ )FSFT B TIPS PG DPEF UP EP UIJT TUPSJOH UIF IJTUPSZ PG UIF LJOHT JTMBOE QPTJUJPOT JO UIF WFDUPS +*.\$/ )0(Ǿ2 &. ʚǶ ǎ ǒ +*.\$/\$*). ʚǶ - +ǿǍǢ)0(Ǿ2 &.Ȁ 0-- )/ ʚǶ ǎǍ !*- ǿ \$ \$) ǎǣ)0(Ǿ2 &. Ȁ ȃ ȕ - *- 0-- )/ +*.\$/\$*) +*.\$/\$*).ȁ\$Ȃ ʚǶ 0-- )/ ȕ !'\$+ *\$) /* " ) -/ +-*+*.' +-*+*.' ʚǶ 0-- )/ ʔ .(+' ǿ ǿǶǎǢǎȀ Ǣ .\$5 ʙǎ Ȁ ȕ )*2 (& .0- # '**+. -*0) /# -#\$+ '"* \$! ǿ +-*+*.' ʚ ǎ Ȁ +-*+*.' ʚǶ ǎǍ \$! ǿ +-*+*.' ʛ ǎǍ Ȁ +-*+*.' ʚǶ ǎ ȕ (*1 Ǩ +-*Ǿ(*1 ʚǶ +-*+*.'ȅ0-- )/ 0-- )/ ʚǶ \$! '. ǿ -0)\$!ǿǎȀ ʚ +-*Ǿ(*1 Ǣ +-*+*.' Ǣ 0-- )/ Ȁ Ȅ
17. ### Markov’s chain of visits 0 200 400 600 800 1000

2 4 6 8 10 week island
18. ### Markov’s chain of visits 0 200 400 600 800 1000

2 4 6 8 10 week island
19. ### 2 4 6 8 10 0 5 10 15 20

25 island number of weeks after 100 weeks 2 4 6 8 10 0 20 40 60 80 100 island number of weeks after 500 weeks 2 4 6 8 10 0 100 200 300 400 island number of weeks after 2000 weeks 0 2000 4000 6000 8000 10000 2 4 6 8 10 week island
20. ### Markov’s chain of visits • Converges to correct proportions, in

the long run • No matter which island starts • As long as proposals are symmetric • Example of Metropolis algorithm 2 4 6 8 10 0 500 1000 1500 island number of weeks after 10000 weeks
21. ### Metropolis and MCMC • Usual use is to draw samples

from a posterior distribution • “Islands”: parameter values • “Population size”: proportional to posterior probability • Works for any number of dimensions (parameters) • Works for continuous as well as discrete parameters
22. ### Metropolis and MCMC • Metropolis: Simple version of Markov chain

Monte Carlo (MCMC) • Metropolis, Rosenbluth, Rosenbluth, Teller, and Teller (1953) into radicals (by collision, or possibly radiation, as in aromatic hydrocarbons). (b) The molecules dissociate, but the resulting radi- cals recombine without escaping from the liquid cage. (c) The molecules dissociate and escape from the cage. In this case we would not expect them to move more than a few molecular diameters through the dense medium before being thermalized. In accordance with the notation introduced by Burton, Magee, and Samuel,22 the molecules following 22 Burton, Magee, and Samuel, J. Chern. Phys. 20, 760 (1952). THE JOURNAL OF CHEMICAL PHYSICS tions that the ionized H2 0 molecules will become the H2 0t molecules, but this is not likely to be a complete correspondence. In conclusion we would like to emphasize that the qualitative result of this section is not critically de- pendent on the exact values of the physical parameters used. However, this treatment is classical, and a correct treatment must be wave mechanical; therefore the result of this section cannot be taken as an a priori theoretical prediction. The success of the radical diffu- sion model given above lends some plausibility to the occurrence of electron capture as described by this crude calculation. Further work is clearly needed. VOLUME 21, NUMBER 6 JUNE, 1953 Equation of State Calculations by Fast Computing Machines NICHOLAS METROPOLIS, ARIANNA W. ROSENBLUTH, MARSHALL N. ROSENBLUTH, AND AUGUSTA H. TELLER, Los Alamos Scientific Laboratory, Los Alamos, New Mexico AND EDWARD TELLER, * Department of Physics, University of Chicago, Chicago, Illinois (Received March 6, 1953) A general method, suitable for fast computing machines, for investigatiflg such properties as equations of state for substances consisting of interacting individual molecules is described. The method consists of a modified Monte Carlo integration over configuration space. Results for the two-dimensional rigid-sphere system have been obtained on the Los Alamos MANIAC and are presented here. These results are compared to the free volume equation of state and to a four-term virial coefficient expansion.
23. ### MANIAC: Mathematical Analyzer, Numerical Integrator, and Computer MANIAC: 1000 pounds

5 kilobytes of memory 70k multiplications/sec Your laptop: 4–7 pounds 2–8 million kilobytes Billions of multiplications/sec
24. ### Metropolis and MCMC • Metropolis: Simple version of Markov chain

Monte Carlo (MCMC) • Chain: Sequence of draws from distribution • Markov chain: History doesn’t matter, just where you are now • Monte Carlo: Random simulation Andrei Andreyevich Markov (Ма́рков) (1856–1922)
25. ### Why MCMC? • Sometimes can’t write an integrated posterior •

Even when can, often cannot use it • Many problems are like this: Multilevel models, networks, phylogenies, spatial models • Optimization not a good strategy in high dimensions — must have full distribution • MCMC is not fancy. It is old and essential.
26. ### MCMC strategies • Metropolis: Granddaddy of them all • Metropolis-Hastings

(MH): More general • Gibbs sampling (GS): Efficient version of MH • Metropolis and Gibbs are “guess and check” strategies • Hamiltonian Monte Carlo (HMC) fundamentally different • New methods being developed, but future belongs to the gradient

31. ### Metropolis gets stuck   ."3,07 \$)"*/ .0/5& \$"3-0 -1.5

-1.0 -0.5 0.0 0.5 1.0 1.5 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 a1 a2 step size 0.1, accept rate 0.62 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 a1 a2 step size 0.25, accept rate 0.34 'ĶĴłĿĲ ƑƋ .FUSPQPMJT DIBJOT VOEFS IJHI DPSSFMBUJPO 'JMMFE QPJOUT JOEJ DBUF BDDFQUFE QSPQPTBMT 0QFO QPJOUT BSF SFKFDUFE QSPQPTBMT #PUI QMPUT TIPX  QSPQPTBMT VOEFS EJČFSFOU QSPQPTBM EJTUSJCVUJPO TUFQ TJ[FT -Fę Figure 9.3 small steps, slow walk bigs steps, low accept rate
32. ### Hamiltonian Monte Carlo • Problem with Gibbs sampling (GS) •

High dimension spaces are concentrated • GS gets stuck, degenerates towards random walk • Inefficient because re-explores • Hamiltonian dynamics to the rescue • represent parameter state as particle • flick it around frictionless log-posterior • record positions • no more “guess and check” • all proposals are good proposals William Rowan Hamilton (1805–1865) Commemorated on Irish Euro coin
33. ### Hamiltonian parable • King Monty’s kingdom is a narrow valley

N–S • Population distribution inversely proportional to altitude • Algorithm: • Start driving randomly N or S at random speed • Car speeds up as it goes downhill • Car slows as it goes uphill, might turn around • Drive for pre-specified duration, then stop • Repeat • Stopping positions will be proportional to population North South
34. ### Hamiltonian parable • King Monty’s kingdom is a narrow valley

N–S • Population distribution inversely proportional to altitude • Algorithm: • Start driving randomly N or S at random speed • Car speeds up as it goes downhill • Car slows as it goes uphill, might turn around • Drive for pre-specified duration, then stop • Repeat • Stopping positions will be proportional to population
35. ### 0 100 200 300 time position 0 south north 'ĶĴłĿĲ

Ƒƍ ,JOH .POUZT 3PZBM %SJWF ćF KPVSOFZ CFHJOT BU UJNF  PO UIF GBS MFę ćF WFIJDMF JT HJWFO B SBOEPN NPNFOUVN BOE B SBOEPN EJSFD UJPO FJUIFS OPSUI UPQ PS TPVUI CPUUPN  ćF UIJDLOFTT PG UIF QBUI TIPXT NPNFOUVN BU FBDI NPNFOU ćF WFIJDMF USBWFMT MPTJOH NPNFOUVN VQIJMM PS HBJOJOH JU EPXOIJMM "ęFS B ĕYFE BNPVOU PG UJNF UIFZ TUPQ BOE NBLF Figure 9.5
36. ### Hamiltonian Monte Carlo • Location is parameter values • Really

simulate motion on frictionless surface • Surface is minus-log-posterior • Series of simulations, each starting from previous • Stopping points comprise valid MCMC samples -3 -2 -1 0 1 2 3 1 2 3 4 5 x -log(dnorm(x)) -3 -2 -1 0 1 2 3 0.0 0.1 0.2 0.3 0.4 x dnorm(x)

39. ### Figure 9.6 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -0.3

mux -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -0.3 mux -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 a1 a2 1 2 3 4 Posterior correlation -0.9 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 a1 a2 50 trajectories 'ĶĴłĿĲ ƑƎ ).\$ USBKFDUPSJFT 5PQMFę 8JUI UIF SJHIU DPNCJOBUJPO PG MFBQGSPH TUFQT BOE TUFQ TJ[F UIF JOEJWJEVBM QBUIT KVNQ BSPVOE BOE QSPEVDF IJHIMZ JOEFQFOEFOU TBNQMFT GSPN UIF QPTUFSJPS 5PQSJHIU 8JUI UIF XSPOH

42. ### Hamiltonian Monte Carlo • Why does HMC work much better?

• Doesn’t get stuck — follows gradient • Extra variables (momentum, energy) provide diagnostics • But also requires more • Gradients — curvature of log-posterior • “Mass” of particle • Number of leaps in a single trajectory • Size of individual leaps • These need to be tuned right • Gradients are unique to each model -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -0.3 -0.2 -0. mux 1 3 4 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 a1 a2 1 2 3 4 Posterior correlation -0.9 'ĶĴłĿĲ ƑƎ ).\$ USBKFDUPSJFT 5P MFBQGSPH TUFQT BOE TUFQ TJ[F UIF JOEJ IJHIMZ JOEFQFOEFOU TBNQMFT GSPN UI DPNCJOBUJPO TFRVFOUJBM TBNQMFT DBO DIBJO JO UIF UPQSJHIU XJMM TUJMM XPSL MFę ).\$ SFBMMZ TIJOFT XIFO UIF Q
43. ###   ."3,07 \$)"*/ .0/5& \$"3-0 3 DPEF  ȕ

 ) . /* - /0-) ) "Ƕ'*"Ƕ+-*\$'\$/4 (4Ǒ ʚǶ !0)/\$*)ǿ , Ǣ ʙǍ Ǣ ʙǎ Ǣ &ʙǍ Ǣ ʙǎ Ȁ ȃ (04 ʚǶ ,ȁǎȂ (03 ʚǶ ,ȁǏȂ  ʚǶ .0(ǿ )*-(ǿ4Ǣ(04ǢǎǢ'*"ʙȀ Ȁ ʔ .0(ǿ )*-(ǿ3Ǣ(03ǢǎǢ'*"ʙȀ Ȁ ʔ )*-(ǿ(04ǢǢǢ'*"ʙȀ ʔ )*-(ǿ(03Ǣ&ǢǢ'*"ʙȀ - /0-)ǿ Ƕ Ȁ Ȅ /PX UIF HSBEJFOU GVODUJPO SFRVJSFT UXP QBSUJBM EFSJWBUJWFT -VDLJMZ (BVTTJBO EFSJWBUJWFT BSF WFSZ DMFBO ćF EFSJWBUJWF PG UIF MPHBSJUIN PG BOZ VOJWBSJBUF (BVTTJBO XJUI NFBO B BOE TUBOEBSE EFWJBUJPO C XJUI SFTQFDU UP B JT ∂ MPH /(Z|B, C) ∂B = Z − B C "OE TJODF UIF EFSJWBUJWF PG B TVN JT B TVN PG EFSJWBUJWFT UIJT JT BMM XF OFFE UP XSJUF UIF HSBEJFOUT ∂6 ∂µY = ∂ MPH /(Y|µY, ) ∂µY + ∂ MPH /(µY|, .) ∂µY = J YJ − µY  +  − µY . "OE UIF HSBEJFOU GPS µZ IBT UIF TBNF GPSN /PX JO DPEF GPSN 3 DPEF  ȕ "-\$ )/ !0)/\$*) ȕ )  1 /*- *! +-/\$'  -\$1/\$1 . *!  2\$/# - .+ / /* 1 /*- , (4Ǿ"-Ǒ ʚǶ !0)/\$*)ǿ , Ǣ ʙǍ Ǣ ʙǎ Ǣ &ʙǍ Ǣ ʙǎ Ȁ ȃ (04 ʚǶ ,ȁǎȂ (03 ʚǶ ,ȁǏȂ ǎ ʚǶ .0(ǿ 4 Ƕ (04 Ȁ ʔ ǿ Ƕ (04ȀȅʟǏ ȕȅ(04 Ǐ ʚǶ .0(ǿ 3 Ƕ (03 Ȁ ʔ ǿ& Ƕ (03ȀȅʟǏ ȕȅ(04 - /0-)ǿ ǿ Ƕǎ Ǣ ǶǏ Ȁ Ȁ ȕ ) "/\$1  ) -"4 \$. ) "Ƕ'*"Ƕ+-* Ȅ ȕ / ./ / . /ǡ. ǿǔȀ 4 ʚǶ -)*-(ǿǒǍȀ 3 ʚǶ -)*-(ǿǒǍȀ 3 ʚǶ .ǡ)0( -\$ǿ.' ǿ3ȀȀ 4 ʚǶ .ǡ)0( -\$ǿ.' ǿ4ȀȀ ćF HSBEJFOU GVODUJPO BCPWF JTOU UPP CBE GPS UIJT NPEFM #VU JU DBO CF UFSSJGZJOH GPS B SFBTPOBCMZ DPNQMFY NPEFM ćBU JT XIZ UPPMT MJLF 4UBO CVJME UIF HSBEJFOUT EZOBNJDBMMZ VTJOH UIF NPEFM EFĕOJ UJPO /PX XF BSF SFBEZ UP WJTJU UIF IFBSU 5P VOEFSTUBOE TPNF PG UIF EFUBJMT IFSF ZPV TIPVME SFBE 3BEGPSE /FBMT DIBQUFS JO UIF )BOECPPL PG .BSLPW \$IBJO .POUF \$BSMP "SNFE XJUI UIF MPHQPTUFSJPS BOE HSBEJFOU GVODUJPOT IFSFT UIF DPEF UP QSPEVDF 'ĶĴłĿĲ ƑƎ 3 DPEF  '\$--4ǿ.#+ Ȁ ȕ !*- !)4 --*2.  ʚǶ '\$./ǿȀ ɶ, ʚǶ ǿǶǍǡǎǢǍǡǏȀ +- ʚǶ Ǎǡǐ +'*/ǿ  Ǣ 4'ʙǫ(04ǫ Ǣ 3'ʙǫ(03ǫ Ǣ 3'\$(ʙǿǶ+-Ǣ+-Ȁ Ǣ 4'\$(ʙǿǶ+-Ǣ+-Ȁ Ȁ ./ + ʚǶ ǍǡǍǐ ʚǶ ǎǎ ȕ ǍǡǍǐȅǏǕ !*- Ƕ/0-). ǶǶǶ ǎǎ !*- 2*-&\$)" 3(+' )Ǿ.(+' . ʚǶ Ǒ +/#Ǿ*' ʚǶ *'ǡ'+#ǿǫ'&ǫǢǍǡǒȀ +*\$)/.ǿ ɶ,ȁǎȂ Ǣ ɶ,ȁǏȂ Ǣ +#ʙǑ Ǣ *'ʙǫ'&ǫ Ȁ !*- ǿ \$ \$) ǎǣ)Ǿ.(+' . Ȁ ȃ  ʚǶ Ǐǿ (4Ǒ Ǣ (4Ǿ"-Ǒ Ǣ ./ + Ǣ Ǣ ɶ, Ȁ  )".*-50/*"/ .0/5& \$"3-0  \$! ǿ )Ǿ.(+' . ʚ ǎǍ Ȁ ȃ !*- ǿ % \$) ǎǣ Ȁ ȃ Ǎ ʚǶ .0(ǿɶ+/-%ȁ%ǢȂʟǏȀȅǏ ȕ &\$) /\$ ) -"4 '\$) .ǿ ɶ/-%ȁ%ǣǿ%ʔǎȀǢǎȂ Ǣ ɶ/-%ȁ%ǣǿ%ʔǎȀǢǏȂ Ǣ *'ʙ+/#Ǿ*' Ǣ '2ʙǎʔǏȉ Ǎ Ȁ Ȅ +*\$)/.ǿ ɶ/-%ȁǎǣ ʔǎǢȂ Ǣ +#ʙǎǓ Ǣ *'ʙǫ2#\$/ ǫ Ǣ  3ʙǍǡǐǒ Ȁ --*2.ǿ ɶ/-%ȁ ǢǎȂ Ǣ ɶ/-%ȁ ǢǏȂ Ǣ ɶ/-%ȁ ʔǎǢǎȂ Ǣ ɶ/-%ȁ ʔǎǢǏȂ Ǣ --ǡ' )"/#ʙǍǡǐǒ Ǣ --ǡ% ʙ Ǎǡǔ Ȁ / 3/ǿ ɶ/-%ȁ ʔǎǢǎȂ Ǣ ɶ/-%ȁ ʔǎǢǏȂ Ǣ \$ Ǣ  3ʙǍǡǕ Ǣ +*.ʙǑ Ǣ *!!. /ʙǍǡǑ Ȁ Ȅ +*\$)/.ǿ ɶ/-%ȁ ʔǎǢǎȂ Ǣ ɶ/-%ȁ ʔǎǢǏȂ Ǣ +#ʙ\$! '. ǿ ɶ +/ʙʙǎ Ǣ ǎǓ Ǣ ǎ Ȁ Ǣ *'ʙ\$! '. ǿ .ǿɶ ȀʛǍǡǎ Ǣ ǫ- ǫ Ǣ ǫ'&ǫ Ȁ Ȁ Ȅ ćF GVODUJPO Ǐ JT CVJMU JOUP - /#\$)&\$)" *U JT CBTFE VQPO POF PG 3BEGPSE /FBMT FYBNQMF TDSJQUT *U JTOU BDUVBMMZ UPP DPNQMJDBUFE -FUT UPVS UISPVHI JU POF TUFQ BU B UJNF UP UBLF UIF NBHJD BXBZ ćJT GVODUJPO SVOT B TJOHMF USBKFDUPSZ BOE TP QSPEVDFT B TJOHMF TBNQMF :PV OFFE UP VTF JU SFQFBUFEMZ UP CVJME B DIBJO ćBUT XIBU UIF MPPQ BCPWF EPFT ćF ĕSTU DIVOL PG UIF GVODUJPO DIPPTFT SBOEPN NPNFOUVNUIF ĘJDL PG UIF QBSUJDMFBOE JOJUJBMJ[FT UIF USBKFDUPSZ 3 DPEF  Ǐ ʚǶ !0)/\$*) ǿǢ "-ǾǢ +.\$'*)Ǣ Ǣ 0-- )/Ǿ,Ȁ ȃ , ʙ 0-- )/Ǿ, + ʙ -)*-(ǿ' )"/#ǿ,ȀǢǍǢǎȀ ȕ -)*( !'\$& Ƕ + \$. (*( )/0(ǡ 0-- )/Ǿ+ ʙ + ȕ &  #'! ./ + !*- (*( )/0( / /#  "\$))\$)" + ʙ + Ƕ +.\$'*) ȉ "-Ǿǿ,Ȁ ȅ Ǐ ȕ \$)\$/\$'\$5 **&& +\$)" Ƕ .1 . /-% /*-4 ,/-% ʚǶ (/-\$3ǿǢ)-*2ʙ ʔǎǢ)*'ʙ' )"/#ǿ,ȀȀ +/-% ʚǶ ,/-% ,/-%ȁǎǢȂ ʚǶ 0-- )/Ǿ, +/-%ȁǎǢȂ ʚǶ + ćFO UIF BDUJPO DPNFT JO B MPPQ PWFS MFBQGSPH TUFQT TUFQT BSF UBLFO VTJOH UIF HSBEJFOU UP DPNQVUF B MJOFBS BQQSPYJNBUJPO PG UIF MPHQPTUFSJPS TVSGBDF BU FBDI QPJOU 3 DPEF  ȕ '/ -)/ !0'' ./ +. !*- +*.\$/\$*) ) (*( )/0( !*- ǿ \$ \$) ǎǣ Ȁ ȃ , ʙ , ʔ +.\$'*) ȉ + ȕ 0'' ./ + !*- /# +*.\$/\$*) ȕ &  !0'' ./ + !*- /# (*( )/0(Ǣ 3 +/ / ) *! /-% /*-4 \$! ǿ \$Ǧʙ Ȁ ȃ + ʙ + Ƕ +.\$'*) ȉ "-Ǿǿ,Ȁ +/-%ȁ\$ʔǎǢȂ ʚǶ + Ȅ ,/-%ȁ\$ʔǎǢȂ ʚǶ , Ȅ /PUJDF IPX UIF TUFQ TJ[F +.\$'*) JT BEEFE UP UIF QPTJUJPO BOE NPNFOUVN WFDUPST *U JT JO UIJT XBZ UIBU UIF QBUI JT POMZ BO BQQSPYJNBUJPO CFDBVTF JU JT B TFSJFT PG MJOFBS KVNQT OPU BO BDUVBM TNPPUI DVSWF ćJT DBO IBWF JNQPSUBOU DPOTFRVFODFT JG UIF MPHQPTUFSJPS CFOET TIBSQMZ BOE UIF TJNVMBUJPO KVNQT PWFS B CFOE "MM UIBU SFNBJOT JT DMFBO VQ FOTVSF UIF QSPQPTBM JT TZNNFUSJD TP UIF .BSLPW DIBJO JT WBMJE BOE EFDJEF XIFUIFS UP BDDFQU PS SFKFDU UIF QSPQPTBM 3 DPEF  ȕ &  #'! ./ + !*- (*( )/0( / /# ) + ʙ + Ƕ +.\$'*) ȉ "-Ǿǿ,Ȁ ȅ Ǐ +/-%ȁ ʔǎǢȂ ʚǶ +
44. ### Figure 9.6  )".*-50/*"/ .0/5& \$"3-0  -0.3 -0.2 -0.1

0.0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 mux muy 1 2 3 4 2D Gaussian, L = 11 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 mux muy 2D Gaussian, L = 28 1.5 Posterior correlation -0.9 1.5 50 trajectories The U-Turn Problem

47. ### Stan is NUTS • No U-Turn Sampler (NUTS2): Adaptive Hamiltonian

Monte Carlo • Implemented in Stan (rstan: mc-stan.org) • Stan figures out gradient for you • autodiff, back-propagation Formula Stan model C++ model Reusable Sampler

54. ### HMC Praxis • Back to terrain ruggedness • Re-approximate posterior

• Practical details • What to expect from healthy Markov chains
55. ### One hand QUAP’ing ɶ-0"" Ǿ./ ʚǶ ɶ-0""  ȅ (3ǿɶ-0""

Ȁ ɶ\$ ʚǶ \$! '. ǿ ɶ*)/Ǿ!-\$ʙʙǎ Ǣ ǎ Ǣ Ǐ Ȁ 4P ZPV SFNFNCFS UIF PME XBZ XFSF HPJOH UP SFQFBU UIF QSPDFEVSF GPS ĕUUJOH UIF JOUFSBDUJPO NPEFM ćJT NPEFM BJNT UP QSFEJDU MPH (%1 XJUI UFSSBJO SVHHFEOFTT DPOUJOFOU BOE UIF JOUFSBDUJPO PG UIF UXP )FSFT UIF XBZ UP EP JU XJUI ,0+ KVTU MJLF CFGPSF 3 DPEF  (Ǖǡǒ ʚǶ ,0+ǿ '\$./ǿ '*"Ǿ"+Ǿ./ ʡ )*-(ǿ (0 Ǣ .\$"( Ȁ Ǣ (0 ʚǶ ȁ\$Ȃ ʔ ȁ\$Ȃȉǿ -0"" Ǿ./ Ƕ ǍǡǏǎǒ Ȁ Ǣ ȁ\$Ȃ ʡ )*-(ǿ ǎ Ǣ Ǎǡǎ Ȁ Ǣ ȁ\$Ȃ ʡ )*-(ǿ Ǎ Ǣ Ǎǡǐ Ȁ Ǣ .\$"( ʡ  3+ǿ ǎ Ȁ Ȁ Ǣ /ʙ Ȁ +- \$.ǿ (Ǖǡǒ Ǣ  +/#ʙǏ Ȁ ( ) . ǒǡǒʉ ǖǑǡǒʉ ȁǎȂ ǍǡǕǖ ǍǡǍǏ ǍǡǕǓ Ǎǡǖǎ ȁǏȂ ǎǡǍǒ ǍǡǍǎ ǎǡǍǐ ǎǡǍǔ ȁǎȂ Ǎǡǎǐ ǍǡǍǔ ǍǡǍǎ ǍǡǏǒ ȁǏȂ ǶǍǡǎǑ ǍǡǍǒ ǶǍǡǏǐ ǶǍǡǍǓ .\$"( Ǎǡǎǎ ǍǡǍǎ ǍǡǎǍ ǍǡǎǏ +VTU BT ZPV TBX JO UIF QSFWJPVT DIBQUFS  1SFQBSBUJPO #VU OPX XFMM BMTP ĕU UIJT NPEFM VTJOH )BNJMUPOJBO .POUF \$BSMP ćJT NFBOT UIFSF XJMM CF OP NPSF RVBESBUJD BQQSPYJNBUJPOJG UIF QPTUFSJPS EJTUSJCVUJPO JT OPO
56. ###   ."3,07 \$)"*/ .0/5& \$"3-0 EPOU IBWF UP EP

UIJT #VU EPJOH TP BWPJET DPNNPO QSPCMFNT 'PS FYBNQMF JG BOZ PG UIF VOVTFE WBSJBCMFT IBWF NJTTJOH WBMVFT  UIFO 4UBO XJMM SFGVTF UP XPSL 8FWF BMSFBEZ QSFUSBOTGPSNFE BMM UIF WBSJBCMFT /PX XF OFFE B TMJN MJTU PG UIF WBSJBCMFT XF XJMM VTF DPEF  /Ǿ.'\$( ʚǶ '\$./ǿ '*"Ǿ"+Ǿ./ ʙ ɶ'*"Ǿ"+Ǿ./Ǣ -0"" Ǿ./ ʙ ɶ-0"" Ǿ./Ǣ \$ ʙ .ǡ\$)/ " -ǿ ɶ\$ Ȁ Ȁ ./-ǿ/Ǿ.'\$(Ȁ \$./ *! ǐ ɶ '*"Ǿ"+Ǿ./ǣ )0( ȁǎǣǎǔǍȂ ǍǡǕǕ ǍǡǖǓǒ ǎǡǎǓǓ ǎǡǎǍǑ Ǎǡǖǎǒ ǡǡǡ ɶ -0"" Ǿ./ ǣ )0( ȁǎǣǎǔǍȂ ǍǡǎǐǕ Ǎǡǒǒǐ ǍǡǎǏǑ ǍǡǎǏǒ ǍǡǑǐǐ ǡǡǡ ɶ \$ ǣ \$)/ ȁǎǣǎǔǍȂ ǎ Ǐ Ǐ Ǐ Ǐ Ǐ Ǐ Ǐ Ǐ ǎ ǡǡǡ *U JT CFUUFS UP VTF B '\$./ UIBO B /ǡ!-( CFDBVTF UIF FMFNFOUT JO B '\$./ DBO CF BOZ MFOHUI *O B /ǡ!-( BMM UIF FMFNFOUT NVTU CF UIF TBNF MFOHUI 8JUI TPNF NPEFMT UP DPNF MBUFS MJLF NVMUJMFWFM NPEFMT JU JTOU VOVTVBM UP IBWF WBSJBCMFT PG EJČFSFOU MFOHUIT  4BNQMJOH GSPN UIF QPTUFSJPS /PX QSPWJEFE ZPV IBWF UIF -./) QBDLBHF JOTUBMMFE NDTUBOPSH ZPV DBO HFU TBNQMFT GSPN UIF QPTUFSJPS EJTUSJCVUJPO XJUI UIJT DPEF DPEF  (ǖǡǎ ʚǶ 0'(ǿ Hamiltonian Flows • Interface to Stan: ulam QMFT ZPV NBOBHFE UP HFU #/ JT B DPNQMJDBUFE FTUJNBUF PG UIF DPOWFSHFODF PG UIF .BSLPW DIBJOT UP UIF UBSHFU EJTUSJCVUJPO *U TIPVME BQQSPBDI  GSPN BCPWF XIFO BMM JT XFMM  4BNQMJOH BHBJO JO QBSBMMFM ćF FYBNQMF TP GBS JT B WFSZ FBTZ QSPCMFN GPS .\$.\$ 4P FWFO UIF EFGBVMU  TBNQMFT JT FOPVHI GPS BDDVSBUF JOGFSFODF *O GBDU BT GFX BT  FČFDUJWF TBNQMFT JT VTVBMMZ QMFOUZ GPS B HPPE BQQSPYJNBUJPO PG UIF QPTUFSJPS #VU XF BMTP XBOU UP SVO NVMUJQMF DIBJOT GPS SFBTPOT XFMM EJTDVTT JO NPSF EFQUI JO UIF OFYU TFDUJPOT ćFSF XJMM CF TQFDJĕD BEWJDF JO 4FDUJPO  QBHF   'PS OPX JUT XPSUI OPUJOH UIBU ZPV DBO FBTJMZ QBSBMMFMJ[F UIPTF DIBJOT BT XFMM ćFZ DBO BMM SVO BU UIF TBNF UJNF JOTUFBE PG JO TFRVFODF 4P BT MPOH BT ZPVS DPNQVUFS IBT GPVS DPSFT JU QSPCBCMZ EPFT JU XPOU UBLF MPOHFS UP SVO GPVS DIBJOT UIBO POF DIBJO 5P SVO GPVS JOEFQFOEFOU .BSLPW DIBJOT GPS UIF NPEFM BCPWF BOE UP EJTUSJCVUF UIFN BDSPTT TFQBSBUF DPSFT JO ZPVS DPNQVUFS KVTU JODSFBTF UIF OVNCFS PG #\$). BOE BEE B *- . BSHVNFOU 3 DPEF  (ǖǡǎ ʚǶ 0'(ǿ '\$./ǿ '*"Ǿ"+Ǿ./ ʡ )*-(ǿ (0 Ǣ .\$"( Ȁ Ǣ (0 ʚǶ ȁ\$Ȃ ʔ ȁ\$Ȃȉǿ -0"" Ǿ./ Ƕ ǍǡǏǎǒ Ȁ Ǣ ȁ\$Ȃ ʡ )*-(ǿ ǎ Ǣ Ǎǡǎ Ȁ Ǣ ȁ\$Ȃ ʡ )*-(ǿ Ǎ Ǣ Ǎǡǐ Ȁ Ǣ .\$"( ʡ  3+ǿ ǎ Ȁ Ȁ Ǣ /ʙ/Ǿ.'\$( Ǣ #\$).ʙǑ Ǣ *- .ʙǑ Ǣ \$/ -ʙǎǍǍǍ Ȁ ćFSF BSF B CVODI PG PQUJPOBM BSHVNFOUT UIBU BMMPX VT UP UVOF BOE DVTUPNJ[F UIF QSPDFTT 8FMM CSJOH UIFN VQ BT UIFZ BSF OFFEFE 'PS OPX LFFQ JO NJOE UIBU .#*2 XJMM SFNJOE ZPV PG UIF NPEFM GPSNVMB BOE BMTP IPX MPOH FBDI DIBJO UPPL UP SVO
57. ### Hamiltonian Flows • What happens when you use ulam? •

Translates formula into raw Stan model code • Stan then builds a custom NUTS sampler • Sampler runs • Samples fed back to R ćFTF FTUJNBUFT BSF WFSZ TJNJMBS UP UIF RVBESBUJD BQQSPYJNBUJPO #VU OPUF UIBU UIFSF BSF UXP OFX DPMVNOT )Ǿ !! BOE #/ ćFTF DPMVNOT QSPWJEF .\$.\$ EJBHOPTUJD DSJUFSJB UP IFMQ ZPV UFMM IPX XFMM UIF TBNQMJOH XPSLFE 8FMM EJTDVTT UIFN JO EFUBJM MBUFS JO UIF DIBQUFS 'PS OPX JUT FOPVHI UP LOPX UIBU )Ǿ !! JT B DSVEF FTUJNBUF PG UIF OVNCFS PG JOEFQFOEFOU TBN QMFT ZPV NBOBHFE UP HFU #/ JT B DPNQMJDBUFE FTUJNBUF PG UIF DPOWFSHFODF PG UIF .BSLPW DIBJOT UP UIF UBSHFU EJTUSJCVUJPO *U TIPVME BQQSPBDI  GSPN BCPWF XIFO BMM JT XFMM  4BNQMJOH BHBJO JO QBSBMMFM ćF FYBNQMF TP GBS JT B WFSZ FBTZ QSPCMFN GPS .\$.\$ 4P FWFO UIF EFGBVMU  TBNQMFT JT FOPVHI GPS BDDVSBUF JOGFSFODF *O GBDU BT GFX BT  FČFDUJWF TBNQMFT JT VTVBMMZ QMFOUZ GPS B HPPE BQQSPYJNBUJPO PG UIF QPTUFSJPS #VU XF BMTP XBOU UP SVO NVMUJQMF DIBJOT GPS SFBTPOT XFMM EJTDVTT JO NPSF EFQUI JO UIF OFYU TFDUJPOT ćFSF XJMM CF TQFDJĕD BEWJDF JO 4FDUJPO  QBHF   'PS OPX JUT XPSUI OPUJOH UIBU ZPV DBO FBTJMZ QBSBMMFMJ[F UIPTF DIBJOT BT XFMM ćFZ DBO BMM SVO BU UIF TBNF UJNF JOTUFBE PG JO TFRVFODF 4P BT MPOH BT ZPVS DPNQVUFS IBT GPVS DPSFT JU QSPCBCMZ EPFT JU XPOU UBLF MPOHFS UP SVO GPVS DIBJOT UIBO POF DIBJO 5P SVO GPVS JOEFQFOEFOU .BSLPW DIBJOT GPS UIF NPEFM BCPWF BOE UP EJTUSJCVUF UIFN BDSPTT TFQBSBUF DPSFT JO ZPVS DPNQVUFS KVTU JODSFBTF UIF OVNCFS PG #\$). BOE BEE B *- . BSHVNFOU 3 DPEF  (ǖǡǎ ʚǶ 0'(ǿ '\$./ǿ '*"Ǿ"+Ǿ./ ʡ )*-(ǿ (0 Ǣ .\$"( Ȁ Ǣ (0 ʚǶ ȁ\$Ȃ ʔ ȁ\$Ȃȉǿ -0"" Ǿ./ Ƕ ǍǡǏǎǒ Ȁ Ǣ ȁ\$Ȃ ʡ )*-(ǿ ǎ Ǣ Ǎǡǎ Ȁ Ǣ ȁ\$Ȃ ʡ )*-(ǿ Ǎ Ǣ Ǎǡǐ Ȁ Ǣ .\$"( ʡ  3+ǿ ǎ Ȁ Ȁ Ǣ /ʙ/Ǿ.'\$( Ǣ #\$).ʙǑ Ǣ *- .ʙǑ Ǣ \$/ -ʙǎǍǍǍ Ȁ ćFSF BSF B CVODI PG PQUJPOBM BSHVNFOUT UIBU BMMPX VT UP UVOF BOE DVTUPNJ[F UIF QSPDFTT 8FMM CSJOH UIFN VQ BT UIFZ BSF OFFEFE 'PS OPX LFFQ JO NJOE UIBU .#*2 XJMM SFNJOE ZPV PG
58. ### Hamiltonian Flows /ʙ/Ǿ.'\$( Ǣ #\$).ʙǑ Ǣ *- .ʙǑ Ǣ \$/

-ʙǎǍǍǍ Ȁ ćFSF BSF B CVODI PG PQUJPOBM BSHVNFOUT UIBU BMMPX VT UP UVOF BOE DVTUPNJ[ 8FMM CSJOH UIFN VQ BT UIFZ BSF OFFEFE 'PS OPX LFFQ JO NJOE UIBU .#*2 XJMM UIF NPEFM GPSNVMB BOE BMTP IPX MPOH FBDI DIBJO UPPL UP SVO .#*2ǿ (ǖǡǎ Ȁ (\$'/*)\$) *)/ -'* ++-*3\$(/\$*) ǏǍǍǍ .(+' . !-*( Ǒ #\$). (+'\$)" 0-/\$*). ǿ. *).Ȁǣ 2-(0+ .(+' /*/' #\$)ǣǎ ǍǡǎǏ ǍǡǍǕ ǍǡǏǍ #\$)ǣǏ ǍǡǎǏ ǍǡǍǕ Ǎǡǎǖ #\$)ǣǐ ǍǡǎǏ ǍǡǍǕ ǍǡǏǍ #\$)ǣǑ ǍǡǎǏ ǍǡǍǕ ǍǡǏǍ *-(0'ǣ '*"Ǿ"+Ǿ./ ʡ )*-(ǿ(0Ǣ .\$"(Ȁ (0 ʚǶ ȁ\$Ȃ ʔ ȁ\$Ȃ ȉ ǿ-0"" Ǿ./ Ƕ ǍǡǏǎǒȀ (0 ʚǶ ȁ\$Ȃ ʔ ȁ\$Ȃȉǿ -0"" Ǿ./ Ƕ ǍǡǏǎǒ Ȁ Ǣ ȁ\$Ȃ ʡ )*-(ǿ ǎ Ǣ Ǎǡǎ Ȁ Ǣ ȁ\$Ȃ ʡ )*-(ǿ Ǎ Ǣ Ǎǡǐ Ȁ Ǣ .\$"( ʡ  3+ǿ ǎ Ȁ Ȁ Ǣ /ʙ/Ǿ.'\$( Ǣ #\$).ʙǑ Ǣ *- .ʙǑ Ǣ \$/ -ʙǎǍǍǍ Ȁ ćFSF BSF B CVODI PG PQUJPOBM BSHVNFOUT UIBU BMMPX VT UP UVOF BOE DVTUPNJ[F UI 8FMM CSJOH UIFN VQ BT UIFZ BSF OFFEFE 'PS OPX LFFQ JO NJOE UIBU .#*2 XJMM SFN UIF NPEFM GPSNVMB BOE BMTP IPX MPOH FBDI DIBJO UPPL UP SVO .#*2ǿ (ǖǡǎ Ȁ (\$'/*)\$) *)/ -'* ++-*3\$(/\$*) ǏǍǍǍ .(+' . !-*( Ǒ #\$). (+'\$)" 0-/\$*). ǿ. *).Ȁǣ 2-(0+ .(+' /*/' #\$)ǣǎ ǍǡǎǏ ǍǡǍǕ ǍǡǏǍ #\$)ǣǏ ǍǡǎǏ ǍǡǍǕ Ǎǡǎǖ #\$)ǣǐ ǍǡǎǏ ǍǡǍǕ ǍǡǏǍ #\$)ǣǑ ǍǡǎǏ ǍǡǍǕ ǍǡǏǍ • Num samples = total minus warmup • Default warmup is half
59. ### Hamiltonian Flows • n_eff: number of effective samples • can

be larger than actual samples! • Rhat: Convergence diagnostic — “1” is good ȁ\$Ȃ ʡ )*-(ǿǍǢ ǍǡǐȀ .\$"( ʡ  3+ǿǎȀ ćFSF XFSF  TBNQMFT GSPN BMM  DIBJOT CFDBVTF FBDI  TBNQMF DIBJO ĕTU IBMG PG UIF TBNQMFT UP BEBQU 4PNFUIJOH DVSJPVT IBQQFOT XIFO XF MPP 3 DPEF  +- \$.ǿ (ǖǡǎ Ǣ Ǐ Ȁ ( ) . ǒǡǒʉ ǖǑǡǒʉ )Ǿ !! #/ .\$"( Ǎǡǎǎ ǍǡǍǎ ǍǡǎǍ ǍǡǎǏ ǏǓǑǎ ǎ ȁǎȂ Ǎǡǎǐ ǍǡǍǔ ǍǡǍǏ ǍǡǏǓ ǏǑǕǑ ǎ ȁǏȂ ǶǍǡǎǑ ǍǡǍǒ ǶǍǡǏǐ ǶǍǡǍǓ ǏǒǑǓ ǎ ȁǎȂ ǍǡǕǖ ǍǡǍǏ ǍǡǕǓ Ǎǡǖǎ ǐǐǐǎ ǎ ȁǏȂ ǎǡǍǒ ǍǡǍǎ ǎǡǍǐ ǎǡǍǔ ǐǏǑǐ ǎ *G UIFSF XFSF POMZ  TBNQMFT JO UPUBM IPX XF DBO IBWF NPSF UIBO  GPS FBDI QBSBNFUFS *UT OP NJTUBLF ćF BEBQUJWF /654 TBNQMFS UIBU 4UB JU DBO BDUVBMMZ QSPEVDF TFRVFOUJBM TBNQMFT UIBU BSF CFUUFS UIBO VODPSSFMB DBO FYQMPSF UIF QPTUFSJPS EJTUSJCVUJPO TP FďDJFOUMZ UIBU JU DBO CFBU SBOEP JO BDUJPO  7JTVBMJ[BUJPO #Z QMPUUJOH UIF TBNQMFT ZPV DBO HFU B EJSFDU BQQ (BVTTJBO RVBESBUJD UIF BDUVBM QPTUFSJPS EFOTJUZ IBT UVSOFE PVU UP CF 6TF
60. ### Figure 9.7 sigma -0.1 0.1 0.3 0.84 0.88 0.92 0.09

0.11 0.13 -0.1 0.1 0.3 -0.03 b.1 0 0 b.2 -0.30 -0.15 0.00 0.84 0.88 0.92 0 0.11 0 a.1 0.09 0.11 0.13 -0.08 0.06 -0.30 -0.15 0.00 -0.05 0.04 1.01 1.04 1.07 1.01 1.04 1.07 a.2 UIF NPEFM PCKFDU TP UIBU 3 LOPXT UP EJTQMBZ QBSBNFUFS OBNFT BOE QBSBNFUFS DPSSF EF  +\$-.ǿ (ǖǡǎ Ȁ 'ĶĴłĿĲ ƑƏ TIPXT UIF SFTVMUJOH QMPU ćJT JT B QBJST QMPU TP JUT TUJMM B NBUSJY PG CJWBSJBU QMPUT #VU OPX BMPOH UIF EJBHPOBM UIF TNPPUIFE IJTUPHSBN PG FBDI QBSBNFUFS JT TIPX XJUI JUT OBNF "OE JO UIF MPXFS USJBOHMF PG UIF NBUSJY CFMPX UIF EJBHPOBM UIF DPS CFUXFFO FBDI QBJS PG QBSBNFUFST JT TIPXO XJUI TUSPOHFS DPSSFMBUJPOT JOEJDBUFE CZ TJ[F 'PS UIJT NPEFM BOE UIFTF EBUB UIF SFTVMUJOH QPTUFSJPS EJTUSJCVUJPO JT RVJUF OFBSM WBSJBUF (BVTTJBO ćF EFOTJUZ GPS .\$"( JT DFSUBJOMZ TLFXFE JO UIF FYQFDUFE EJSFDUJ PUIFSXJTF UIF RVBESBUJD BQQSPYJNBUJPO EPFT BMNPTU BT XFMM BT )BNJMUPOJBO .POU ćJT JT B WFSZ TJNQMF LJOE PG NPEFM TUSVDUVSF PG DPVSTF XJUI (BVTTJBO QSJPST TP BO NBUFMZ RVBESBUJD QPTUFSJPS TIPVME CF OP TVSQSJTF -BUFS XFMM TFF TPNF NPSF FYPUJD Q EJTUSJCVUJPOT 0WFSUIJOLJOH 4UBO NFTTBHFT 8IFO ZPV ĕU B NPEFM VTJOH (+Ǐ./) 3 XJMM ĕSTU USBOT NPEFM GPSNVMB JOUP B 4UBO MBOHVBHF NPEFM ćFO JU TFOET UIBU NPEFM UP 4UBO ćF NFTT TFF JO ZPVS 3 DPOTPMF BSF TUBUVT VQEBUFT GSPN 4UBO 4UBO ĕSTU BHBJO USBOTMBUFT UIF NPEFM JOUP \$ DPEF ćBU DPEF JT UIFO TFOU UP B \$ DPNQJMFS UP CVJME BO FYFDVUBCMF ĕMF JT B TQ TBNQMJOH FOHJOF GPS ZPVS NPEFM ćFO 4UBO GFFET UIF EBUB BOE TUBSUJOH WBMVFT UP UIBU FYFDVU BOE JG BMM HPFT XFMM TBNQMJOH CFHJOT :PV XJMM TFF 4UBO DPVOU UISPVHI UIF JUFSBUJPOT %VSJOH T ZPV NJHIU PDDBTJPOBMMZ TFF B TDBSZ MPPLJOH XBSOJOH TPNFUIJOH MJLF UIJT
61. ### Check the chains • Sometimes it doesn’t work • Good

chains: • Converge to same target distribution • Once there, explore efficiently • Different ways to check • Trace plots • Convergence diagnostics (n_eff, Rhat) • Special warnings (divergent transitions)
62. ### Check the chains • Trace plot: Plot of sequential samples

in chain • Shows some problems, but not all • Want to see “hairy caterpillar”  &"4: ).\$ ƽ  'ĶĴłĿĲ ƐƎ 5SBDF QMPU PG UIF .BSLPW DIBJO GSPN UIF SVHHFEOFTT NPEFM warmup 'PS OPX MFUT NFFU UIF NPTU CSPBEMZ VTFGVM UPPM GPS EJBHOPT ĽĹļŁ " USBDF QMPU NFSFMZ QMPUT UIF TBNQMFT JO TFRVFOUJBM PSEFS .BSLPWT QBUI UISPVHI UIF JTMBOET JO UIF NFUBQIPS BU UIF TUBSU PG U USBDF QMPU PG FBDI QBSBNFUFS JT PęFO UIF CFTU UIJOH GPS EJBHOPTJOH PODF ZPV DPNF UP SFDPHOJ[F B IFBMUIZ GVODUJPOJOH .BSLPW DIBJO QSPWJEF B MPU PG QFBDF PG NJOE " USBDF QMPU JTOU UIF MBTU UIJOH BOB PVUQVU #VU JUT OFBSMZ BMXBZT UIF ĕSTU *O UIF UFSSBJO SVHHFEOFTT FYBNQMF UIF USBDF QMPU TIPXT B WF XJUI 3 DPEF  /- +'*/ǿ (ǖǡǎ Ȁ ćF SFTVMU JT TIPXO JO 'ĶĴłĿĲ ƑƐ "DUVBMMZ UIF ĕHVSF TIPXT UIF U :PV DBO HFU UIJT CZ BEEJOH #\$).ʙǎ UP UIF DBMM :PV DBO UIJOL FBDI QBSBNFUFS BT UIF QBUI UIF DIBJO UPPL UISPVHI FBDI EJNFOTJP
63. ### 500 1000 1500 2000 8.8 9.0 9.2 9.4 9.6 n_eff

= 267 a 500 1000 1500 2000 −0.4 −0.2 0.0 n_eff = 265 bR 500 1000 1500 2000 −2.5 −2.0 −1.5 n_eff = 296 bA 500 1000 1500 2000 0.0 0.2 0.4 0.6 0.8 n_eff = 294 bAR 500 1000 1500 2000 0.80 0.90 1.00 1.10 n_eff = 471 sigma “Hairy caterpillar ocular inspection test”
64. ### Warmup • What is “warmup”? • Adaptation to posterior for

efficient sampling • Figures out good step size • Samples during warmup NOT from posterior • Automatically discarded by precis/summary and other functions • Warmup is NOT “burn in” warmup   ."3,07 \$)"*/ 500 1000 1500 2000 8.8 9.0 9.2 9.4 9.6 n_eff = 267 a 500 1000 1500 2000 −2.5 −2.0 −1.5 n_eff = 296 bA .80 0.90 1.00 1.10 n_eff = 471 sigma
65. ###   ."3,07 \$)"*/ .0/5& \$"3-0 ȁ\$Ȃ ʡ )*-(ǿǍǢ ǍǡǐȀ

.\$"( ʡ  3+ǿǎȀ ćFSF XFSF  TBNQMFT GSPN BMM  DIBJOT CFDBVTF FBDI  TBNQMF DIBJO VTFT CZ EFBVMU U ĕTU IBMG PG UIF TBNQMFT UP BEBQU 4PNFUIJOH DVSJPVT IBQQFOT XIFO XF MPPL BU UIF TVNNBS 3 DPEF  +- \$.ǿ (ǖǡǎ Ǣ Ǐ Ȁ ( ) . ǒǡǒʉ ǖǑǡǒʉ )Ǿ !! #/ .\$"( Ǎǡǎǎ ǍǡǍǎ ǍǡǎǍ ǍǡǎǏ ǏǓǑǎ ǎ ȁǎȂ Ǎǡǎǐ ǍǡǍǔ ǍǡǍǏ ǍǡǏǓ ǏǑǕǑ ǎ ȁǏȂ ǶǍǡǎǑ ǍǡǍǒ ǶǍǡǏǐ ǶǍǡǍǓ ǏǒǑǓ ǎ ȁǎȂ ǍǡǕǖ ǍǡǍǏ ǍǡǕǓ Ǎǡǖǎ ǐǐǐǎ ǎ ȁǏȂ ǎǡǍǒ ǍǡǍǎ ǎǡǍǐ ǎǡǍǔ ǐǏǑǐ ǎ *G UIFSF XFSF POMZ  TBNQMFT JO UPUBM IPX XF DBO IBWF NPSF UIBO  FČFDUJWF TBNQ Convergence diagnostics • n_eff: “effective” number of samples • n_eff/n < 0.1, be alarmed • R-hat • R-hat: crudely, ratio of variance between chains to variance within chains • Should approach 1 • Both may mislead
66. ### A wild chain • Two observations: {–1,1} • Estimate mean

and standard deviation DBO SFBDI  FWFO GPS BO JOWBMJE DIBJO 4P WJFX JU QFSIBQT BT B TJHOBM PG EBOHFS CVU OFWFS PG TBGFUZ 'PS DPOWFOUJPOBM NPEFMT UIFTF NFUSJDT UZQJDBMMZ XPSL XFMM  5BNJOH B XJME DIBJO 0OF DPNNPO QSPCMFN XJUI TPNF NPEFMT JT UIBU UIFSF BSF CSPBE ĘBU SFHJPOT PG UIF QPTUFSJPS EFOTJUZ ćJT IBQQFOT NPTU PęFO BT ZPV NJHIU HVFTT XIFO POF VTFT ĘBU QSJPST ćF QSPCMFN UIJT DBO HFOFSBUF JT B XJME XBOEFSJOH .BSLPW DIBJO UIBU FSSBUJDBMMZ TBNQMFT FYUSFNFMZ QPTJUJWF BOE FYUSFNFMZ OFHBUJWF QBSBNFUFS WBMVFT -FUT MPPL BU B TJNQMF FYBNQMF ćF DPEF CFMPX USJFT UP FTUJNBUF UIF NFBO BOE TUBOEBSE EFWJBUJPO PG UIF UXP (BVTTJBO PCTFSWBUJPOT − BOE  #VU JU VTFT UPUBMMZ ĘBU QSJPST 3 DPEF  4 ʚǶ ǿǶǎǢǎȀ . /ǡ. ǿǎǎȀ (ǖǡǏ ʚǶ 0'(ǿ '\$./ǿ 4 ʡ )*-(ǿ (0 Ǣ .\$"( Ȁ Ǣ (0 ʚǶ '+# Ǣ '+# ʡ )*-(ǿ Ǎ Ǣ ǎǍǍǍ Ȁ Ǣ .\$"( ʡ  3+ǿ ǍǡǍǍǍǎ Ȁ Ȁ Ǣ /ʙ'\$./ǿ4ʙ4Ȁ Ǣ #\$).ʙǏ Ȁ /PX MFUT MPPL BU UIF +- \$. PVUQVU 3 DPEF  +- \$.ǿ (ǖǡǏ Ȁ ( ) . ǒǡǒʉ ǖǑǡǒʉ )Ǿ !! #/
67. ### A wild chain CSPBE ĘBU SFHJPOT PG UIF QPTUFSJPS EFOTJUZ

ćJT IBQQFOT NPTU PęFO BT ZPV NJHIU HVFTT XIFO POF VTFT ĘBU QSJPST ćF QSPCMFN UIJT DBO HFOFSBUF JT B XJME XBOEFSJOH .BSLPW DIBJO UIBU FSSBUJDBMMZ TBNQMFT FYUSFNFMZ QPTJUJWF BOE FYUSFNFMZ OFHBUJWF QBSBNFUFS WBMVFT -FUT MPPL BU B TJNQMF FYBNQMF ćF DPEF CFMPX USJFT UP FTUJNBUF UIF NFBO BOE TUBOEBSE EFWJBUJPO PG UIF UXP (BVTTJBO PCTFSWBUJPOT − BOE  #VU JU VTFT UPUBMMZ ĘBU QSJPST 3 DPEF  4 ʚǶ ǿǶǎǢǎȀ . /ǡ. ǿǎǎȀ (ǖǡǏ ʚǶ 0'(ǿ '\$./ǿ 4 ʡ )*-(ǿ (0 Ǣ .\$"( Ȁ Ǣ (0 ʚǶ '+# Ǣ '+# ʡ )*-(ǿ Ǎ Ǣ ǎǍǍǍ Ȁ Ǣ .\$"( ʡ  3+ǿ ǍǡǍǍǍǎ Ȁ Ȁ Ǣ /ʙ'\$./ǿ4ʙ4Ȁ Ǣ #\$).ʙǏ Ȁ /PX MFUT MPPL BU UIF +- \$. PVUQVU 3 DPEF  +- \$.ǿ (ǖǡǏ Ȁ ( ) . ǒǡǒʉ ǖǑǡǒʉ )Ǿ !! #/ .\$"( ǒǖǓǡǖǖ ǎǒǑǏǡǕǒ Ǐǡǎǔ ǏǑǓǎǡǑǑ ǒǔ ǎǡǍǓ '+# ǶǒǖǡǑǒ ǐǒǒǡǒǐ ǶǕǎǎǡǑǒ ǐǒǖǡǑǖ ǐǏ ǎǡǍǒ 8IPB ćJT QPTUFSJPS DBOU CF SJHIU ćF NFBO PG − BOE  JT [FSP TP XFSF IPQJOH UP HFU B NFBO WBMVF GPS '+# BSPVOE [FSP *OTUFBE XF HFU DSB[Z WBMVFT BOE JNQMBVTJCMZ XJEF JO UFSWBMT *OGFSFODF GPS .\$"( JT OP CFUUFS ćF )Ǿ !! BOE #/ EJBHOPTUJDT EPOU MPPL HPPE FJUIFS 8F IBWF  TBNQMFT UP XPSL XJUI IFSF CVU UIF FTUJNBUFE FČFDUJWF TBNQMF TJ[FT BSF XIFO POF VTFT ĘBU QSJPST ćF QSPCMFN UIJT DBO HFOFSBUF JT B XJME XBOEFSJOH .BSLPW DIBJO UIBU FSSBUJDBMMZ TBNQMFT FYUSFNFMZ QPTJUJWF BOE FYUSFNFMZ OFHBUJWF QBSBNFUFS WBMVFT -FUT MPPL BU B TJNQMF FYBNQMF ćF DPEF CFMPX USJFT UP FTUJNBUF UIF NFBO BOE TUBOEBSE EFWJBUJPO PG UIF UXP (BVTTJBO PCTFSWBUJPOT − BOE  #VU JU VTFT UPUBMMZ ĘBU QSJPST 3 DPEF  4 ʚǶ ǿǶǎǢǎȀ . /ǡ. ǿǎǎȀ (ǖǡǏ ʚǶ 0'(ǿ '\$./ǿ 4 ʡ )*-(ǿ (0 Ǣ .\$"( Ȁ Ǣ (0 ʚǶ '+# Ǣ '+# ʡ )*-(ǿ Ǎ Ǣ ǎǍǍǍ Ȁ Ǣ .\$"( ʡ  3+ǿ ǍǡǍǍǍǎ Ȁ Ȁ Ǣ /ʙ'\$./ǿ4ʙ4Ȁ Ǣ #\$).ʙǏ Ȁ /PX MFUT MPPL BU UIF +- \$. PVUQVU 3 DPEF  +- \$.ǿ (ǖǡǏ Ȁ ( ) . ǒǡǒʉ ǖǑǡǒʉ )Ǿ !! #/ .\$"( ǒǖǓǡǖǖ ǎǒǑǏǡǕǒ Ǐǡǎǔ ǏǑǓǎǡǑǑ ǒǔ ǎǡǍǓ '+# ǶǒǖǡǑǒ ǐǒǒǡǒǐ ǶǕǎǎǡǑǒ ǐǒǖǡǑǖ ǐǏ ǎǡǍǒ 8IPB ćJT QPTUFSJPS DBOU CF SJHIU ćF NFBO PG − BOE  JT [FSP TP XFSF IPQJOH UP HFU B NFBO WBMVF GPS '+# BSPVOE [FSP *OTUFBE XF HFU DSB[Z WBMVFT BOE JNQMBVTJCMZ XJEF JO UFSWBMT *OGFSFODF GPS .\$"( JT OP CFUUFS ćF )Ǿ !! BOE #/ EJBHOPTUJDT EPOU MPPL HPPE FJUIFS 8F IBWF  TBNQMFT UP XPSL XJUI IFSF CVU UIF FTUJNBUFE FČFDUJWF TBNQMF TJ[FT BSF  BOE 
68. ### A wild chain '+# ʡ )*-(ǿ Ǎ Ǣ ǎǍǍǍ Ȁ

Ǣ .\$"( ʡ  3+ǿ ǍǡǍǍǍǎ Ȁ Ȁ Ǣ /ʙ'\$./ǿ4ʙ4Ȁ Ǣ #\$).ʙǏ Ȁ /PX MFUT MPPL BU UIF +- \$. PVUQVU 3 DPEF  +- \$.ǿ (ǖǡǏ Ȁ ( ) . ǒǡǒʉ ǖǑǡǒʉ )Ǿ !! #/ .\$"( ǒǖǓǡǖǖ ǎǒǑǏǡǕǒ Ǐǡǎǔ ǏǑǓǎǡǑǑ ǒǔ ǎǡǍǓ '+# ǶǒǖǡǑǒ ǐǒǒǡǒǐ ǶǕǎǎǡǑǒ ǐǒǖǡǑǖ ǐǏ ǎǡǍǒ 8IPB ćJT QPTUFSJPS DBOU CF SJHIU ćF NFBO PG − BOE  JT [FSP TP XFSF IPQJOH UP HFU B NFBO WBMVF GPS '+# BSPVOE [FSP *OTUFBE XF HFU DSB[Z WBMVFT BOE JNQMBVTJCMZ XJEF JO UFSWBMT *OGFSFODF GPS .\$"( JT OP CFUUFS ćF )Ǿ !! BOE #/ EJBHOPTUJDT EPOU MPPL HPPE FJUIFS 8F IBWF  TBNQMFT UP XPSL XJUI IFSF CVU UIF FTUJNBUFE FČFDUJWF TBNQMF TJ[FT BSF  BOE  :PV TIPVME BMTP TFF B XBSOJOH MJLF -)\$)" ( .." .ǣ ǎǣ # - 2 - ǔǍ \$1 -" )/ /-).\$/\$*). !/ - 2-(0+ǡ )- .\$)" +/Ǿ '/ *1 Ǎǡǖǒ (4 # '+ǡ  #//+ǣȅȅ(Ƕ./)ǡ*-"ȅ(\$.ȅ2-)\$)".ǡ#/('ȕ\$1 -" )/Ƕ/-).\$/\$*).Ƕ!/ -Ƕ2-(0+ ćFSF JT VTFGVM BEWJDF BU UIF 63- ćF RVJDL WFSTJPO JT UIBU 4UBO EFUFDUFE QSPCMFNT JO FY QMPSJOH BMM PG UIF QPTUFSJPS ćFSF BSF QPSUJPOT PG JU UIBU JU NJTTFE ćFTF BSF ıĶŃĲĿĴĲĻŁ ŁĿĮĻŀĶŁĶļĻŀ *MM HJWF B NPSF UIPSPVHI FYQMBOBUJPO JO B MBUFS DIBQUFS 'PS TJNQMFS NPE FMT JODSFBTJOH UIF +/Ǿ '/ DPOUSPM QBSBNFUFS XJMM VTVBMMZ SFNPWF UIFN ćJT JT FY QMBJOFE NPSF JO UIF 0WFSUIJOLJOH CPY BU UIF FOE PG UIJT TFDUJPO :PV DBO USZ BEEJOH *)Ƕ /PX MFUT MPPL BU UIF +- \$. PVUQVU 3  +- \$.ǿ (ǖǡǏ Ȁ ( ) . ǒǡǒʉ ǖǑǡǒʉ )Ǿ !! #/ .\$"( ǒǖǓǡǖǖ ǎǒǑǏǡǕǒ Ǐǡǎǔ ǏǑǓǎǡǑǑ ǒǔ ǎǡǍǓ '+# ǶǒǖǡǑǒ ǐǒǒǡǒǐ ǶǕǎǎǡǑǒ ǐǒǖǡǑǖ ǐǏ ǎǡǍǒ 8IPB ćJT QPTUFSJPS DBOU CF SJHIU ćF NFBO PG − BOE  JT [FSP TP XFSF IPQJOH UP HFU B NFBO WBMVF GPS '+# BSPVOE [FSP *OTUFBE XF HFU DSB[Z WBMVFT BOE JNQMBVTJCMZ XJEF JO UFSWBMT *OGFSFODF GPS .\$"( JT OP CFUUFS ćF )Ǿ !! BOE #/ EJBHOPTUJDT EPOU MPPL HPPE FJUIFS 8F IBWF  TBNQMFT UP XPSL XJUI IFSF CVU UIF FTUJNBUFE FČFDUJWF TBNQMF TJ[FT BSF  BOE  :PV TIPVME BMTP TFF B XBSOJOH MJLF -)\$)" ( .." .ǣ ǎǣ # - 2 - ǔǍ \$1 -" )/ /-).\$/\$*). !/ - 2-(0+ǡ )- .\$)" +/Ǿ '/ *1 Ǎǡǖǒ (4 # '+ǡ  #//+ǣȅȅ(Ƕ./)ǡ*-"ȅ(\$.ȅ2-)\$)".ǡ#/('ȕ\$1 -" )/Ƕ/-).\$/\$*).Ƕ!/ -Ƕ2-(0+ ćFSF JT VTFGVM BEWJDF BU UIF 63- ćF RVJDL WFSTJPO JT UIBU 4UBO EFUFDUFE QSPCMFNT JO FY QMPSJOH BMM PG UIF QPTUFSJPS ćFSF BSF QPSUJPOT PG JU UIBU JU NJTTFE ćFTF BSF ıĶŃĲĿĴĲĻŁ ŁĿĮĻŀĶŁĶļĻŀ *MM HJWF B NPSF UIPSPVHI FYQMBOBUJPO JO B MBUFS DIBQUFS 'PS TJNQMFS NPE FMT JODSFBTJOH UIF +/Ǿ '/ DPOUSPM QBSBNFUFS XJMM VTVBMMZ SFNPWF UIFN ćJT JT FY QMBJOFE NPSF JO UIF 0WFSUIJOLJOH CPY BU UIF FOE PG UIJT TFDUJPO :PV DBO USZ BEEJOH *)Ƕ /-*'ʙ'\$./ǿ+/Ǿ '/ʙǍǡǖǖȀ UP UIF 0'( DBMM0'(T EFGBVMU JT  #VU JU XPOU IFMQ NVDI JO UIJT TQFDJĕD DBTF ćJT QSPCMFN SVOT EFFQFS XJUI UIF NPEFM JUTFMG :PV TIPVME BMTP TFF B TFDPOE XBSOJOH • What the what is a divergent transition? • Hamiltonian approximation “broke” • Chain has trouble exploring some part of posterior
69. ### Figure 9.9 200 400 600 800 1000 0 10000 30000

n_eff = 57 sigma 200 400 600 800 1000 -2000 0 1000 n_eff = 32 alpha 200 400 600 800 1000 1 2 3 4 5 6 7 n_eff = 317 sigma 200 400 600 800 1000 -4 -2 0 2 4 6 n_eff = 284 alpha 'ĶĴłĿĲ ƑƑ %JBHOPTJOH BOE IFBMJOH B TJDL .BSLPW DIBJO 5PQ SPX 5SBDF QMPU GSPN UXP JOEFQFOEFOU DIBJOT EFĕOFE CZ NPEFM (ǖǡǏ ćFTF DIBJOT
70. ### A wild chain • Problem is flat priors • Flat

means flat forever • Much probability out to thousands • Also a problem in BUGS/JAGS • Fix with weakly informative priors PDDBTJPOBMMZ UP FYUSFNF WBMVFT ćJT JT OPU B IFBMUIZ QBJS PG DIBJOT BOE EF VTFGVM TBNQMFT NF UIJT QBSUJDVMBS DIBJO CZ VTJOH XFBLMZ JOGPSNBUJWF QSJPST ćF SFBTPO ESJęT XJMEMZ JO CPUI EJNFOTJPOT JT UIBU UIFSF JT WFSZ MJUUMF EBUB KVTU UXP ĘBU QSJPST ćF ĘBU QSJPST TBZ UIBU FWFSZ QPTTJCMF WBMVF PG UIF QBSBNFUFS F B QSJPSJ 'PS QBSBNFUFST UIBU DBO UBLF B QPUFOUJBMMZ JOĕOJUF OVNCFS PG UIJT NFBOT UIF .BSLPW DIBJO OFFET UP PDDBTJPOBMMZ TBNQMF TPNF QSFUUZ BVTJCMF WBMVFT MJLF OFHBUJWF  NJMMJPO ćFTF FYUSFNF ESJęT PWFSXIFMN JLFMJIPPE XFSF TUSPOHFS UIFO UIF DIBJO XPVME CF ĕOF CFDBVTF JU XPVME  UBLF NVDI JOGPSNBUJPO JO UIF QSJPS UP TUPQ UIJT GPPMJTIOFTT FWFO XJUIPVU TF UIJT NPEFM ZJ ∼ /PSNBM(µ, σ) µ = α α ∼ /PSNBM(, ) σ ∼ &YQPOFOUJBM() 1000 _eff = 57 200 400 600 800 1000 -2000 0 1000 n_eff = 32 alpha 1000 eff = 317 200 400 600 800 1000 -4 -2 0 2 4 6 n_eff = 284 alpha IFBMJOH B TJDL .BSLPW DIBJO 5PQ SPX 5SBDF IBJOT EFĕOFE CZ NPEFM (ǖǡǏ ćFTF DIBJOT PU CF VTFE GPS JOGFSFODF #PUUPN SPX "EEJOH FF (ǖǡǐ DMFBST VQ UIF DPOEJUJPO SJHIU BXBZ PS JOGFSFODF
71. ### A tame chain FBTJMZ PWFSDPNFT UIFTF QSJPST :FU UIF QPTUFSJPS

DBOOPU CF TVDDFTTGVMMZ BQ QSPYJNBUFE XJUIPVU UIFN *WF KVTU BEEFE XFBLMZ JOGPSNBUJWF QSJPST GPS α BOE σ 8FMM QMPU UIFTF QSJPST JO B NPNFOU TP ZPV XJMM CF BCMF UP TFF KVTU IPX XFBL UIFZ BSF #VU MFUT SFBQQSPYJNBUF UIF QPTUFSJPS ĕSTU 3 DPEF  . /ǡ. ǿǎǎȀ (ǖǡǐ ʚǶ 0'(ǿ '\$./ǿ 4 ʡ )*-(ǿ (0 Ǣ .\$"( Ȁ Ǣ (0 ʚǶ '+# Ǣ '+# ʡ )*-(ǿ ǎ Ǣ ǎǍ Ȁ Ǣ .\$"( ʡ  3+ǿ ǎ Ȁ Ȁ Ǣ /ʙ'\$./ǿ4ʙ4Ȁ Ǣ #\$).ʙǏ Ȁ +- \$.ǿ (ǖǡǐ Ȁ ( ) . ǒǡǒʉ ǖǑǡǒʉ )Ǿ !! #/ .\$"( ǎǡǒǐ Ǎǡǔǒ ǍǡǓǓ ǏǡǕǓ ǐǎǔ ǎ '+# ǍǡǍǑ ǎǡǎǑ Ƕǎǡǔǎ ǎǡǕǍ ǏǕǑ ǎ ćBUT NVDI CFUUFS 5BLF B MPPL BU UIF CPUUPN SPX JO 'ĶĴłĿĲ ƑƑ ćJT USBDF QMPU MPPLT IFBMUIZ #PUI DIBJOT BSF TUBUJPOBSZ BSPVOE UIF TBNF WBMVFT BOE NJYJOH JT HPPE /P NPSF XJME EFUPVST JOUP UIF UIPVTBOET "OE UIPTF EJWFSHFOU USBOTJUJPOT TIPVME CF HPOF 5P BQQSFDJBUF XIBU IBT IBQQFOFE UBLF B MPPL BU UIF QSJPST EBTIFE BOE QPTUFSJPST CMVF JO 'ĶĴłĿĲ ƑƉƈ #PUI UIF (BVTTJBO QSJPS GPS α BOE UIF FYQPOFOUJBM QSJPS GPS σ DPOUBJO WFSZ HSBEVBM EPXOIJMM TMPQFT ćFZ BSF TP HSBEVBM UIBU FWFO XJUI POMZ UXP PCTFSWBUJPOT BT JO
72. ### Figure 9.9 200 400 600 800 1000 0 10000 30000

n_eff = 57 sigma 200 400 600 800 1000 -2000 0 1000 n_eff = 32 alpha 200 400 600 800 1000 1 2 3 4 5 6 7 n_eff = 317 sigma 200 400 600 800 1000 -4 -2 0 2 4 6 n_eff = 284 alpha 'ĶĴłĿĲ ƑƑ %JBHOPTJOH BOE IFBMJOH B TJDL .BSLPW DIBJO 5PQ SPX 5SBDF QMPU GSPN UXP JOEFQFOEFOU DIBJOT EFĕOFE CZ NPEFM (ǖǡǏ ćFTF DIBJOT
73. ### A tame chain Even with only 2 observations, these priors

have no effect on inference! Except to allow you to make inferences...  \$"3& "/% '&&%*/( 0' :063 ."3,07 \$)"*/  -15 -10 -5 0 5 10 15 0.0 0.1 0.2 0.3 0.4 alpha Density posterior prior 0 2 4 6 8 10 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 sigma Density 'ĶĴłĿĲ ƑƉƈ 1SJPS EBTIFE BOE QPTUFSJPS CMVF GPS UIF NPEFM XJUI XFBLMZ JOGPSNBUJWF QSJPST (ǖǡǐ &WFO XJUI POMZ UXP PCTFSWBUJPOT UIF MJLFMJIPPE FBTJMZ PWFSDPNFT UIFTF QSJPST :FU UIF QPTUFSJPS DBOOPU CF TVDDFTTGVMMZ BQ QSPYJNBUFE XJUIPVU UIFN Figure 9.10
74. ### The Folk Theorem of Statistical Computing “When you have computational

problems, often there’s a problem with your model.” –Andrew Gelman
75. ###  \$"3& "/% '&&%*/( 0' :063 ."3,07 \$)"*/  4

ʡ )*-(ǿ (0 Ǣ .\$"( Ȁ Ǣ (0 ʚǶ ǎ ʔ Ǐ Ǣ ǎ ʡ )*-(ǿ Ǎ Ǣ ǎǍǍǍ ȀǢ Ǐ ʡ )*-(ǿ Ǎ Ǣ ǎǍǍǍ ȀǢ .\$"( ʡ  3+ǿ ǎ Ȁ Ȁ Ǣ /ʙ'\$./ǿ4ʙ4Ȁ Ǣ #\$).ʙǏ Ȁ +- \$.ǿ (ǖǡǑ Ȁ ( ) . ǒǡǒʉ ǖǑǡǒʉ )Ǿ !! #/ .\$"( ǎǡǍǒ ǍǡǍǕ Ǎǡǖǎ ǎǡǎǔ ǒ ǎǡǐǏ Ǐ ǏǎǡǑǓ Ǐǖǎǡǖǔ ǶǑǔǐǡǔǔ Ǒǔǒǡǒǖ Ǐ ǏǡǍǎ ǎ ǶǏǎǡǏǔ ǏǖǎǡǖǓ ǶǑǔǒǡǒǑ ǑǔǐǡǖǓ Ǐ ǏǡǍǎ ćPTF FTUJNBUFT MPPL TVTQJDJPVT BOE UIF )Ǿ !! BOE #/ WBMVFT BSF UFSSJCMF ćF NFBOT GPS Unidentified chains Ǣ .ʙǎ Ȁ OPX UIF SJHIU BOTXFS ćFO XF ĕU UIJT NPEFM ZJ ∼ /PSNBM(µ, σ) µ = α + α σ ∼ &YQPOFOUJBM() P QBSBNFUFST α BOE α XIJDI DBOOPU CF JEFOUJĕFE 0OMZ UIFJS TIPVME CF BCPVU [FSP BęFS FTUJNBUJPO BJO BOE TFF XIBU IBQQFOT ćJT DIBJO JT HPJOH UP UBLF NVDI T #VU JU TIPVME TUJMM ĕOJTI BęFS B GFX NJOVUFT TVDI QBSBNFUFST MPPL MJLF JOTJEF PG B .BSLPW DI UIFN JO QSJODJQMF CZ VTJOH B MJUUMF QSJPS JOGPSNBUJ DIBJOT QSPEVDFE JO UIJT FYBNQMF XJMM FYIJCJU DIB UIF TBNF QBUUFSO JO ZPVS PXO NPEFMT ZPVMM IBWF 5P DPOTUSVDU B OPOJEFOUJĕBCMF NPEFM XF ĕST JBO EJTUSJCVUJPO XJUI NFBO [FSP BOE TUBOEBSE EFW 3 DPEF  . /ǡ. ǿǑǎȀ 4 ʚǶ -)*-(ǿ ǎǍǍ Ǣ ( )ʙǍ Ǣ .ʙǎ Ȁ #Z TJNVMBUJOH UIF EBUB XF LOPX UIF SJHIU BOTXFS ZJ ∼ /PSNBM( µ = α + α σ ∼ &YQPOFO ćF MJOFBS NPEFM DPOUBJOT UXP QBSBNFUFST α BOE TVN DBO CF JEFOUJĕFE BOE JU TIPVME CF BCPVU [FSP -FUT SVO UIF .BSLPW DIBJO BOE TFF XIBU IB MPOHFS UIBO UIF QSFWJPVT POFT #VU JU TIPVME TUJMM ĕ 3 DPEF  (ǖǡǑ ʚǶ 0'(ǿ '\$./ǿ #Z TJNVMBUJOH UIF EBUB XF LOPX UIF SJHIU BOTXFS ćFO XF ĕU UIJT NPEFM ZJ ∼ /PSNBM(µ, σ) µ = α + α σ ∼ &YQPOFOUJBM() ćF MJOFBS NPEFM DPOUBJOT UXP QBSBNFUFST α BOE α XIJDI DBOOPU CF JEFOUJĕFE 0OMZ UIFJS TVN DBO CF JEFOUJĕFE BOE JU TIPVME CF BCPVU [FSP BęFS FTUJNBUJPO -FUT SVO UIF .BSLPW DIBJO BOE TFF XIBU IBQQFOT ćJT DIBJO JT HPJOH UP UBLF NVDI MPOHFS UIBO UIF QSFWJPVT POFT #VU JU TIPVME TUJMM ĕOJTI BęFS B GFX NJOVUFT 3 DPEF  (ǖǡǑ ʚǶ 0'(ǿ '\$./ǿ
76. ### Figure 9.11 200 400 600 800 1000 0.9 1.1 1.3

n_eff = 5 sigma 200 400 600 800 1000 -500 0 500 n_eff = 2 a2 200 400 600 800 1000 -500 0 500 n_eff = 2 a1 200 400 600 800 1000 0.9 1.1 1.3 n_eff = 287 sigma 200 400 600 800 1000 -20 0 10 20 n_eff = 244 a2 200 400 600 800 1000 -20 0 10 20 n_eff = 245 a1 'ĶĴłĿĲ ƑƉƉ -Fę DPMVNO " DIBJO XJUI XBOEFSJOH QBSBNFUFST ǎ BOE Ǐ
77. ### Unidentified chains CVU JU XPOU IFMQ NVDI ćFSF JT TPNFUIJOH

TFSJPVTMZ XSPOH IFSF -PPLJOH BU UIF USBDF QMPU SFWFBMT NPSF ćF MFę DPMVNO JO 'ĶĴłĿĲ ƑƉƉ TIPXT UXP .BSLPW DIBJOT GSPN UIF NPEFM BCPWF ćFTF DIBJOT EP OPU MPPL MJLF UIFZ BSF TUBUJPOBSZ OPS EP UIFZ TFFN UP CF NJYJOH WFSZ XFMM *OEFFE XIFO ZPV TFF B QBUUFSO MJLF UIJT JU JT SFBTPO UP XPSSZ %POU VTF UIFTF TBNQMFT "HBJO XFBL QSJPST DBO SFTDVF VT /PX UIF NPEFM ĕUUJOH DPEF JT 3 DPEF  (ǖǡǒ ʚǶ 0'(ǿ '\$./ǿ 4 ʡ )*-(ǿ (0 Ǣ .\$"( Ȁ Ǣ (0 ʚǶ ǎ ʔ Ǐ Ǣ ǎ ʡ )*-(ǿ Ǎ Ǣ ǎǍ ȀǢ Ǐ ʡ )*-(ǿ Ǎ Ǣ ǎǍ ȀǢ .\$"( ʡ  3+ǿ ǎ Ȁ Ȁ Ǣ /ʙ'\$./ǿ4ʙ4Ȁ Ǣ #\$).ʙǏ Ȁ +- \$.ǿ (ǖǡǒ Ȁ ( ) . ǒǡǒʉ ǖǑǡǒʉ )Ǿ !! #/ .\$"( ǎǡǍǑ ǍǡǍǕ Ǎǡǖǐ ǎǡǎǕ ǏǕǔ ǎ Ǐ ǍǡǏǏ ǔǡǏǐ ǶǎǍǡǓǔ ǎǏǡǏǓ ǏǑǑ ǎ ǎ ǶǍǡǍǐ ǔǡǏǏ ǶǎǏǡǍǖ ǎǍǡǕǔ ǏǑǒ ǎ
78. ### Figure 9.11 200 400 600 800 1000 0.9 1.1 1.3

n_eff = 5 sigma 200 400 600 800 1000 -500 0 500 n_eff = 2 a2 200 400 600 800 1000 -500 0 500 n_eff = 2 a1 200 400 600 800 1000 0.9 1.1 1.3 n_eff = 287 sigma 200 400 600 800 1000 -20 0 10 20 n_eff = 244 a2 200 400 600 800 1000 -20 0 10 20 n_eff = 245 a1 'ĶĴłĿĲ ƑƉƉ -Fę DPMVNO " DIBJO XJUI XBOEFSJOH QBSBNFUFST ǎ BOE Ǐ
79. ### Homeward • Updated book PDF (20 Jan) & rethinking 1.82

• data(Wines2012): Interactions and Markov chains • Next week: Maximizing Entropy for Inferential Justice Generalized Linear Models (GLMs)