Tutorial 09: Bootstrap and Pareto smoothed importance sampling

Finn Lindgren

Introduction

In this lab session you will explore

• Exchangeability test for prediction scores
• Pareto smoothed importance sampling

1. Open one of your github lab repository clone projects
2. During this lab, you can work in a .R file, but working with code chunks in a .Rmd is recommended.
3. Make sure you update the StatCompLab package to version 21.9.0 or higher.

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Solution: The accompanying Tutorial09Solutions tutorial/vignette documents contain the solutions explicitly, to make it easier to review the material after the workshops.

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Filement model prediction comparison

Leave-one-out prediction

Revisit the Tutorial 6 and load the 3D printer filament data:

data("filament1", package = "StatCompLab")

In tutorial 6, the two models A and B were estimated with

Predictions and scores were done with

pred_A <- filament1_predict(fit_A, newdata = filament1)
pred_B <- filament1_predict(fit_B, newdata = filament1)
score_A <- cbind(pred_A, filament1) %>%
  mutate(
    se = proper_score("se", Actual_Weight, mean = mean),
    ds = proper_score("ds", Actual_Weight, mean = mean, sd = sd),
    interval = proper_score("interval", Actual_Weight,
                            lwr = lwr, upr = upr, alpha = 0.1)
  )
score_B <- cbind(pred_B, filament1) %>%
  mutate(
    se = proper_score("se", Actual_Weight, mean = mean),
    ds = proper_score("ds", Actual_Weight, mean = mean, sd = sd),
    interval = proper_score("interval", Actual_Weight,
                            lwr = lwr, upr = upr, alpha = 0.1)
  )

as well as with a 50% estimation/prediction data split.

Implement a function leave1out(data, model) that performs leave-one-out cross validation for the selected model. the two models; for each $i=1,\dots,N$, estimate the model parameters using $\{(x_j,y_j), j\neq i\}$, compute the prediction information based on $x_i$ for prediction model $F_i$, and compute the required scores $S(F_i,y_i)$.

The output should be a data.frame with $N$ rows that extends the original data frame with four additional columns mean, sd, se and ds of leave-one-out prediction means, standard deviations, and prediction scores for the Squared Error and Dawid-Sebastiani scores.

The following code should work if your code is correct:

score_A <- leave1out(filament1, model = "A")
score_B <- leave1out(filament1, model = "B")
ggplot() +
  geom_point(aes(CAD_Weight, se, colour = "A"), data = score_A) +
  geom_point(aes(CAD_Weight, se, colour = "B"), data = score_B)
ggplot() +
  geom_point(aes(CAD_Weight, ds, colour = "A"), data = score_A) +
  geom_point(aes(CAD_Weight, ds, colour = "B"), data = score_B)

Exchangeability test

We want to investigate whether one model is better at predicting than the other. If they are equivalent, it should not matter, on average, if we randomly swap the A and B prediction scores within each leave-one-out score pair $(S^A_i, S^B_i)$. Use the test statistic $\frac{1}{N}\sum_{i=1}^N (S^A_i - S^B_i)$, and make a Monte Carlo estimate of the p-value for a test of exchangeability between model predictions from A and from B against the alternative hypothesis that B is better than A. First compute the test statistic for the two prediction scores.

Hints: For this particular test statistic, one possible approach is to first compute the pairwise score differences and then generate randomisation samples by sampling random sign changes. Compute the test statistic for each randomly altered set of values and compare with the original test statistic. See the lecture on exchangeability for more information. Start with $J=1000$ iterations when testing your code. Increase to 10000 for more precise results.

Pareto smoothed importance sampling

Explore the Pareto smoothed importance sampling (PSIS) method. Use a t-distribution with 10 degrees of freedom, and importance sampling distribution $x\sim \mathsf{Normal}(0, \sigma^2)$ for different values of $\sigma$. Use $N=100$ or more samples.

Use the loo::psis() function to compute Pareto smoothed importance log-weights and diagnostics. What effect does different values of $\sigma$ have on the $k$ parameter estimate? (the $\hat{k}$ value is in the $diagnostics$pareto_k field of the loo::psis() output.) Plot the original and smoothed log-weights as a function of $x$, and with stat_ecdf, for different values of $\sigma$, including $\sigma=1$ and $\sigma=2$.