Yes, but Did It Work?: Evaluating Variational InferenceYes, but Did It Work?: Evaluating Variational...

Yes but Did It Work Evaluating Variational Inference

Yuling Yao 1 Aki Vehtari 2 Daniel Simpson 3 Andrew Gelman 1

AbstractWhile itrsquos always possible to compute a varia-tional approximation to a posterior distributionit can be difficult to discover problems with thisapproximation We propose two diagnostic al-gorithms to alleviate this problem The Pareto-smoothed importance sampling (PSIS) diagnosticgives a goodness of fit measurement for joint dis-tributions while simultaneously improving theerror in the estimate The variational simulation-based calibration (VSBC) assesses the averageperformance of point estimates

1 IntroductionVariational Inference (VI) including a large family of pos-terior approximation methods like stochastic VI (Hoffmanet al 2013) black-box VI (Ranganath et al 2014) automaticdifferentiation VI (ADVI Kucukelbir et al 2017) and manyother variants has emerged as a widely-used method forscalable Bayesian inference These methods come with fewtheoretical guarantees and itrsquos difficult to assess how wellthe computed variational posterior approximates the trueposterior

Instead of computing expectations or sampling draws fromthe posterior p(θ | y) variational inference fixes a fam-ily of approximate densities Q and finds the member qlowast

minimizing the Kullback-Leibler (KL) divergence to thetrue posterior KL (q(θ) p(θ | y)) This is equivalent tomaximizing the evidence lower bound (ELBO)

ELBO(q) =

intΘ

(log p(θ y)minus log q(θ)) q(θ)dθ (1)

There are many situations where the VI approximation isflawed This can be due to the slow convergence of the

1Department of Statistics Columbia University NY USA2Helsinki Institute for Information Technology Department ofComputer Science Aalto University Finland 3Department of Sta-tistical Sciences University of Toronto Canada Correspondenceto Yuling Yao ltyy2618columbiaedugt

Proceedings of the 35 th International Conference on MachineLearning Stockholm Sweden PMLR 80 2018 Copyright 2018by the author(s)

optimization problem the inability of the approximationfamily to capture the true posterior the asymmetry of thetrue distribution the fact that the direction of the KL diver-gence under-penalizes approximation with too-light tails orall these reasons We need a diagnostic algorithm to testwhether the VI approximation is useful

There are two levels of diagnostics for variational inferenceFirst the convergence test should be able to tell if the ob-jective function has converged to a local optimum Whenthe optimization problem (1) is solved through stochasticgradient descent (SGD) the convergence can be assessedby monitoring the running average of ELBO changes Re-searchers have introduced many convergence tests based onthe asymptotic property of stochastic approximations (egSielken 1973 Stroup amp Braun 1982 Pflug 1990 Wada ampFujisaki 2015 Chee amp Toulis 2017) Alternatively Bleiet al (2017) suggest monitoring the expected log predictivedensity by holding out an independent test dataset Afterconvergence the optimum is still an approximation to thetruth This paper is focusing on the second level of VI di-agnostics whether the variational posterior qlowast(θ) is closeenough to the true posterior p(θ|y) to be used in its place

Purely relying on the objective function or the equivalentELBO does not solve the problem An unknown multi-plicative constant exists in p(θ y) prop p(θ | y) that changeswith reparametrization making it meaningless to compareELBO across two approximations Moreover the ELBO isa quantity on an uninterpretable scale that is itrsquos not clear atwhat value of the ELBO we can begin to trust the variationalposterior This makes it next to useless as a method to assesshow well the variational inference has fit

In this paper we propose two diagnostic methods that assessrespectively the quality of the entire variational posterior fora particular data set and the average bias of a point estimateproduced under correct model specification

The first method is based on generalized Pareto distributiondiagnostics used to assess the quality of a importance sam-pling proposal distribution in Pareto smoothed importancesampling (PSIS Vehtari et al 2017) The benefit of PSISdiagnostics is two-fold First we can tell the discrepancybetween the approximate and the true distribution by theestimated continuous k value When it is larger than a pre-specified threshold users should be alert of the limitation

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018

Evaluating Variational Inference

of current variational inference computation and considerfurther tuning it or turn to exact sampling like Markov chainMonte Carlo (MCMC) Second in the case when k is smallthe fast convergence rate of the importance-weighted MonteCarlo integration guarantees a better estimation accuracy Insuch sense the PSIS diagnostics could also be viewed as apost-adjustment for VI approximations Unlike the second-order correction Giordano et al (2017) which relies on anun-testable unbiasedness assumption we make diagnosticsand adjustment at the same time

The second diagnostic considers only the quality of themedian of the variational posterior as a point estimate (inGaussian mean-field VI this corresponds to the modal es-timate) This diagnostic assesses the average behavior ofthe point estimate under data from the model and can in-dicate when a systemic bias is present The magnitude ofthat bias can be monitored while computing the diagnosticThis diagnostic can also assess the average calibration ofunivariate functionals of the parameters revealing if theposterior is under-dispersed over-dispersed or biased Thisdiagnostic could be used as a partial justification for usingthe second-order correction of Giordano et al (2017)

2 Is the Joint Distribution Good EnoughIf we can draw a sample (θ1 θS) from p(θ|y) the ex-pectation of any integrable function Ep[h(θ)] can be esti-

mated by Monte Carlo integrationsumSs=1 h(θs)S

SrarrinfinminusminusminusminusminusrarrEp [h(θ)] Alternatively given samples (θ1 θS) froma proposal distribution q(θ) the importance sampling (IS)estimate is

(sumSs=1 h(θs)rs

)sumSs=1 rs where the impor-

tance ratios rs are defined as

rs =p(θs y)

q(θs) (2)

In general with a sample (θ1 θS) drawn from the varia-tional posterior q(θ) we consider a family of estimates withthe form

Ep[h(θ)] asympsumSs=1 h(θs)wssumS

s=1 ws (3)

which contains two extreme cases

1 When ws equiv 1 estimate (3) becomes the plain VI esti-mate that is we completely trust the VI approximationIn general this will be biased to an unknown extentand inconsistent However this estimator has smallvariance

2 When ws = rs (3) becomes importance samplingThe strong law of large numbers ensures it is consistent

as S rarr infin and with small O(1S) bias due to self-normalization But the IS estimate may have a large orinfinite variance

There are two questions to be answered First can we find abetter bias-variance trade-off than both plain VI and IS

Second VI approximation q(θ) is not designed for an op-timal IS proposal for it has a lighter tail than p(θ|y) as aresult of entropy penalization which lead to a heavy righttail of rs A few large-valued rs dominates the summationbringing in large uncertainty But does the finite sampleperformance of IS or stabilized IS contain the informationabout the dispensary measure between q(θ) and p(θ|y)

21 Pareto Smoothed Importance Sampling

The solution to the first question is the Pareto smoothedimportance sampling (PSIS) We give a brief review andmore details can be found in Vehtari et al (2017)

A generalized Pareto distribution with shape parameter kand location-scale parameter (micro τ) has the density

p(y|micro σ k) =

1

σ

(1 + k

(y minus microσ

))minus 1kminus1

k 6= 0

1

σexp

(y minus microσ

) k = 0

PSIS stabilizes importance ratios by fitting a generalizedPareto distribution using the largest M samples of ri whereM is empirically set as min(S5 3

radicS) It then reports the

estimated shape parameter k and replaces the M largest rsby their expected value under the fitted generalized Paretodistribution The other importance weights remain un-changed We further truncate all weights at the raw weightmaximum max(rs) The resulted smoothed weights aredenoted by ws based on which a lower variance estimationcan be calculated through (3)

Pareto smoothed importance sampling can be considered asBayesian version of importance sampling with prior on thelargest importance ratios It has smaller mean square errorsthan plain IS and truncated-IS (Ionides 2008)

22 Using PSIS as a Diagnostic Tool

The fitted shape parameter k turns out to provide the desireddiagnostic measurement between the true posterior p(θ|y)and the VI approximation q(θ) A generalized Pareto dis-tribution with shape k has finite moments up to order 1kthus any positive k value can be viewed as an estimate to

k = inf

kprime gt 0 Eq

(p(θ|y)

q(θ)

) 1kprime

ltinfin

(4)


k is invariant under any constant multiplication of p or qwhich explains why we can suppress the marginal likeli-hood (normalizing constant) p(y) and replace the intractablep(θ|y) with p(θ y) in (2)

After log transformation (4) can be interpreted as Renyidivergence (Renyi et al 1961) with order α between p(θ|y)and q(θ)

k = infkprime gt 0 D 1

kprime(p||q) ltinfin

whereDα (p||q) =1

αminus 1log

intΘ

p(θ)αq(θ)1minusαdθ

It is well-defined since Renyi divergence is monotonic in-creasing on order α Particularly when k gt 05 the χ2

divergence χ(p||q) becomes infinite and when k gt 1D1(p||q) = KL(p q) = infin indicating a disastrous VIapproximation despite the fact that KL(q p) is always min-imized among the variational family The connection toRenyi divergence holds when k gt 0 When k lt 0 itpredicts the importance ratios are bounded from above

This also illustrates the advantage of a continuous k estimatein our approach over only testing the existence of secondmoment of Eq(qp)2 (Epifani et al 2008 Koopman et al2009) ndash it indicates if the Renyi divergence between q and pis finite for all continuous order α gt 0

Meanwhile the shape parameter k determines the finitesample convergence rate of both IS and PSIS adjusted es-timate Geweke (1989) shows when Eq[r(θ)

2] lt infin andEq[(r(θ)h(θ)

)2] ltinfin hold (both conditions can be tested

by k in our approach) the central limit theorem guaran-tees the square root convergence rate Furthermore whenk lt 13 then the Berry-Essen theorem states faster con-vergence rate to normality (Chen et al 2004) Cortes et al(2010) and Cortes et al (2013) also link the finite sampleconvergence rate of IS with the number of existing momentsof importance ratios

PSIS has smaller estimation error than the plain VI esti-mate which we will experimentally verify this in Section4 A large k indicates the failure of finite sample PSIS so itfurther indicates the large estimation error of VI approxima-tion Therefore even when the researchersrsquo primary goal isnot to use variational approximation q as an PSIS proposalthey should be alert by a large k which tells the discrepancybetween the VI approximation result and the true posterior

According to empirical study in Vehtari et al (2017) we setthe threshold of k as follows

bull If k lt 05 we can invoke the central limit theorem tosuggest PSIS has a fast convergence rate We concludethe variational approximation q is close enough to thetrue density We recommend further using PSIS to

Algorithm 1 PSIS diagnostic1 Input the joint density function p(θ y) number of

posterior samples S number of tail samples M 2 Run variational inference to p(θ|y) obtain VI approxi-

mation q(θ)3 Sample (θs s = 1 S) from q(θ)4 Calculate the importance ratio rs = p(θs y)q(θs)5 Fit generalized Pareto distribution to the M largest rs6 Report the shape parameter k7 if k lt 07 then8 Conclude VI approximation q(θ) is close enough to

the unknown truth p(θ|y)9 Recommend further shrinking errors by PSIS

10 else11 Warn users that the VI approximation is not reliable12 end if

adjust the estimator (3) and calculate other divergencemeasures

bull If 05 lt k lt 07 we still observe practically usefulfinite sample convergence rates and acceptable MonteCarlo error for PSIS It indicates the variational ap-proximation q is not perfect but still useful Again werecommend PSIS to shrink errors

bull If k gt 07 the PSIS convergence rate becomes im-practically slow leading to a large mean square er-ror and a even larger error for plain VI estimate Weshould consider tuning the variational methods (egre-parametrization increase iteration times increasemini-batch size decrease learning rate etal) or turn-ing to exact MCMC Theoretically k is always smallerthan 1 for Eq [p(θ|y)q(θ)] = p(y) lt infin while inpractice finite sample estimate k may be larger than 1which indicates even worse finite sample performance

The proposed diagnostic method is summarized in Algo-rithm 1

23 Invariance Under Re-Parametrization

Re-parametrization is common in variational inference Par-ticularly the reparameterization trick (Rezende et al 2014)rewrites the objective function to make gradient calculationeasier in Monte Carlo integrations

A nice property of PSIS diagnostics is that the k quantity isinvariant under any re-parametrization Suppose ξ = T (θ)is a smooth transformation then the density ratio of ξ underthe target p and the proposal q does not change

p(ξ)

q(ξ)=p(Tminus1(ξ)

)|detJξT

minus1(ξ)|q (Tminus1(ξ)) |detJξTminus1(ξ)|

=p (θ)

q(θ)


Therefore p(ξ)q(ξ) and p(θ)q(θ) have the same distri-bution under q making it free to choose any convenientparametrization form when calculating k

However if the re-parametrization changes the approxima-tion family then it will change the computation result andPSIS diagnostics will change accordingly Finding the op-timal parametrization form such that the re-parametrizedposterior distribution lives exactly in the approximation fam-ily

p(T (ξ)) = p(Tminus1(ξ)

)|JξTminus1(ξ)| isin Q

can be as hard as finding the true posterior The PSIS diag-nostic can guide the choice of re-parametrization by simplycomparing the k quantities of any parametrization Section43 provides a practical example

24 Marginal PSIS Diagnostics Do Not Work

As dimension increases the VI posterior tends to be furtheraway from the truth due to the limitation of approximationfamilies As a result k increases indicating inefficiencyof importance sampling This is not the drawback of PSISdiagnostics Indeed when the focus is the joint distribu-tion such behaviour accurately reflects the quality of thevariational approximation to the joint posterior

Denoting the one-dimensional true and approximatemarginal density of the i-th coordinate θi as p(θi|y) andq(θi) the marginal k for θi can be defined as

ki = inf

0 lt kprime lt 1 Eq

(p(θi|y)

q(θi)

) 1kprime

ltinfin

The marginal ki is never larger (and usually smaller) thanthe joint k in (4)

Proposition 1 For any two distributions p and q withsupport Θ and the margin index i if there is a num-ber α gt 1 satisfying Eq (p(θ)q(θ))

αlt infin then

Eq (p(θi)q(θi))αltinfin

Proposition 1 demonstrates why the importance samplingis usually inefficient in high dimensional sample space inthat the joint estimation is ldquoworserdquo than any of the marginalestimation

Should we extend the PSIS diagnostics to marginal distri-butions We find two reasons why the marginal PSIS diag-nostics can be misleading Firstly unlike the easy accessto the unnormalized joint posterior distribution p(θ y) thetrue marginal posterior density p(θi|y) is typically unknownotherwise one can conduct one-dimensional sampling easilyto obtain the the marginal samples Secondly a smaller kidoes not necessary guarantee a well-performed marginalestimation The marginal approximations in variational in-ference can both over-estimate and under-estimate the tail

thickness of one-dimensional distributions the latter situa-tion gives rise to a smaller ki Section 43 gives an examplewhere the marginal approximations with extremely smallmarginal k have large estimation errors This does not hap-pen in the joint case as the direction of the Kullback-Leiblerdivergence qlowast(θ) strongly penalizes too-heavy tails whichmakes it unlikely that the tails of the variational posteriorare significantly heavier than the tails of the true posterior

3 Assessing the Average Performance of thePoint Estimate

The proposed PSIS diagnostic assesses the quality of theVI approximation to the full posterior distribution It isoften observed that while the VI posterior may be a poorapproximation to the full posterior point estimates that arederived from it may still have good statistical properties Inthis section we propose a new method for assessing thecalibration of the center of a VI posterior

31 The Variational Simulation-Based Calibration(VSBC) Diagnostic

This diagnostic is based on the proposal of Cook et al (2006)for validating general statistical software They noted that ifθ(0) sim p(θ) and y sim p(y | θ(0)) then

Pr(yθ(0))

(Prθ|y(θ lt θ(0)) le middot)

)= Unif[01]([0 middot])

To use the observation of Cook et al (2006) to assess the per-formance of a VI point estimate we propose the followingprocedure Simulate M gt 1 data sets yjMj=1 as follows

Simulate θ(0)j sim p(θ) and then simulate y(j) sim p(y | θ

(0)j )

where y(j) has the same dimension as y For each ofthese data sets construct a variational approximation top(θ | yj) and compute the marginal calibration probabilities

pij = Prθ|y(j)

(θi le [θ

(0)j ]i

)

To apply the full procedure of Cook et al (2006) we wouldneed to test dim(θ) histograms for uniformity however thiswould be too stringent a check as like our PSIS diagnosticthis test is only passed if the variational posterior is a goodapproximation to the true posterior Instead we followan observation of Anderson (1996) from the probabilisticforecasting validation literature and note that asymmetryin the histogram for pi indicates bias in the variationalapproximation to the marginal posterior θi | y

The VSBC diagnostic tests for symmetry of the marginal cal-ibration probabilities around 05 and either by visual inspec-tion of the histogram or by using a Kolmogorov-Smirnov(KS) test to evaluate whether pi and 1minus pi have the samedistribution When θ is a high-dimensional parameter itis important to interpret the results of any hypothesis tests


Algorithm 2 VSBC marginal diagnostics1 Input prior density p(θ) data likelihood p(y | θ)

number of replications M parameter dimensions K2 for j = 1 M do3 Generate θ(0)

j from prior p(θ)

4 Generate a size-n dataset(y(j)

)from p(y | θ(0)

j )5 Run variational inference using dataset y(j) obtain a

VI approximation distribution qj(middot)6 for i = 1 K do7 Label θ(0)

ij as the i-th marginal component of θ(0)j

Label θlowasti as the i-th marginal component of θlowast8 Calculate pij = Pr(θ

(0)ij lt θlowasti | θlowast sim qj)

9 end for10 end for11 for i = 1 K do12 Test if the distribution of pijMj=1 is symmetric13 If rejected the VI approximation is biased in its i-th

margin14 end for

through a multiple testing lens

32 Understanding the VSBC Diagnostic

Unlike the PSIS diagnostic which focuses on a the perfor-mance of variational inference for a fixed data set y theVSBC diagnostic assesses the average calibration of thepoint estimation over all datasets that could be constructedfrom the model Hence the VSBC diagnostic operatesunder a different paradigm to the PSIS diagnostic and werecommend using both as appropriate

There are two disadvantages to this type of calibration whencompared to the PSIS diagnostic As is always the casewhen interpreting hypothesis tests just because somethingworks on average doesnrsquot mean it will work for a particularrealization of the data The second disadvantage is that thisdiagnostic does not cover the case where the observed datais not well represented by the model We suggest interpret-ing the diagnostic conservatively if a variational inferencescheme fails the diagnostic then it will not perform well onthe model in question If the VI scheme passes the diagnos-tic it is not guaranteed that it will perform well for real dataalthough if the model is well specified it should do well

The VSBC diagnostic has some advantages compared tothe PSIS diagnostic It is well understood that for complexmodels the VI posterior can be used to produce a good pointestimate even when it is far from the true posterior In thiscase the PSIS diagnostic will most likely indicate failureThe second advantage is that unlike the PSIS diagnostic theVSBC diagnostic considers one-dimensional marginals θi(or any functional h(θ)) which allows for a more targetedinterrogation of the fitting procedure

With stronger assumptions The VSBC test can be formal-ized as in Proposition 2

Proposition 2 Denote θ as a one-dimensional parameterthat is of interest Suppose in addition we have (i) theVI approximation q is symmetric (ii) the true posteriorp(θ|y) is symmetric If the VI estimation q is unbiased ieEθsimq(θ|y) θ = Eθsimp(θ|y) θ then the distribution of VSBCp-value is symmetric Otherwise if the VI estimation ispositivelynegatively biased then the distribution of VSBCp-value is rightleft skewed

The symmetry of the true posterior is a stronger assumptionthan is needed in practice for this result to hold In theforecast evaluation literature as well as the literature onposterior predictive checks the symmetry of the histogramis a commonly used heuristic to assess the potential bias ofthe distribution In our tests we have seen the same thingoccurs the median of the variational posterior is close tothe median of the true posterior when the VSBC histogramis symmetric We suggest again that this test be interpretedconservatively if the histogram is not symmetric then theVI is unlikely to have produced a point estimate close to themedian of the true posterior

4 ApplicationsBoth PSIS and VSBC diagnostics are applicable to anyvariational inference algorithm Without loss of generalitywe implement mean-field Gaussian automatic differentiationvariational inference (ADVI) in this section

41 Linear Regression

Consider a Bayesian linear regression y sim N(Xβ σ2) withprior βiKi=1 sim N(0 1) σ sim gamma(5 5) We fix sam-ple size n = 10000 and number of regressors K = 100

Figure 1 visualizes the VSBC diagnostic showing the dis-tribution of VSBC p-values of the first two regression coef-ficients β1 β2 and log σ based on M = 1000 replicationsThe two sided Kolmogorov-Smirnov test for p and 1minus p isonly rejected for pσ suggesting the VI approximation is inaverage marginally unbiased for β1 and β2 while σ is over-estimated as pσ is right-skewed The under-estimation ofposterior variance is reflected by the U-shaped distributions

Using one randomly generated dataset in the same problemthe PSIS k is 061 indicating the joint approximation isclose to the true posterior However the performance ofADVI is sensitive to the stopping time as in any other opti-mization problems As displayed in the left panel of Figure2 changing the threshold of relative ELBO change froma conservative 10minus5 to the default recommendation 10minus2

increases k to 44 even though 10minus2 works fine for manyother simpler problems In this example we can also view k


0 05 10

2

4

6 KSminustest p= 027

dens

ity

β1

pβ1

0 05 1

p= 008

pβ2

β2

0 05 1

p= 000 reject

pσ

log σ

Figure 1 VSBC diagnostics for β1 β2 and log σ in the Bayesianlinear regression example The VI estimation overestimates σ aspσ is right-skewed while β1 and β2 is unbiased as the two-sidedKS-test is not rejected

10minus4 10minus3 10minus20

5

7

1

2

k ha

t

relative tolerance

0 20 40 600

5

7

1

2

k ha

t

running time (s)

NUTS sampling

time=2300

0 05 10

01

02R

MS

E

k hat

Raw ADVI

IS

PSIS

Figure 2 ADVI is sensitive to the stopping time in the linear re-gression example The default 001 threshold lead to a fake con-vergence which can be diagnosed by monitoring PSIS k PSISadjustment always shrinks the estimation errors

as a convergence test The right panel shows k diagnoses es-timation error which eventually become negligible in PSISadjustment when k lt 07 To account for the uncertaintyof stochastic optimization and k estimation simulations arerepeated 100 times

42 Logistic Regression

Next we run ADVI to a logistic regression Y simBernoulli

(logitminus1(βX)

)with a flat prior on β We gener-

ate X = (x1 xn) from N(0 (1minus ρ)IKtimesK + ρ1KtimesK)such that the correlation in design matrix is ρ and ρ ischanged from 0 to 099 The first panel in Figure 3 showsPSIS k increases as the design matrix correlation increasesIt is not monotonic because β is initially negatively corre-lated when X is independent A large ρ transforms into alarge correlation for posterior distributions in β making itharder to be approximated by a mean-field family as canbe diagnosed by k In panel 2 we calculate mean log pre-dictive density (lpd) of VI approximation and true posteriorusing 200 independent test sets Larger ρ leads to worsemean-field approximation while prediction becomes eas-ier Consequently monitoring lpd does not diagnose the VIbehavior it increases (misleadingly suggesting better fit)as ρ increases In this special case VI has larger lpd thanthe true posterior due to the VI under-dispersion and themodel misspecification Indeed if viewing lpd as a functionh(β) it is the discrepancy between VI lpd and true lpd thatreveals the VI performance which can also be diagnosedby k Panel 3 shows a sharp increase of lpd discrepancyaround k = 07 consistent with the empirical threshold wesuggest

0 05 10

5

7

1

correlations

k hat

lpd of true posterior

lpd of VI

minus6

minus5

minus4

minus3

0 05 1

correlations

mean log predictive density

02 05 07 1

0

1

2

k hat

VI lpd minus true lpd

Figure 3 In the logistic regression example as the correlation indesign matrix increase the correlation in parameter space alsoincreases leading to larger k Such flaw is hard to tell from theVI log predictive density (lpd) as a larger correlation makes theprediction easier k diagnose the discrepancy of VI lpd and trueposterior lpd with a sharp jump at 07

03 05 07 090

25

5

RM

SE

1st Moment

k hat

Raw ADVIISPSIS

03 05 07 090

25

50

k hat

RM

SE

2nd Moment

Figure 4 In the logistic regression with varying correlations thek diagnoses the root mean square of first and second momenterrors No estimation is reliable when k gt 07 Meanwhile PSISadjustment always shrinks the VI estimation errors

Figure 4 compares the first and second moment root meansquare errors (RMSE) ||Epβ minus Eqlowastβ||2 and ||Epβ2 minusEqlowastβ

2||2 in the previous example using three estimates(a) VI without post-adjustment (b) VI adjusted by vanillaimportance sampling and (c) VI adjusted by PSIS

PSIS diagnostic accomplishes two tasks here (1) A small kindicates that VI approximation is reliable When k gt 07all estimations are no longer reasonable so the user shouldbe alerted (2) It further improves the approximation usingPSIS adjustment leading to a quicker convergence rate andsmaller mean square errors for both first and second momentestimation Plain importance sampling has larger RMSE forit suffers from a larger variance

43 Re-parametrization in a Hierarchical Model

The Eight-School Model (Gelman et al 2013 Section 55)is the simplest Bayesian hierarchical normal model Eachschool reported the treatment effect mean yi and standarddeviation σi separately There was no prior reason to believethat any of the treatments were more effective than any otherso we model them as independent experiments

yj |θj sim N(θj σ2j ) θj |micro τ sim N(micro τ2) 1 le j le 8

micro sim N(0 5) τ sim halfminusCauchy(0 5)

where θj represents the treatment effect in school j and microand τ are the hyper-parameters shared across all schools


0

5

1

7

2 3 4 5 6 7 8θ1 micro τ

marginal and joint k hat diagnoistics

k ha

t

variables

centered joint k =10

0

5

1

7

2 3 4 5 6 7 8θ1 micro τ


variables

nonminuscentered

joint k =64

θ1

minus2 0 2log τ

minus20

0

20

40

Joints of τ and θ1

NUTS

centered ADVI

nonminuscentered ADVI

θ1

2

3

45

67

8microτ

θ1

234

5

67

8

microτ(nonminuscentered)

(centered)minus2

0

2

minus3 0 3

bias of posterior mean

bias

of p

oste

rior

sd

point estimation error

underminus dispersed

overminus dispersed

Figure 5 The upper two panels shows the joint and marginal PSISdiagnostics of the eight-school example The centered parame-terization has k gt 07 for it cannot capture the funnel-shapeddependency between τ and θ The bottom-right panel shows thebias of posterior mean and standard errors of marginal distribu-tions Positive bias of τ leads to over-dispersion of θ

In this hierarchical model the conditional variance of θ isstrongly dependent on the standard deviation τ as shown bythe joint sample of micro and log τ in the bottom-left corner inFigure 5 The Gaussian assumption in ADVI cannot capturesuch structure More interestingly ADVI over-estimates theposterior variance for all parameters θ1 through θ8 as shownby positive biases of their posterior standard deviation inthe last panel In fact the posterior mode is at τ = 0 whilethe entropy penalization keeps VI estimation away from itleading to an overestimation due to the funnel-shape Sincethe conditional expectation E[θi|τ y σ] =

(σminus2j + τminus2

)minus1

is an increasing function on τ a positive bias of τ producesover-dispersion of θ

The top left panel shows the marginal and joint PSIS di-agnostics The joint k is 100 much beyond the thresholdwhile the marginal k calculated through the true marginaldistribution for all θ are misleadingly small due to the over-dispersion

Alerted by such large k researchers should seek some im-provements such as re-parametrization The non-centeredparametrization extracts the dependency between θ and τthrough a transformation θlowast = (θ minus micro)τ

yj |θj sim N(micro+ τθlowastj σ2j ) θlowastj sim N(0 1)

There is no general rule to determine whether non-centeredparametrization is better than the centered one and thereare many other parametrization forms Finding the optimalparametrization can be as hard as finding the true posteriorbut k diagnostics always guide the choice of parametriza-

centeredθ1

KSminustest p= 034

0 05 1

0

2

4

pθ1

p=000 rejectτ

centered

0 05 1

0

5

10

pτ

p=000 rejectτ

nonminuscentered

0 05 1

0

2

4

pτ

Figure 6 In the eight-school example the VSBC diagnostic veri-fies VI estimation of θ1 is unbiased as the distribution of pθ1 issymmetric τ is overestimated in the centered parametrization andunderestimated in the non-centered one as told by the right leftskewness of pτ

tion As shown by the top right panel in Figure 5 the jointk for the non-centered ADVI decreases to 064 which indi-cated the approximation is not perfect but reasonable andusable The bottom-right panel demonstrates that the re-parametrized ADVI posterior is much closer to the truthand has smaller biases for both first and second momentestimations

We can assess the marginal estimation using VSBC diagnos-tic as summarized in Figure 6 In the centered parametriza-tion the point estimation for θ1 is in average unbiased asthe two-sided KS-test is not rejected The histogram for τis right-skewed for we can reject one-sided KS-test withthe alternative to be pτ being stochastically smaller thanpτ Hence we conclude τ is over-estimated in the centeredparameterization On the contrast the non-centered τ isnegatively biased as diagnosed by the left-skewness of pτ Such conclusion is consistent with the bottom-right panel inFigure 5

To sum up this example illustrates how the Gaussian fam-ily assumption can be unrealistic even for a simple hier-archical model It also clarifies VI posteriors can be bothover-dispersed and under-dispersed depending crucially onthe true parameter dependencies Nevertheless the recom-mended PSIS and VSBC diagnostics provide a practicalsummary of the computation result

44 Cancer Classification Using Horseshoe Priors

We illustrate how the proposed diagnostic methods workin the Leukemia microarray cancer dataset that containsD = 7129 features and n = 72 observations Denote y1n

as binary outcome and XntimesD as the predictor the logisticregression with a regularized horseshoe prior (Piironen ampVehtari 2017) is given by

y|β sim Bernoulli(logitminus1 (Xβ)

) βj |τ λ c sim N(0 τ2λ2

j )

λj sim C+(0 1) τ sim C+(0 τ0) c2 sim InvminusGamma(2 8)

where τ gt 0 and λ gt 0 are global and local shrinkageparameters and λ2

j = c2λ2j(c2 + τ2λ2

j

) The regularized


horseshoe prior adapts to the sparsity and allows us to spec-ify a minimum level of regularization to the largest values

ADVI is computationally appealing for it only takes a fewminutes while MCMC sampling takes hours on this datasetHowever PSIS diagnostic gives k = 98 for ADVI sug-gesting the VI approximation is not even close to the trueposterior Figure 7 compares the ADVI and true posteriordensity of β1834 log λ1834 and τ The Gaussian assumptionmakes it impossible to recover the bimodal distribution ofsome β

0 5 10 15

β1834

0

3

6

pos

terio

r de

snity

VI

NUTS

minus5 5 15

log λ1834

0

15

3

minus11 minus8 minus5

log τ

0

5

1

Figure 7 The comparison of ADVI and true posterior density ofθ1834 log λ1834 and τ in the horseshoe logistic regression ADVImisses the right mode of log λ making β prop λ become a spike

0 05 10

2

4

6KSminustest p= 033

dens

ity

β1834

pβ1834

0 05 1

p= 000 reject

plog λ1834

log λ1834

0 05 1

p= 000 reject

plog τ

log τ

Figure 8 VSBC test in the horseshoe logistic regression It tells thepositive bias of τ and negative bias of λ1834 β1834 is in averageunbiased for its symmetric prior

The VSBC diagnostics as shown in Figure 8 tell the neg-ative bias of local shrinkage λ1834 from the left-skewnessof plog λ1834

which is the consequence of the right-missingmode For compensation the global shrinkage τ is over-estimated which is in agreement with the right-skewnessof plog τ β1834 is in average unbiased even though it isstrongly underestimated from in Figure 7 This is becauseVI estimation is mostly a spike at 0 and its prior is symmet-ric As we have explained passing the VSBC test means theaverage unbiasedness and does not ensure the unbiasednessfor a specific parameter setting This is the price that VSBCpays for averaging over all priors

5 Discussion51 The Proposed Diagnostics are Local

As no single diagnostic method can tell all problems theproposed diagnostic methods have limitations The PSISdiagnostic is limited when the posterior is multimodal asthe samples drawn from q(θ) may not cover all the modesof the posterior and the estimation of k will be indifferentto the unseen modes In this sense the PSIS diagnostic is

a local diagnostic that will not detect unseen modes Forexample imagine the true posterior is p = 08N(0 02) +02N(3 02) with two isolated modes Gaussian family VIwill converge to one of the modes with the importance ratioto be a constant number 08 or 02 Therefore k is 0 failingto penalize the missing density In fact any divergencemeasure based on samples from the approximation such asKL(q p) is local

The bi-modality can be detected by multiple over-dispersedinitialization It can also be diagnosed by other divergencemeasures such as KL(p q) = Ep log(qp) which is com-putable through PSIS by letting h = log(qp)

In practice a marginal missing mode will typically lead tolarge joint discrepancy that is still detectable by k such asin Section 44

The VSBC test however samples the true parameter fromthe prior distribution directly Unless the prior is too restric-tive the VSBC p-value will diagnose the potential missingmode

52 Tailoring Variational Inference for ImportanceSampling

The PSIS diagnostic makes use of stabilized IS to diag-nose VI By contrast can we modify VI to give a better ISproposal

Geweke (1989) introduce an optimal proposal distributionbased on split-normal and split-t implicitly minimizingthe χ2 divergence between q and p Following this ideawe could first find the usual VI solution and then switchGaussian to Student-t with a scale chosen to minimize theχ2 divergence

More recently some progress is made to carry out varia-tional inference based on Renyi divergence (Li amp Turner2016 Dieng et al 2017) But a big α say α = 2 is onlymeaningful when the proposal has a much heavier tail thanthe target For example a normal family does not containany member having finite χ2 divergence to a Student-t dis-tribution leaving the optimal objective function defined byDieng et al (2017) infinitely large

There are several research directions First our proposeddiagnostics are applicable to these modified approximationmethods Second PSIS re-weighting will give a more re-liable importance ratio estimation in the Renyi divergencevariational inference Third a continuous k and the cor-responding α are more desirable than only fixing α = 2as the latter one does not necessarily have a finite resultConsidering the role k plays in the importance sampling wecan optimize the discrepancy Dα(q||p) and α gt 0 simulta-neously We leave this for future research


AcknowledgementsThe authors acknowledge support from the Office of NavalResearch grants N00014-15-1-2541 and N00014-16-P-2039the National Science Foundation grant CNS-1730414 andthe Academy of Finland grant 313122

ReferencesAnderson J L A method for producing and evaluating

probabilistic forecasts from ensemble model integrationsJournal of Climate 9(7)1518ndash1530 1996

Blei D M Kucukelbir A and McAuliffe J D Varia-tional inference A review for statisticians Journal ofthe American Statistical Association 112(518)859ndash8772017

Chee J and Toulis P Convergence diagnostics for stochas-tic gradient descent with constant step size arXiv preprintarXiv171006382 2017

Chen L H Shao Q-M et al Normal approximationunder local dependence The Annals of Probability 32(3)1985ndash2028 2004

Cook S R Gelman A and Rubin D B Validation ofsoftware for Bayesian models using posterior quantilesJournal of Computational and Graphical Statistics 15(3)675ndash692 2006

Cortes C Mansour Y and Mohri M Learning bounds forimportance weighting In Advances in neural informationprocessing systems pp 442ndash450 2010

Cortes C Greenberg S and Mohri M Relative deviationlearning bounds and generalization with unbounded lossfunctions arXiv preprint arXiv13105796 2013

Dieng A B Tran D Ranganath R Paisley J andBlei D Variational inference via chi upper bound mini-mization In Advances in Neural Information ProcessingSystems pp 2729ndash2738 2017

Epifani I MacEachern S N Peruggia M et al Case-deletion importance sampling estimators Central limittheorems and related results Electronic Journal of Statis-tics 2774ndash806 2008

Gelman A Carlin J B Stern H S Dunson D BVehtari A and Rubin D B Bayesian data analysisCRC press 2013

Geweke J Bayesian inference in econometric models us-ing Monte Carlo integration Econometrica 57(6)1317ndash1339 1989

Giordano R Broderick T and Jordan M I Covari-ances robustness and variational Bayes arXiv preprintarXiv170902536 2017

Hoffman M D and Gelman A The No-U-Turn sampleradaptively setting path lengths in Hamiltonian MonteCarlo Journal of Machine Learning Research 15(1)1593ndash1623 2014

Hoffman M D Blei D M Wang C and Paisley JStochastic variational inference The Journal of MachineLearning Research 14(1)1303ndash1347 2013

Ionides E L Truncated importance sampling Journal ofComputational and Graphical Statistics 17(2)295ndash3112008

Koopman S J Shephard N and Creal D Testing theassumptions behind importance sampling Journal ofEconometrics 149(1)2ndash11 2009

Kucukelbir A Tran D Ranganath R Gelman A andBlei D M Automatic differentiation variational infer-ence Journal of Machine Learning Research 18(14)1ndash45 2017

Li Y and Turner R E Renyi divergence variational in-ference In Advances in Neural Information ProcessingSystems pp 1073ndash1081 2016

Pflug G C Non-asymptotic confidence bounds for stochas-tic approximation algorithms with constant step sizeMonatshefte fur Mathematik 110(3)297ndash314 1990

Piironen J and Vehtari A Sparsity information and reg-ularization in the horseshoe and other shrinkage priorsElectronic Journal of Statistics 11(2)5018ndash5051 2017

Ranganath R Gerrish S and Blei D Black box varia-tional inference In Artificial Intelligence and Statisticspp 814ndash822 2014

Renyi A et al On measures of entropy and informationIn Proceedings of the Fourth Berkeley Symposium onMathematical Statistics and Probability Volume 1 Con-tributions to the Theory of Statistics The Regents of theUniversity of California 1961

Rezende D J Mohamed S and Wierstra D Stochas-tic backpropagation and approximate inference in deepgenerative models In Proceedings of the 31st Interna-tional Conference on Machine Learning (ICML-14) pp1278ndash1286 2014

Sielken R L Stopping times for stochastic approximationprocedures Probability Theory and Related Fields 26(1)67ndash75 1973


Stan Development Team Stan modeling language usersguide and reference manual httpmc-stanorg2017 Version 217

Stroup D F and Braun H I On a new stopping rulefor stochastic approximation Probability Theory andRelated Fields 60(4)535ndash554 1982

Vehtari A Gelman A and Gabry J Pareto smoothedimportance sampling arXiv preprint arXiv1507026462017

Vehtari A Gabry J Yao Y and Gelman A loo Efficientleave-one-out cross-validation and waic for bayesian mod-els 2018 URL httpsCRANR-projectorgpackage=loo R package version 200

Wada T and Fujisaki Y A stopping rule for stochasticapproximation Automatica 601ndash6 2015 ISSN 0005-1098

Supplement to ldquoYes but Did It Work Evaluating Variational Inferencerdquo

A Sketch of ProofsA1 Proof to Proposition 1 Marginal k in PSIS diagnostic

Proposition 1 For any two distributions p and q with support Θ and the margin index i if there is a number α gt 1satisfying Eq (p(θ)q(θ))

αltinfin then Eq (p(θi)q(θi))

αltinfin

Proof Without loss of generality we could assume Θ = RK otherwise a smooth transformation is conducted

For any 1 le i le K p(θminusi|θi) and q(θminusi|θi) define the conditional distribution of (θ1 θiminus1 θi+1 θK) isin RKminus1

given θi under the true posterior p and the approximation q separately

For any given index α gt 1 Jensen inequality yieldsintRKminus1

(p(θminusi|θi)q(θminusi|θi)

)αq(θminusi|θi) ge

(intRKminus1

p(θminusi|θi)q(θminusi|θi)

q(θminusi|θi))α

= 1

Hence intRK

(p(θ)

q(θ)

)αq(θ)dθ =

intRKminus1

intR

(p(θi)p(θminusi|θi)q(θi)q(θminusi|θi)

)αq(θi)q(θminusi|θi)dθidθminusi

=

intR

(intRKminus1


)αq(θminusi|θi)dθminusi

)(p(θi)

q(θi)

)αq(θi)dθi

geintR

(p(θi)

q(θi)

)αq(θi)dθi

A2 Proof to Proposition 2 Symmetry in VSBC-Test

Proposition 2 For a one-dimensional parameter θ that is of interest Suppose in addition we have(i) the VI approximation q is symmetric(ii) the true posterior p(θ|y) is symmetricIf the VI estimation q is unbiased ie

Eθsimq(θ|y) θ = Eθsimp(θ|y) θforally

Then the distribution of VSBC p-value is symmetricIf the VI estimation is positivelynegatively biased then the distribution of VSBC p-value is rightleft skewed

In the proposition we write q(θ|y) to emphasize that the VI approximation also depends on the observed data

Proof First as the same logic in (Cook et al 2006) when θ(0) is sampled from its prior p(θ) and simulated data y sampledfrom likelihood p(y|θ(0)) (y θ(0)) represents a sample from the joint distribution p(y θ) and therefore θ(0) can be viewedas a draw from p(θ|y) the true posterior distribution of θ with y being observed

We denote q(θ(0)) as the VSBC p-value of the sample θ(0) Also denote Qx(f) as the xminusquantile (x isin [0 1]) of anydistribution f To prove the result we need to show

1minus Pr(q(θ(0)) lt x) = Pr(q(θ(0)) lt 1minus x)forallx isin [0 1]

Supplement to ldquoEvaluating Variational Inferencerdquo

LHS = Pr(q(θ(0)) gt x

)= Pr

(θ(0) gt Qx (q(θ|y))

)

RHS = Pr(θ(0) lt Q1minusx (q(θ|y))

)= Pr

(θ(0) lt 2Eq(θ|y)θ minusQx (q(θ|y))

)= Pr

(θ(0) lt 2Ep(θ|y)θ minusQx (q(θ|y))

)= Pr


)= LHS

The first equation above uses the symmetry of q(θ|y) the second equation comes from the the unbiasedness condition Thethird is the result of the symmetry of p(θ|y)

If the VI estimation is positively biased Eθsimq(θ|y) θ gt Eθsimp(θ|y) θforally then we change the second equality sign into aless-than sign

B Details of Simulation ExamplesIn this section we give more detailed description of the simulation examples in the manuscript We use Stan (StanDevelopment Team 2017) to implement both automatic differentiation variational inference (ADVI) and Markov chainMonte Carlo (MCMC) sampling We implement Pareto smoothing through R package ldquoloordquo (Vehtari et al 2018) We alsoprovide all the source code in httpsgithubcomyao-ylEvaluating-Variational-Inference

B1 Linear and Logistic Regressions

In Section 41 We start with a Bayesian linear regression y sim N(Xβ σ2) without intercept The prior is set as βidi=1 simN(0 1) σ sim gamma(05 05) We fix sample size n = 10000 and number of regressors d = 100 Figure IX displays theStan code

1 data 2 int ltlower=0gt n number of observations we fix n=10000 in the simulation3 int ltlower=0gt d number of predictor variables fix d=1004 matrix [nd] x predictors5 vector [n] y outcome6 7 parameters 8 vector [d] b linear regression coefficient9 real ltlower=0gt sigma linear regression std

10 11 model 12 y sim normal(x b sigma)13 b sim normal(01) prior for regression coefficient14 sigma sim gamma(0505) prior for regression std15 16

Figure IX Stan code for linear regressions

We find ADVI can be sensitive to the stopping time Part of the reason is the objective function itself is evaluated throughMonte Carlo samples producing large uncertainty In the current version of Stan ADVI computes the running average andrunning median of the relative ELBO norm changes Should either number fall below a threshold tol rel obj with thedefault value to be 001 the algorithm is considered converged

In Figure 1 of the main paper we run VSBC test on ADVI approximation ADVI is deliberately tuned in a conservative wayThe convergence tolerance is set as tol rel obj=10minus4 and the learning rate is η = 005 The predictor X105times102 is fixed


in all replications and is generated independently from N(0 1) To avoid multiple-comparison problem we pre-register thefirst and second coefficients β1 β2 and log σ before the test The VSBC diagnostic is based on M = 1000 replications

In Figure 2 we independently generate each coordinate of β from N(0 1) and set a relatively large variance σ = 2 Thepredictor X is generated independently from N(0 1) and y is sampled from the normal likelihood We vary the thresholdtol rel obj from 001 to 10minus5 and show the trajectory of k diagnostics The k estimation IS and PSIS adjustment areall calculated from S = 5times 104 posterior samples We ignore the ADVI posterior sampling time The actual running time isbased on a laptop experiment result (25 GHz processor 8 cores)The exact sampling time is based on the No-U-Turn Sampler(NUTS Hoffman amp Gelman 2014) in Stan with 4 chains and 3000 iterations in each chain We also calculate the root meansquare errors (RMSE) of all parameters ||Ep[(β σ)]minus Eq[(β σ)]||L2 where (β σ) represents the combined vector of all βand σ To account for the uncertainty k running time and RMSE takes the average of 50 repeated simulations

1 data 2 int ltlower=0gt n number of observations3 int ltlower=0gt d number of predictor variables4 matrix [nd] x predictors we vary its correlation during simulations5 intltlower=0upper=1gt y[n] binary outcome6 7 parameters 8 vector[d] beta9

10 model 11 y sim bernoulli_logit(xbeta)12 13

Figure X Stan code for logistic regressions

Figure 3 and 4 in the main paper is a simulation result of a logistic regression

Y sim Bernoulli(logitminus1(βX)

)with a flat prior on β We vary the correlation in design matrix by generating X from N(0 (1minus ρ)Idtimesd + ρ1dtimesd) where1dtimesd represents the d by d matrix with all elements to be 1 In this experiment we fix a small number n = 100 and d = 2since the main focus is parameter correlations We compare k with the log predictive density which is calculated from100 independent test data The true posterior is from NUTS in Stan with 4 chains and 3000 iterations each chain Thek estimation IS and PSIS adjustment are calculated from 105 posterior samples To account for the uncertainty k logpredictive density and RMSE are the average of 50 repeated experiments

B2 Eight-School Model

The eight-school model is named after Gelman et al (2013 section 55) The study was performed for the EducationalTesting Service to analyze the effects of a special coaching program on studentsrsquo SAT-V (Scholastic Aptitude Test Verbal)scores in each of eight high schools The outcome variable in each study was the score of a standardized multiple choice testEach school i separately analyzed the treatment effect and reported the mean yi and standard deviation of the treatmenteffect estimation σi as summarized in Table 1

There was no prior reason to believe that any of the eight programs was more effective than any other or that some were moresimilar in effect to each other than to any other Hence we view them as independent experiments and apply a Bayesianhierarchical normal model

yj |θj sim N(θj σj) θj sim N(micro τ) 1 le j le 8


where θj represents the underlying treatment effect in school j while micro and τ are the hyper-parameters that are sharedacross all schools

There are two parametrization forms being discussed centered parameterization and non-centered parameterizationListing XI and XII give two Stan codes separately The true posterior is from NUTS in Stan with 4 chains and 3000 iterations


School Index j Estimated Treatment Effect yi Standard Deviation of Effect Estimate σj1 28 152 8 103 -3 164 7 115 -1 96 1 117 8 108 12 18

Table 1 School-level observed effects of special preparation on SAT-V scores in eight randomized experiments Estimates are based onseparate analyses for the eight experiments

1 data 2 intltlower=0gt J number of schools3 real y[J] estimated treatment4 realltlower=0gt sigma[J] std of estimated effect5 6

7 parameters 8 real theta[J] treatment effect in school j9 real mu hyper-parameter of mean

10 realltlower=0gt tau hyper-parameter of sdv11 12 model 13 theta sim normal(mu tau)14 y sim normal(theta sigma)15 mu sim normal(0 5) a non-informative prior16 tau sim cauchy(0 5)17 18

Figure XI Stan code for centered parametrization in the eight-school model It leads to strong dependency between tau and theta

each chain The k estimation and PSIS adjustment are calculated from S = 105 posterior samples The marginal k iscalculated by using the NUTS density which is typically unavailable for more complicated problems in practice

The VSBC test in Figure 6 is based onM = 1000 replications and we pre-register the first treatment effect θ1 and group-levelstandard error log τ before the test

As discussed in Section 32 VSBC assesses the average calibration of the point estimation Hence the result depends on thechoice of prior For example if we instead set the prior to be

micro sim N(0 50) τ sim N+(0 25)

which is essentially flat in the region of interesting part of the likelihood and more in agreement with the prior knowledgethen the result of VSBC test change to Figure XIII Again the skewness of p-values verifies VI estimation of θ1 is in averageunbiased while τ is biased in both centered and non-centered parametrization

B3 Cancer Classification Using Horseshoe Priors

In Section 43 of the main paper we replicate the cancer classification under regularized horseshoe prior as first introducedby Piironen amp Vehtari (2017)

The Leukemia microarray cancer classification dataset 1 It contains n = 72 observations and d = 7129 features Xntimesd Xis standardized before any further process The outcome y1n is binary so we can fit a logistic regression

yi|β sim Bernoulli

logitminus1

dsumj=1

βjxij + β0

1The Leukemia classification dataset can be downloaded from httpfeatureselectiocnasuedudatasetsphp


1 data 2 intltlower=0gt J number of schools3 real y[J] estimated treatment4 realltlower=0gt sigma[J] std of estimated effect5 6 parameters 7 vector[J] theta_trans transformation of theta8 real mu hyper-parameter of mean9 realltlower=0gt tau hyper-parameter of sd

10 11 transformed parameters12 vector[J] theta original theta13 theta=theta_transtau+mu14 15 model 16 theta_trans simnormal (01)17 y sim normal(theta sigma)18 mu sim normal(0 5) a non-informative prior19 tau sim cauchy(0 5)20 21

Figure XII Stan code for non-centered parametrization in the eight-school model It extracts the dependency between tau and theta

centeredθ1

KSminustest p= 031

0 05 1

0

5

10

pθ1

p=000 rejectτ

centered

0 05 1

0

5

10

pτ

p=000 rejectτ

nonminuscentered

0 05 1

0

5

10

pτ

Figure XIII The VSBC diagnostic of the eight-school example under a non-informative prior micro sim N(0 50) τ sim N+(0 25) The skewnessof p-values verifies VI estimation of θ1 is in average unbiased while τ is biased in both centered and non-centered parametrization

There are far more predictors than observations so we expect only a few of predictors to be related and therefore have aregression coefficient distinguishable from zero Further many predictors are correlated making it necessary to have aregularization

To this end we apply the regularized horseshoe prior which is a generalization of horseshoe prior

βj |τ λ c sim N(0 τ2λ2j ) c2 sim InvminusGamma(2 8)

λj sim HalfminusCauchy(0 1) τ |τ0 sim HalfminusCauchy(0 τ0)

The scale of the global shrinkage is set according to the recommendation τ0 = 2(n12(dminus 1)

)minus1There is no reason to

shrink intercept so we put β0 sim N(0 10) The Stan code is summarized in Figure XIV

We first run NUTS in Stan with 4 chains and 3000 iterations each chain We manually pick β1834 the coefficient that has thelargest posterior mean The posterior distribution of it is bi-modal with one spike at 0

ADVI is implemented using the same parametrization and we decrease the learning rate η to 01 and the thresholdtol rel obj to 0001

The k estimation is based on S = 104 posterior samples Since k is extremely large indicating VI is far away from the trueposterior and no adjustment will work we do not further conduct PSIS


1 data 2 intltlower=0gt n number of observations3 intltlower=0gt d number of predictors4 intltlower=0upper=1gt y[n] outputs5 matrix[nd] x inputs6 realltlower=0gt scale_icept prior std for the intercept7 realltlower=0gt scale_global scale for the half-t prior for tau8 realltlower=0gt slab_scale9 realltlower=0gt slab_df

10 11 parameters 12 real beta0 intercept13 vector[d] z auxiliary parameter14 realltlower=0gt tau global shrinkage parameter15 vectorltlower=0gt[d] lambda local shrinkage parameter16 realltlower=0gt caux auxiliary17 18 transformed parameters 19 realltlower=0gt c20 vector[d] beta regression coefficients21 vector[n] f latent values22 vectorltlower=0gt[d] lambda_tilde23 c = slab_scale sqrt(caux)24 lambda_tilde = sqrt( cˆ2 square(lambda) (cˆ2 + tauˆ2 square(lambda)) )25 beta = z lambda_tildetau26 f = beta0 + xbeta27 28 model 29 z sim normal(01)30 lambda sim cauchy(01)31 tau sim cauchy(0 scale_global)32 caux sim inv_gamma(05slab_df 05slab_df)33 beta0 sim normal(0scale_icept)34 y sim bernoulli_logit(f)35

Figure XIV Stan code for regularized horseshoe logistic regression

In the VSBC test we pre-register that pre-chosen coefficient β1834 log λ1834 and global shrinkage log τ before the test TheVSBC diagnostic is based on M=1000 replications


of current variational inference computation and considerfurther tuning it or turn to exact sampling like Markov chainMonte Carlo (MCMC) Second in the case when k is smallthe fast convergence rate of the importance-weighted MonteCarlo integration guarantees a better estimation accuracy Insuch sense the PSIS diagnostics could also be viewed as apost-adjustment for VI approximations Unlike the second-order correction Giordano et al (2017) which relies on anun-testable unbiasedness assumption we make diagnosticsand adjustment at the same time

The second diagnostic considers only the quality of themedian of the variational posterior as a point estimate (inGaussian mean-field VI this corresponds to the modal es-timate) This diagnostic assesses the average behavior ofthe point estimate under data from the model and can in-dicate when a systemic bias is present The magnitude ofthat bias can be monitored while computing the diagnosticThis diagnostic can also assess the average calibration ofunivariate functionals of the parameters revealing if theposterior is under-dispersed over-dispersed or biased Thisdiagnostic could be used as a partial justification for usingthe second-order correction of Giordano et al (2017)

2 Is the Joint Distribution Good EnoughIf we can draw a sample (θ1 θS) from p(θ|y) the ex-pectation of any integrable function Ep[h(θ)] can be esti-

mated by Monte Carlo integrationsumSs=1 h(θs)S

SrarrinfinminusminusminusminusminusrarrEp [h(θ)] Alternatively given samples (θ1 θS) froma proposal distribution q(θ) the importance sampling (IS)estimate is

(sumSs=1 h(θs)rs

)sumSs=1 rs where the impor-

tance ratios rs are defined as

rs =p(θs y)

q(θs) (2)

In general with a sample (θ1 θS) drawn from the varia-tional posterior q(θ) we consider a family of estimates withthe form

Ep[h(θ)] asympsumSs=1 h(θs)wssumS

s=1 ws (3)

which contains two extreme cases

1 When ws equiv 1 estimate (3) becomes the plain VI esti-mate that is we completely trust the VI approximationIn general this will be biased to an unknown extentand inconsistent However this estimator has smallvariance

2 When ws = rs (3) becomes importance samplingThe strong law of large numbers ensures it is consistent

as S rarr infin and with small O(1S) bias due to self-normalization But the IS estimate may have a large orinfinite variance

There are two questions to be answered First can we find abetter bias-variance trade-off than both plain VI and IS

Second VI approximation q(θ) is not designed for an op-timal IS proposal for it has a lighter tail than p(θ|y) as aresult of entropy penalization which lead to a heavy righttail of rs A few large-valued rs dominates the summationbringing in large uncertainty But does the finite sampleperformance of IS or stabilized IS contain the informationabout the dispensary measure between q(θ) and p(θ|y)

21 Pareto Smoothed Importance Sampling

The solution to the first question is the Pareto smoothedimportance sampling (PSIS) We give a brief review andmore details can be found in Vehtari et al (2017)

A generalized Pareto distribution with shape parameter kand location-scale parameter (micro τ) has the density

p(y|micro σ k) =

1

σ

(1 + k

(y minus microσ

))minus 1kminus1

k 6= 0

1

σexp

(y minus microσ

) k = 0

PSIS stabilizes importance ratios by fitting a generalizedPareto distribution using the largest M samples of ri whereM is empirically set as min(S5 3

radicS) It then reports the

estimated shape parameter k and replaces the M largest rsby their expected value under the fitted generalized Paretodistribution The other importance weights remain un-changed We further truncate all weights at the raw weightmaximum max(rs) The resulted smoothed weights aredenoted by ws based on which a lower variance estimationcan be calculated through (3)

Pareto smoothed importance sampling can be considered asBayesian version of importance sampling with prior on thelargest importance ratios It has smaller mean square errorsthan plain IS and truncated-IS (Ionides 2008)

22 Using PSIS as a Diagnostic Tool

The fitted shape parameter k turns out to provide the desireddiagnostic measurement between the true posterior p(θ|y)and the VI approximation q(θ) A generalized Pareto dis-tribution with shape k has finite moments up to order 1kthus any positive k value can be viewed as an estimate to

k = inf

kprime gt 0 Eq

(p(θ|y)

q(θ)

) 1kprime

ltinfin

(4)






whereDα (p||q) =1

αminus 1log

intΘ
























p(ξ)

q(ξ)=p(Tminus1(ξ)

)|detJξT


=p (θ)

q(θ)










ki = inf

0 lt kprime lt 1 Eq

(p(θi|y)

q(θi)

) 1kprime

ltinfin



αlt infin then









Pr(yθ(0))





(0)j )


pij = Prθ|y(j)

(θi le [θ

(0)j ]i

)






j from prior p(θ)


)from p(y | θ(0)







margin14 end for
















0 05 10

2

4


dens

ity

β1

pβ1

0 05 1

p= 008

pβ2

β2

0 05 1

p= 000 reject

pσ

log σ



5

7

1

2

k ha

t

relative tolerance

0 20 40 600

5

7

1

2

k ha

t

running time (s)

NUTS sampling

time=2300

0 05 10

01

02R

MS

E

k hat

Raw ADVI

IS

PSIS





(logitminus1(βX)



0 05 10

5

7

1

correlations

k hat


lpd of VI

minus6

minus5

minus4

minus3

0 05 1

correlations


02 05 07 1

0

1

2

k hat



03 05 07 090

25

5

RM

SE

1st Moment

k hat

Raw ADVIISPSIS

03 05 07 090

25

50

k hat

RM

SE

2nd Moment











0

5

1

7

2 3 4 5 6 7 8θ1 micro τ


k ha

t

variables


0

5

1

7

2 3 4 5 6 7 8θ1 micro τ


variables

nonminuscentered

joint k =64

θ1

minus2 0 2log τ

minus20

0

20

40


NUTS

centered ADVI


θ1

2

3

45

67

8microτ

θ1

234

5

67

8


(centered)minus2

0

2

minus3 0 3


bias

of p

oste

rior

sd



overminus dispersed




)minus1






centeredθ1

KSminustest p= 034

0 05 1

0

2

4

pθ1

p=000 rejectτ

centered

0 05 1

0

5

10

pτ

p=000 rejectτ

nonminuscentered

0 05 1

0

2

4

pτ










j )



j = c2λ2j(c2 + τ2λ2

j

) The regularized




0 5 10 15

β1834

0

3

6

pos

terio

r de

snity

VI

NUTS

minus5 5 15

log λ1834

0

15

3


log τ

0

5

1


0 05 10

2

4

6KSminustest p= 033

dens

ity

β1834

pβ1834

0 05 1

p= 000 reject

plog λ1834

log λ1834

0 05 1

p= 000 reject

plog τ

log τ




















































αltinfin







(intRKminus1


q(θminusi|θi))α

= 1

Hence intRK

(p(θ)

q(θ)

)αq(θ)dθ =

intRKminus1

intR



=

intR

(intRKminus1



)(p(θi)

q(θi)

)αq(θi)dθi

geintR

(p(θi)

q(θi)

)αq(θi)dθi











)= Pr


)


)= Pr


)= Pr


)= Pr


)= LHS










































yi|β sim Bernoulli

logitminus1

dsumj=1

βjxij + β0






centeredθ1

KSminustest p= 031

0 05 1

0

5

10

pθ1

p=000 rejectτ

centered

0 05 1

0

5

10

pτ

p=000 rejectτ

nonminuscentered

0 05 1

0

5

10

pτ






















whereDα (p||q) =1

αminus 1log

intΘ
























p(ξ)

q(ξ)=p(Tminus1(ξ)

)|detJξT


=p (θ)

q(θ)










ki = inf

0 lt kprime lt 1 Eq

(p(θi|y)

q(θi)

) 1kprime

ltinfin



αlt infin then









Pr(yθ(0))





(0)j )


pij = Prθ|y(j)

(θi le [θ

(0)j ]i

)






j from prior p(θ)


)from p(y | θ(0)







margin14 end for
















0 05 10

2

4


dens

ity

β1

pβ1

0 05 1

p= 008

pβ2

β2

0 05 1

p= 000 reject

pσ

log σ



5

7

1

2

k ha

t

relative tolerance

0 20 40 600

5

7

1

2

k ha

t

running time (s)

NUTS sampling

time=2300

0 05 10

01

02R

MS

E

k hat

Raw ADVI

IS

PSIS





(logitminus1(βX)



0 05 10

5

7

1

correlations

k hat


lpd of VI

minus6

minus5

minus4

minus3

0 05 1

correlations


02 05 07 1

0

1

2

k hat



03 05 07 090

25

5

RM

SE

1st Moment

k hat

Raw ADVIISPSIS

03 05 07 090

25

50

k hat

RM

SE

2nd Moment











0

5

1

7

2 3 4 5 6 7 8θ1 micro τ


k ha

t

variables


0

5

1

7

2 3 4 5 6 7 8θ1 micro τ


variables

nonminuscentered

joint k =64

θ1

minus2 0 2log τ

minus20

0

20

40


NUTS

centered ADVI


θ1

2

3

45

67

8microτ

θ1

234

5

67

8


(centered)minus2

0

2

minus3 0 3


bias

of p

oste

rior

sd



overminus dispersed




)minus1






centeredθ1

KSminustest p= 034

0 05 1

0

2

4

pθ1

p=000 rejectτ

centered

0 05 1

0

5

10

pτ

p=000 rejectτ

nonminuscentered

0 05 1

0

2

4

pτ










j )



j = c2λ2j(c2 + τ2λ2

j

) The regularized




0 5 10 15

β1834

0

3

6

pos

terio

r de

snity

VI

NUTS

minus5 5 15

log λ1834

0

15

3


log τ

0

5

1


0 05 10

2

4

6KSminustest p= 033

dens

ity

β1834

pβ1834

0 05 1

p= 000 reject

plog λ1834

log λ1834

0 05 1

p= 000 reject

plog τ

log τ




















































αltinfin







(intRKminus1


q(θminusi|θi))α

= 1

Hence intRK

(p(θ)

q(θ)

)αq(θ)dθ =

intRKminus1

intR



=

intR

(intRKminus1



)(p(θi)

q(θi)

)αq(θi)dθi

geintR

(p(θi)

q(θi)

)αq(θi)dθi











)= Pr


)


)= Pr


)= Pr


)= Pr


)= LHS










































yi|β sim Bernoulli

logitminus1

dsumj=1

βjxij + β0






centeredθ1

KSminustest p= 031

0 05 1

0

5

10

pθ1

p=000 rejectτ

centered

0 05 1

0

5

10

pτ

p=000 rejectτ

nonminuscentered

0 05 1

0

5

10

pτ


























ki = inf

0 lt kprime lt 1 Eq

(p(θi|y)

q(θi)

) 1kprime

ltinfin



αlt infin then









Pr(yθ(0))





(0)j )


pij = Prθ|y(j)

(θi le [θ

(0)j ]i

)






j from prior p(θ)


)from p(y | θ(0)







margin14 end for
















0 05 10

2

4


dens

ity

β1

pβ1

0 05 1

p= 008

pβ2

β2

0 05 1

p= 000 reject

pσ

log σ



5

7

1

2

k ha

t

relative tolerance

0 20 40 600

5

7

1

2

k ha

t

running time (s)

NUTS sampling

time=2300

0 05 10

01

02R

MS

E

k hat

Raw ADVI

IS

PSIS





(logitminus1(βX)



0 05 10

5

7

1

correlations

k hat


lpd of VI

minus6

minus5

minus4

minus3

0 05 1

correlations


02 05 07 1

0

1

2

k hat



03 05 07 090

25

5

RM

SE

1st Moment

k hat

Raw ADVIISPSIS

03 05 07 090

25

50

k hat

RM

SE

2nd Moment











0

5

1

7

2 3 4 5 6 7 8θ1 micro τ


k ha

t

variables


0

5

1

7

2 3 4 5 6 7 8θ1 micro τ


variables

nonminuscentered

joint k =64

θ1

minus2 0 2log τ

minus20

0

20

40


NUTS

centered ADVI


θ1

2

3

45

67

8microτ

θ1

234

5

67

8


(centered)minus2

0

2

minus3 0 3


bias

of p

oste

rior

sd



overminus dispersed




)minus1






centeredθ1

KSminustest p= 034

0 05 1

0

2

4

pθ1

p=000 rejectτ

centered

0 05 1

0

5

10

pτ

p=000 rejectτ

nonminuscentered

0 05 1

0

2

4

pτ










j )



j = c2λ2j(c2 + τ2λ2

j

) The regularized




0 5 10 15

β1834

0

3

6

pos

terio

r de

snity

VI

NUTS

minus5 5 15

log λ1834

0

15

3


log τ

0

5

1


0 05 10

2

4

6KSminustest p= 033

dens

ity

β1834

pβ1834

0 05 1

p= 000 reject

plog λ1834

log λ1834

0 05 1

p= 000 reject

plog τ

log τ




















































αltinfin







(intRKminus1


q(θminusi|θi))α

= 1

Hence intRK

(p(θ)

q(θ)

)αq(θ)dθ =

intRKminus1

intR



=

intR

(intRKminus1



)(p(θi)

q(θi)

)αq(θi)dθi

geintR

(p(θi)

q(θi)

)αq(θi)dθi











)= Pr


)


)= Pr


)= Pr


)= Pr


)= LHS










































yi|β sim Bernoulli

logitminus1

dsumj=1

βjxij + β0






centeredθ1

KSminustest p= 031

0 05 1

0

5

10

pθ1

p=000 rejectτ

centered

0 05 1

0

5

10

pτ

p=000 rejectτ

nonminuscentered

0 05 1

0

5

10

pτ




















j from prior p(θ)


)from p(y | θ(0)







margin14 end for
















0 05 10

2

4


dens

ity

β1

pβ1

0 05 1

p= 008

pβ2

β2

0 05 1

p= 000 reject

pσ

log σ



5

7

1

2

k ha

t

relative tolerance

0 20 40 600

5

7

1

2

k ha

t

running time (s)

NUTS sampling

time=2300

0 05 10

01

02R

MS

E

k hat

Raw ADVI

IS

PSIS





(logitminus1(βX)



0 05 10

5

7

1

correlations

k hat


lpd of VI

minus6

minus5

minus4

minus3

0 05 1

correlations


02 05 07 1

0

1

2

k hat



03 05 07 090

25

5

RM

SE

1st Moment

k hat

Raw ADVIISPSIS

03 05 07 090

25

50

k hat

RM

SE

2nd Moment











0

5

1

7

2 3 4 5 6 7 8θ1 micro τ


k ha

t

variables


0

5

1

7

2 3 4 5 6 7 8θ1 micro τ


variables

nonminuscentered

joint k =64

θ1

minus2 0 2log τ

minus20

0

20

40


NUTS

centered ADVI


θ1

2

3

45

67

8microτ

θ1

234

5

67

8


(centered)minus2

0

2

minus3 0 3


bias

of p

oste

rior

sd



overminus dispersed




)minus1






centeredθ1

KSminustest p= 034

0 05 1

0

2

4

pθ1

p=000 rejectτ

centered

0 05 1

0

5

10

pτ

p=000 rejectτ

nonminuscentered

0 05 1

0

2

4

pτ










j )



j = c2λ2j(c2 + τ2λ2

j

) The regularized




0 5 10 15

β1834

0

3

6

pos

terio

r de

snity

VI

NUTS

minus5 5 15

log λ1834

0

15

3


log τ

0

5

1


0 05 10

2

4

6KSminustest p= 033

dens

ity

β1834

pβ1834

0 05 1

p= 000 reject

plog λ1834

log λ1834

0 05 1

p= 000 reject

plog τ

log τ




















































αltinfin







(intRKminus1


q(θminusi|θi))α

= 1

Hence intRK

(p(θ)

q(θ)

)αq(θ)dθ =

intRKminus1

intR



=

intR

(intRKminus1



)(p(θi)

q(θi)

)αq(θi)dθi

geintR

(p(θi)

q(θi)

)αq(θi)dθi











)= Pr


)


)= Pr


)= Pr


)= Pr


)= LHS










































yi|β sim Bernoulli

logitminus1

dsumj=1

βjxij + β0






centeredθ1

KSminustest p= 031

0 05 1

0

5

10

pθ1

p=000 rejectτ

centered

0 05 1

0

5

10

pτ

p=000 rejectτ

nonminuscentered

0 05 1

0

5

10

pτ


















0 05 10

2

4


dens

ity

β1

pβ1

0 05 1

p= 008

pβ2

β2

0 05 1

p= 000 reject

pσ

log σ



5

7

1

2

k ha

t

relative tolerance

0 20 40 600

5

7

1

2

k ha

t

running time (s)

NUTS sampling

time=2300

0 05 10

01

02R

MS

E

k hat

Raw ADVI

IS

PSIS





(logitminus1(βX)



0 05 10

5

7

1

correlations

k hat


lpd of VI

minus6

minus5

minus4

minus3

0 05 1

correlations


02 05 07 1

0

1

2

k hat



03 05 07 090

25

5

RM

SE

1st Moment

k hat

Raw ADVIISPSIS

03 05 07 090

25

50

k hat

RM

SE

2nd Moment











0

5

1

7

2 3 4 5 6 7 8θ1 micro τ


k ha

t

variables


0

5

1

7

2 3 4 5 6 7 8θ1 micro τ


variables

nonminuscentered

joint k =64

θ1

minus2 0 2log τ

minus20

0

20

40


NUTS

centered ADVI


θ1

2

3

45

67

8microτ

θ1

234

5

67

8


(centered)minus2

0

2

minus3 0 3


bias

of p

oste

rior

sd



overminus dispersed




)minus1






centeredθ1

KSminustest p= 034

0 05 1

0

2

4

pθ1

p=000 rejectτ

centered

0 05 1

0

5

10

pτ

p=000 rejectτ

nonminuscentered

0 05 1

0

2

4

pτ










j )



j = c2λ2j(c2 + τ2λ2

j

) The regularized




0 5 10 15

β1834

0

3

6

pos

terio

r de

snity

VI

NUTS

minus5 5 15

log λ1834

0

15

3


log τ

0

5

1


0 05 10

2

4

6KSminustest p= 033

dens

ity

β1834

pβ1834

0 05 1

p= 000 reject

plog λ1834

log λ1834

0 05 1

p= 000 reject

plog τ

log τ




















































αltinfin







(intRKminus1


q(θminusi|θi))α

= 1

Hence intRK

(p(θ)

q(θ)

)αq(θ)dθ =

intRKminus1

intR



=

intR

(intRKminus1



)(p(θi)

q(θi)

)αq(θi)dθi

geintR

(p(θi)

q(θi)

)αq(θi)dθi











)= Pr


)


)= Pr


)= Pr


)= Pr


)= LHS










































yi|β sim Bernoulli

logitminus1

dsumj=1

βjxij + β0






centeredθ1

KSminustest p= 031

0 05 1

0

5

10

pθ1

p=000 rejectτ

centered

0 05 1

0

5

10

pτ

p=000 rejectτ

nonminuscentered

0 05 1

0

5

10

pτ


















0

5

1

7

2 3 4 5 6 7 8θ1 micro τ


k ha

t

variables


0

5

1

7

2 3 4 5 6 7 8θ1 micro τ


variables

nonminuscentered

joint k =64

θ1

minus2 0 2log τ

minus20

0

20

40


NUTS

centered ADVI


θ1

2

3

45

67

8microτ

θ1

234

5

67

8


(centered)minus2

0

2

minus3 0 3


bias

of p

oste

rior

sd



overminus dispersed




)minus1






centeredθ1

KSminustest p= 034

0 05 1

0

2

4

pθ1

p=000 rejectτ

centered

0 05 1

0

5

10

pτ

p=000 rejectτ

nonminuscentered

0 05 1

0

2

4

pτ










j )



j = c2λ2j(c2 + τ2λ2

j

) The regularized




0 5 10 15

β1834

0

3

6

pos

terio

r de

snity

VI

NUTS

minus5 5 15

log λ1834

0

15

3


log τ

0

5

1


0 05 10

2

4

6KSminustest p= 033

dens

ity

β1834

pβ1834

0 05 1

p= 000 reject

plog λ1834

log λ1834

0 05 1

p= 000 reject

plog τ

log τ




















































αltinfin







(intRKminus1


q(θminusi|θi))α

= 1

Hence intRK

(p(θ)

q(θ)

)αq(θ)dθ =

intRKminus1

intR



=

intR

(intRKminus1



)(p(θi)

q(θi)

)αq(θi)dθi

geintR

(p(θi)

q(θi)

)αq(θi)dθi











)= Pr


)


)= Pr


)= Pr


)= Pr


)= LHS










































yi|β sim Bernoulli

logitminus1

dsumj=1

βjxij + β0






centeredθ1

KSminustest p= 031

0 05 1

0

5

10

pθ1

p=000 rejectτ

centered

0 05 1

0

5

10

pτ

p=000 rejectτ

nonminuscentered

0 05 1

0

5

10

pτ




















0 5 10 15

β1834

0

3

6

pos

terio

r de

snity

VI

NUTS

minus5 5 15

log λ1834

0

15

3


log τ

0

5

1


0 05 10

2

4

6KSminustest p= 033

dens

ity

β1834

pβ1834

0 05 1

p= 000 reject

plog λ1834

log λ1834

0 05 1

p= 000 reject

plog τ

log τ




















































αltinfin







(intRKminus1


q(θminusi|θi))α

= 1

Hence intRK

(p(θ)

q(θ)

)αq(θ)dθ =

intRKminus1

intR



=

intR

(intRKminus1



)(p(θi)

q(θi)

)αq(θi)dθi

geintR

(p(θi)

q(θi)

)αq(θi)dθi











)= Pr


)


)= Pr


)= Pr


)= Pr


)= LHS










































yi|β sim Bernoulli

logitminus1

dsumj=1

βjxij + β0






centeredθ1

KSminustest p= 031

0 05 1

0

5

10

pθ1

p=000 rejectτ

centered

0 05 1

0

5

10

pτ

p=000 rejectτ

nonminuscentered

0 05 1

0

5

10

pτ






















































αltinfin







(intRKminus1


q(θminusi|θi))α

= 1

Hence intRK

(p(θ)

q(θ)

)αq(θ)dθ =

intRKminus1

intR



=

intR

(intRKminus1



)(p(θi)

q(θi)

)αq(θi)dθi

geintR

(p(θi)

q(θi)

)αq(θi)dθi











)= Pr


)


)= Pr


)= Pr


)= Pr


)= LHS










































yi|β sim Bernoulli

logitminus1

dsumj=1

βjxij + β0






centeredθ1

KSminustest p= 031

0 05 1

0

5

10

pθ1

p=000 rejectτ

centered

0 05 1

0

5

10

pτ

p=000 rejectτ

nonminuscentered

0 05 1

0

5

10

pτ



















)= Pr


)


)= Pr


)= Pr


)= Pr


)= LHS










































yi|β sim Bernoulli

logitminus1

dsumj=1

βjxij + β0






centeredθ1

KSminustest p= 031

0 05 1

0

5

10

pθ1

p=000 rejectτ

centered

0 05 1

0

5

10

pτ

p=000 rejectτ

nonminuscentered

0 05 1

0

5

10

pτ
















































yi|β sim Bernoulli

logitminus1

dsumj=1

βjxij + β0






centeredθ1

KSminustest p= 031

0 05 1

0

5

10

pθ1

p=000 rejectτ

centered

0 05 1

0

5

10

pτ

p=000 rejectτ

nonminuscentered

0 05 1

0

5

10

pτ





















centeredθ1

KSminustest p= 031

0 05 1

0

5

10

pθ1

p=000 rejectτ

centered

0 05 1

0

5

10

pτ

p=000 rejectτ

nonminuscentered

0 05 1

0

5

10

pτ

















Yes, but Did It Work?: Evaluating Variational InferenceYes, but Did It Work?: Evaluating Variational...

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Transcript of Yes, but Did It Work?: Evaluating Variational InferenceYes, but Did It Work?: Evaluating Variational...