From 346611d7f03c52598e5d6a532c78f684d02eb416 Mon Sep 17 00:00:00 2001 From: Aki Vehtari Date: Tue, 27 Dec 2022 13:52:04 +0200 Subject: [PATCH 1/5] MixIS vignette by Luca Silva and Giacomo Zanella --- data-raw/voice_data.csv | 127 ++++++++++++++++++++++ data-raw/voice_loo_values.csv | 127 ++++++++++++++++++++++ data/voice.rda | Bin 0 -> 301225 bytes data/voice_loo.rda | Bin 0 -> 1582 bytes vignettes/loo2-mixis.Rmd | 198 ++++++++++++++++++++++++++++++++++ 5 files changed, 452 insertions(+) create mode 100644 data-raw/voice_data.csv create mode 100644 data-raw/voice_loo_values.csv create mode 100644 data/voice.rda create mode 100644 data/voice_loo.rda create mode 100644 vignettes/loo2-mixis.Rmd diff --git a/data-raw/voice_data.csv b/data-raw/voice_data.csv new file mode 100644 index 00000000..208fa77e --- /dev/null +++ b/data-raw/voice_data.csv @@ -0,0 +1,127 @@ 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(2022)](https://arxiv.org/abs/2209.09190). These estimators have shown to be useful in presence of outliers but also, and especially, in high-dimensional settings where the model features many parameters. In these contexts it can happen that a large portion of observations lead to high values of Pareto-$k$ diagnostics and potential instability of PSIS-LOO estimators. + +For this illustration we consider a high-dimensional Bayesian Logistic regression model applied to the _Voice_ dataset. + +## Setup: load packages and set seed + +```{r, warnings=FALSE, message=FALSE} +library("rstan") +library("loo") +library("matrixStats") +options(mc.cores = parallel::detectCores(), parallel=FALSE) +set.seed(24877) +``` + +## Model + +This is the Stan code for a logistic regression model with regularized horseshoe prior. The code includes an if statement to include a code line needed later for the MixIS approach. + +```{r stancode_horseshoe} +stancode_horseshoe <- " +data { + int n; + int p; + int y[n]; + matrix [n,p] X; + real scale_global; + int mixis; +} +transformed data { + real nu_global=1; // degrees of freedom for the half-t priors for tau + real nu_local=1; // degrees of freedom for the half-t priors for lambdas + // (nu_local = 1 corresponds to the horseshoe) + real slab_scale=2;// for the regularized horseshoe + real slab_df=100; // for the regularized horseshoe +} +parameters { + vector[p] z; // for non-centered parameterization + real tau; // global shrinkage parameter + vector [p] lambda; // local shrinkage parameter + real caux; +} +transformed parameters { + vector[p] beta; + { + vector[p] lambda_tilde; // 'truncated' local shrinkage parameter + real c = slab_scale * sqrt(caux); // slab scale + lambda_tilde = sqrt( c^2 * square(lambda) ./ (c^2 + tau^2*square(lambda))); + beta = z .* lambda_tilde*tau; + } +} +model { + vector[n] means=X*beta; + vector[n] log_lik; + target += std_normal_lpdf(z); + target += student_t_lpdf(lambda | nu_local, 0, 1); + target += student_t_lpdf(tau | nu_global, 0, scale_global); + target += inv_gamma_lpdf(caux | 0.5*slab_df, 0.5*slab_df); + for (index in 1:n) { + log_lik[index]= bernoulli_logit_lpmf(y[index] | means[index]); + } + target += sum(log_lik); + if (mixis) { + target += log_sum_exp(-log_lik); + } +} +generated quantities { + vector[n] means=X*beta; + vector[n] log_lik; + for (index in 1:n) { + log_lik[index] = bernoulli_logit_lpmf(y[index] | means[index]); + } +} +" +``` + +## Dataset + +The _LSVT Voice Rehabilitation Data Set_ (see [link](https://archive.ics.uci.edu/ml/datasets/LSVT+Voice+Rehabilitation) for details) has $p=312$ covariates and $n=126$ observations with binary response. We construct data list for Stan. +```{r, results='hide', warning=FALSE, message=FALSE, error=FALSE} +data(voice) +y <- voice$y +X <- voice[2:length(voice)] +n <- dim(X)[1] +p <- dim(X)[2] +p0 <- 10 +scale_global <- 2*p0/(p-p0)/sqrt(n-1) +standata <- list(n = n, p = p, X = as.matrix(X), y = c(y), scale_global = scale_global, mixis = 0) +``` + +Note that in our prior specification we divide the prior variance by the number of covariates $p$. This is often done in high-dimensional contexts to have a prior variance for the linear predictors $X\beta$ that remains bounded as $p$ increases. + +## PSIS estimators and Pareto-$k$ diagnostics + +LOO-CV computations are challenging in this context due to high-dimensionality of the parameter space. To show that, we compute PSIS-LOO estimators, which require sampling from the posterior distribution, and inspect the associated Pareto-$k$ diagnostics. + +```{r, results='hide', warning=FALSE} +chains <- 4 +n_iter <- 2000 +warm_iter <- 1000 +stanmodel <- stan_model(model_code = stancode_horseshoe) +fit_post <- sampling(stanmodel, data = standata, chains = chains, iter = n_iter, warmup = warm_iter, refresh = 0) +loo_post <-loo(fit_post) +``` + +```{r} +print(loo_post) +``` + +As we can see the diagnostics signal either "bad" or "very bad" Pareto-$k$ values for roughly $15-30\%$ of the observations which is a significant portion of the dataset. + +## Mixture estimators + +We now compute the mixture estimators proposed in Silva and Zanella (2022). These require to sample from the following mixture of leave-one-out posteriors +\begin{equation} +q_{mix}(\theta) = \frac{\sum_{i=1}^n p(y_{-i}|\theta)p(\theta)}{\sum_{i=1}^np(y_{-i})}\propto p(\theta|y)\cdot \left(\sum_{i=1}^np(y_i|\theta)^{-1}\right). +\end{equation} +The code to generate a Stan model for the above mixture distribution is the same to the one for the posterior, just enabling one line of code with a _LogSumExp_ contribution to account for the last term in the equation above. +``` + if (mixis) { + target += log_sum_exp(-log_lik); + } +``` + +We sample from the mixture and collect the log-likelihoods term. +```{r, results='hide', warnings=FALSE} +standata$mixis <- 1 +fit_mix <- sampling(stanmodel, data = standata, chains = chains, iter = n_iter, warmup = warm_iter, refresh = 0, pars = "log_lik") +log_lik_mix <- extract(fit_mix)$log_lik +``` + +We now compute the mixture estimators, following the numerically stable implementation in Appendix A.2 of [Silva and Zanella (2022)](https://arxiv.org/abs/2209.09190). The code below makes use of the package "matrixStats". + +```{r} +l_common_mix <- rowLogSumExps(-log_lik_mix) +log_weights <- -log_lik_mix - l_common_mix +elpd_mixis <- logSumExp(-l_common_mix) - rowLogSumExps(t(log_weights)) +``` + +## Comparison with benchmark values obtained with long simulations + +To evaluate the performance of mixture estimators (MixIS) we also generate _benchmark values_, i.e.\ accurate approximations of the LOO predictives $\{p(y_i|y_{-i})\}_{i=1,\dots,n}$, obtained by brute-force sampling from the leave-one-out posteriors directly, getting $90k$ samples from each and discarding the first $10k$ as warmup. This is computationally heavy, hence we have saved the results and we just load them in the current vignette. +```{r} +data(voice_loo) +elpd_loo <- voice_loo$elpd_loo +``` + +We can then compute the root mean squared error (RMSE) of the PSIS and mixture estimators relative to such benchmark values. +```{r} +elpd_psis <- loo_post$pointwise[,1] +print(paste("RMSE(PSIS) =",round( sqrt(mean((elpd_loo-elpd_psis)^2)) ,2))) +print(paste("RMSE(MixIS) =",round( sqrt(mean((elpd_loo-elpd_mixis)^2)) ,2))) +``` +Here mixture estimators provide a $\approx 2-3$x reduction in RMSE. Note that this value would increase with the number of samples drawn from the posterior and mixture, since in this example the RMSE of MixIS will exhibit a CLT-type decay while the one of PSIS will converge at a slower rate (this can be verified by running the above code with a larger sample size; see also Figure 3 of Silva and Zanella (2022) for analogous results). + +We then compare the overall ELPD estimates with the brute force one. +```{r} +elpd_psis <- loo_post$pointwise[,1] +print(paste("ELPD (PSIS)=",round(sum(elpd_psis),2))) +print(paste("ELPD (MixIS)=",round(sum(elpd_mixis),2))) +print(paste("ELPD (brute force)=",round(sum(elpd_loo),2))) +``` + +In this example, MixIS provides a more accurate ELPD estimate closer to the brute force estimate, while PSIS severely overestimates the ELPD. Note that low accuracy of the PSIS ELPD estimate is expected in this example given the large number of large Pareto-$k$ values. In this example, the accuracy of MixIS estimate will also improve with bigger MCMC sample size. + +More generally, mixture estimators can be useful in situations where standard PSIS estimators struggle and return many large Pareto-$k$ values. In these contexts MixIS often provides more accurate LOO-CV and ELPD estimates with a single sampling routine (i.e. with a cost comparable to sampling from the original posterior). + +## References + +Silva L. and Zanella G. (2022). Robust leave-one-out cross-validation for high-dimensional Bayesian models. Preprint at [arXiv:2209.09190](https://arxiv.org/abs/2209.09190) + +Vehtari A., Gelman A., and Gabry J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. *Statistics and Computing*, 27(5), 1413--1432. Preprint at [arXiv:1507.04544](https://arxiv.org/abs/1507.04544) + +Vehtari A., Simpson D., Gelman A., Yao Y., and Gabry J. (2022). Pareto smoothed importance sampling. Preprint at [arXiv:1507.02646](https://arxiv.org/abs/1507.02646) + + + From a6875c0d6f82fe306e22d9deb41ee46b1a911941 Mon Sep 17 00:00:00 2001 From: jgabry Date: Tue, 21 Mar 2023 13:52:24 -0600 Subject: [PATCH 2/5] document voice dataset --- R/datasets.R | 10 ++++++++-- 1 file changed, 8 insertions(+), 2 deletions(-) diff --git a/R/datasets.R b/R/datasets.R index 9f0f20f9..e067a85e 100644 --- a/R/datasets.R +++ b/R/datasets.R @@ -6,7 +6,7 @@ #' on CRAN. #' #' @name loo-datasets -#' @aliases Kline milk +#' @aliases Kline milk voice voice_loo #' #' @details #' Currently the data sets included are: @@ -18,7 +18,8 @@ #' Small dataset from Hinde and Milligan (2011) on primate milk #' composition.This data is discussed in detail in McElreath (2016a,2016b). #' [(Link to variable descriptions)](https://www.rdocumentation.org/packages/rethinking/versions/1.59/topics/milk) -#' +#' * `voice`: +#' Voice rehabilitation data from Tsanas et al. (2014). #' @references #' Hinde and Milligan. 2011. *Evolutionary Anthropology* 20:9-23. #' @@ -30,6 +31,11 @@ #' McElreath, R. (2016b). *Statistical rethinking: A Bayesian course with #' examples in R and Stan*. Chapman & Hall/CRC. #' +#' A. Tsanas, M.A. Little, C. Fox, L.O. Ramig: Objective automatic assessment of +#' rehabilitative speech treatment in Parkinson's disease, IEEE +#' Transactions on Neural Systems and Rehabilitation Engineering, Vol. 22, pp. +#' 181-190, January 2014 +#' #' @examples #' str(Kline) #' str(milk) From b3fc294c946417cd2f7606fc85ef99bb1fa913bb Mon Sep 17 00:00:00 2001 From: jgabry Date: Tue, 21 Mar 2023 13:52:47 -0600 Subject: [PATCH 3/5] Update loo-datasets.Rd --- man/loo-datasets.Rd | 9 +++++++++ 1 file changed, 9 insertions(+) diff --git a/man/loo-datasets.Rd b/man/loo-datasets.Rd index 2360af7e..88002300 100644 --- a/man/loo-datasets.Rd +++ b/man/loo-datasets.Rd @@ -4,6 +4,8 @@ \alias{loo-datasets} \alias{Kline} \alias{milk} +\alias{voice} +\alias{voice_loo} \title{Datasets for loo examples and vignettes} \description{ Small datasets for use in \strong{loo} examples and vignettes. The \code{Kline} @@ -22,6 +24,8 @@ McElreath (2016a,2016b). \href{https://www.rdocumentation.org/packages/rethinkin Small dataset from Hinde and Milligan (2011) on primate milk composition.This data is discussed in detail in McElreath (2016a,2016b). \href{https://www.rdocumentation.org/packages/rethinking/versions/1.59/topics/milk}{(Link to variable descriptions)} +\item \code{voice}: +Voice rehabilitation data from Tsanas et al. (2014). } } \examples{ @@ -39,4 +43,9 @@ R package version 1.59. McElreath, R. (2016b). \emph{Statistical rethinking: A Bayesian course with examples in R and Stan}. Chapman & Hall/CRC. + +A. Tsanas, M.A. Little, C. Fox, L.O. Ramig: Objective automatic assessment of +rehabilitative speech treatment in Parkinson's disease, IEEE +Transactions on Neural Systems and Rehabilitation Engineering, Vol. 22, pp. +181-190, January 2014 } From 99c7e7ec989592c335bf4f4f03f459112cbbc5f0 Mon Sep 17 00:00:00 2001 From: Aki Vehtari Date: Thu, 23 Mar 2023 15:20:25 +0200 Subject: [PATCH 4/5] fix voice_loo data files --- data-raw/voice_loo_values.csv | 2 +- data/voice_loo.rda | Bin 1582 -> 1418 bytes 2 files changed, 1 insertion(+), 1 deletion(-) diff --git a/data-raw/voice_loo_values.csv b/data-raw/voice_loo_values.csv index bcda69db..d5164344 100644 --- a/data-raw/voice_loo_values.csv +++ b/data-raw/voice_loo_values.csv @@ -1,4 +1,4 @@ -"","X","bk_loo" +"","X","elpd_loo" "1",0,-0.18696865672193 "2",1,-0.0245826392867627 "3",2,-0.0176466006534053 diff --git a/data/voice_loo.rda b/data/voice_loo.rda index 372a2d0b964ec9dad106e0bf1df02a8d862a6394..eb09376653249720703535147ab8fa977467c613 100644 GIT binary patch literal 1418 zcmV;51$Fu#iwFP!000001MSxPH%*=ijKWzWP?t9MXeV+Gwp6C0# z=l#CteIGX))_Q4{G(iv~f-EjhkSN$CC>}fQ)@>j#f)`>0nUKO(>O|0CKaOueP!JAQ z_%eea7NgiNg9t)=K1QC)fF#f&Bsn)3&8;GXWasC7hvbkvT7s6Mxx7@g4AGDxT8`*w z1yVxFXeC;ORM2Xqiqw!g(m3#~!gNC&M&x=0VLL;A=786qQOjMgI)WQxoX1L1=p zHlj_)9DR>0&}L+bn8*rkLDt9y*&;h+j~tLA+KRTJ?Z^o^BNya~b|5$8j&>psE@KjNSR=pZ_T4x=OJC<;KwP$0ru62T}0g`zNY9EGD3 zC;~;IljszRLeVIO_*(ac`p+HzzXSihcVMo5u~7FRvlMuK&BjvcRv&b5N+?}ZRnX;`{hqd* z34^5;zI0kD)JD4Ge3qDm9)aYlfvF*A2z0iTO>2OY6=ohchscml}Un}FQNExww1 z(@>(zNMx#zpk0+Z+fVI*yWqV$K{f%ZtPQ(q_QB9(ApM#$H3h;D@x_YiEpSDs+pG9O z7JB}Crl(&e28A1JMfrRIq=$_pY2;wupPxiXO-{q@1TSr}_YAZ+&e~bb*g+Zntkff_ zI1DwmYI_xMVe(F<`s?#)(9A5CqsOGcbGm(WW*h-!RuO|O+r{9%O4~qFfB;JCm6?k%%dD-WsJXl#(jG#M_+nfBD-%dHK8G5N0BLI+|L0*g70KF2*eL_uU8mj8l}HYg|ZG zJ?|M1B7kYN)8-OtqtKu=Qhy;i3Hrt=xhq0n!@Fn~x^{gCaO;!>yqg4E%ieAtb%+83 zwelqip6<{;9-&&LbQgZtQfc05Y|j_VGdpKNr$Lr@>5PNzI6VFJeaJz>Lf0nwmKweT`~;eOwzMShjdK*%UK)pb7rN7>wrFT=X3CS=>wtINCEMWQBuv3r zbZc`NfN*E-R8kgnXoxa3ICwuYxdk;rcOa!TdGzkX3COtE$kL4)hd0S>^oKfeQ2(mY zM^C8??`e2g8jd4ApV`eFMu7kkYFJw?aw~W);|!6xxd|rga$?EFRVHD Y6A_QclSm}uVk4aM4{Jt7bJhp|01H00Pyhe` literal 1582 zcmV+}2GRLKT4*^jL0KkKSvx~SLI45KfB*mg|NsC0|NsC0|NsC0|Ns5}|Nj5q|NsC0 z|NsB@|NGDbzV6*1^d8{V?lO2tlVv?jo}s6yqtw&XJf?=8h|tqP=>XHz)6{8@^wN5O zXf)GkH2@hIGGPq^6Gn!a4?xmrXk^+=lr(5S%0000008A!K8WR8j34jc#rqfBLm?kDBm}s7W0MkNd znF+MYdQ8v)FcFBri~t${CIK)A$$$U=002x4OaM#(03!fFnwch;0GI(V0007DCIA2c z00009fB*mh000001i%0Q000000EVIno~BbxHiBu0&E4p}!gjbfb@ zCeCiI&7IyUFEpp8x5JQ!M}Cl}ho{Hr2q;Ksh^WZv2_ccvgQcb>`jeByCn$!y!@ zT{^GAnrf)uvcn4hOLUO=DSpnLBQq@udD3|N`;u(4SV$dIYQ8Beg?8?T*Y2a8u7U~F zGDI5_p>t-92~zMnGna+p}QX$Q|5pC*|@U6Q2>a2+nAJ>lm3sQ z31u$ms|kpVGAH-%9UI9Nh7h-PY~;j^k^Z>bP9E(Jqv^55R}79vEVW21-@{ul8SgMs z0yX3&PZ0VfvV{ZM-?*G)Ns>J$$aRV(stRpD(Jq;z8Flj97CD8WEY0!H)G&gcinrtK zB*-KEM=;hTuXZ>f0#p%KhQxK!xjQ0mB9$H-6ewvso=K=Q^Oweu5-4c84|2ORYd4iK zjQPGQgq>O27Z+2xTiO@be*hjSQkJ%-b?I+Mz`2gS=NZYk$le_GX!TfA-|e&$J23n< zuP6X;kgz2W{P2J}%s|g47%(=82(TFDd^&?mcz^&%q9Fkh7;jRvu;32V$iFd^6T??x z*RTGBlc_@NmN$`q9d%H|5)_a_Kc%8iZ!wBpVAMy-@~8RmGRv?XgwTo<6KJSG#qE@W zhi;N4+ZV)?m@bs_+HsM{sf^Yw6&G|$g0$d&v9c5?vO$Mit#L%Vs-7|opF?Z{2$Bev z1hEbH0F?@IP)z2~x~>5%1)1RzOQ}vtuKtb~S)()(N+civKoaOv?4n7)Af6UcMP?RM z8W%ystY{`dWmzRnOyhq9k4D=5)r@qzj7GL0We>IjD8b?cU4^85U4(yOxQw_bWdnmT)?gNII{`bOYaKb5k+v{v%+I9m zb5dDFAA!8(znGatezo)z$>ky`UmG!pp+BKqsgr-S7C?9hC8}5xM54_$5k+9FP#!1f zEE{iyB$8e=U7)Cngz0+DeScc=Kwt+$dJ4t@ngBQmLjfZ{5J{AUpngVDVLwXc!V*r$ gH*ZA$fDr)1KyAc;zR Date: Thu, 23 Mar 2023 15:20:54 +0200 Subject: [PATCH 5/5] minor edit to take into account variability in the results --- vignettes/loo2-mixis.Rmd | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/vignettes/loo2-mixis.Rmd b/vignettes/loo2-mixis.Rmd index 36c46e4a..5ae0f628 100644 --- a/vignettes/loo2-mixis.Rmd +++ b/vignettes/loo2-mixis.Rmd @@ -172,7 +172,7 @@ elpd_psis <- loo_post$pointwise[,1] print(paste("RMSE(PSIS) =",round( sqrt(mean((elpd_loo-elpd_psis)^2)) ,2))) print(paste("RMSE(MixIS) =",round( sqrt(mean((elpd_loo-elpd_mixis)^2)) ,2))) ``` -Here mixture estimators provide a $\approx 2-3$x reduction in RMSE. Note that this value would increase with the number of samples drawn from the posterior and mixture, since in this example the RMSE of MixIS will exhibit a CLT-type decay while the one of PSIS will converge at a slower rate (this can be verified by running the above code with a larger sample size; see also Figure 3 of Silva and Zanella (2022) for analogous results). +Here mixture estimator provides a reduction in RMSE. Note that this value would increase with the number of samples drawn from the posterior and mixture, since in this example the RMSE of MixIS will exhibit a CLT-type decay while the one of PSIS will converge at a slower rate (this can be verified by running the above code with a larger sample size; see also Figure 3 of Silva and Zanella (2022) for analogous results). We then compare the overall ELPD estimates with the brute force one. ```{r}