# Qslice package

#### 2024-05-29

The quantile slice sampler of Heiner et al. (2024+), as well as other popular slice samplers, are implemented in the qslice package. Several utility functions for specifying pseudo-target distributions, for diagnostics and for tuning, are also included.

Other implemented methods include the generalized elliptical (Nishihara et al., 2014), latent (Li and Walker, 2023), and stepping-out (Neal, 2003) slice samplers, and independence Metropolis-Hastings sampler.

All sampling functions in the qslice package perform a single scan that can be iteratively called as part of an MCMC routine.

# Slice sampling

Let $$g(x)$$ represent an unnormalized target density. Augment the target with a latent random variable $$V \mid X = x \sim \text{Uniform}(0, g(x))$$, yielding a joint density for $$X$$ and $$V$$ that is proportional to a constant.

Simple slice samplers first draw from the conditional distribution of $$V$$ given above, and then from the conditional of $$X \mid V = v$$, which is uniform on the set $$A = \{ x : v < g(x) \}$$ known as the slice region. If $$A$$ is available analytically, a custom slice sampler can be implemented. The slice samplers available in this package repeatedly sample until a draw on the slice region is found. All use the shrinking method in Neal (2003), which we demonstrate below.

A Gibbs sampler drawing $$V_t \mid X_{t-1}$$ followed by $$X_t \mid V_t$$ produces a sequence of draws $$\{X_t\}$$ whose distribution converges to the marginal density proportional to $$g(x)$$.

## Example

Each sampling method requires a function to evaluate the natural logarithm of a (typically unnormalized) target density. For the sake of illustration, we use a target that is proportional to a gamma density (with shape $$\alpha$$ and rate 1):

$g(x) = x^{\alpha - 1} \exp(-x) \, 1_{(x>0)} \, .$

## Target is a Gamma(shape = alpha, rate = 1) distribution
alpha <- 2.5
ltarget <- function(x) ifelse(x > 0.0, (alpha - 1.0) * log(x) - x, -Inf)

Note that log-target functions in qslice rely on lexical scoping. For example, the parameter alpha is not passed as an argument to function ltarget. If the target represents a full conditional distribution that changes at each iteration of an MCMC algorithm, the user must either: 1. change the value of $$\alpha$$ within the environment in which ltarget is defined or 2. redefine the ltarget function.

We first demonstrate the stepping-out-and-shrinkage sampler of Neal (2003), a general tool that is robust to the choice of its tuning parameter w. The sampling function slice_stepping_out performs a single draw, so we embed it within an MCMC algorithm.

## Set up MCMC
n_iter <- 1e3
x_sample <- numeric(n_iter + 1)
x_sample[1] <- 0.5  # initialize
n_eval <- 0  # count target evaluations

## Run step-and-shrink slice sampler
for (i in 2:(n_iter+1)) {
state <- slice_stepping_out(x = x_sample[i-1],
log_target = ltarget,
w = 2.0)
n_eval <- n_eval + state$nEvaluations x_sample[i] <- state$x
}

## Check samples
n_eval / n_iter  # target evaluations per MCMC iteration
#> [1] 6.679
hist(x_sample, freq = FALSE, n = 30)
curve(dgamma(x, shape = alpha), col = "blue", lwd = 2, add = TRUE)


## Null hypothesis: Samples are iid Gamma(alpha, 1)
ks.test(x_sample[seq(1, n_iter, by = 5)], pgamma, shape = alpha)
#>
#>  Asymptotic one-sample Kolmogorov-Smirnov test
#>
#> data:  x_sample[seq(1, n_iter, by = 5)]
#> D = 0.076299, p-value = 0.1947
#> alternative hypothesis: two-sided

# Quantile slice sampler

The quantile slice sampler uses an approximation to the target density, called a pseudo-target. Let $$\hat{G}(\cdot)$$ represent the distribution function (CDF) of the pseudo-target with inverse (i.e., quantile) function $$\hat{G}^{-1}(\cdot)$$ and density $$\hat{g}(\cdot)$$. We use the probability integral transform to define a new random variable $$\psi = \hat{G}(X) \in [0,1]$$. The original target can be written as $g(x) = \frac{g(x)}{\hat{g}(x)}\hat{g}(x) \, ,$ which after transformation becomes $h(\psi) = \frac{g(\hat{G}^{-1}(\psi))}{\hat{g}(\hat{G}^{-1}(\psi))} \, 1_{(0 \le \psi \le 1)} \, .$ If the pseudo-target approximates the target well, $$h(\psi)$$ will be close to a constant function, yielding an efficient slice sampler. The quantile slice sampler uses target $$h(\psi)$$ and adaptively shrinks (around the previously sampled value of $$\psi$$) to draw a sample belonging the slice region $$A_\psi \subseteq [0,1]$$.

## Example

The quantile slice sampler requires a pseudo-target. This is specified in the qslice package with a list containing two functions: the log-density ld and inverse-CDF (quantile) q functions. qslice offers pseudo-target constructor functions with options for several distribution families. See help(pseudo_list). Here we use a truncated $$t$$ distribution:

pseu <- pseudo_list(family = "t",
params = list(loc = 0, sc = 3, degf = 1),
lb = 0)

When specifying a pseudo-target, please keep in mind that:

1. The support of the pseudo-target should match (or exceed) that of the original target. Note the use of lb = 0 to set the pseudo-target’s lower bound in the call to pseudo_list.
2. The pseudo-target should have tails that are at least as heavy as the original target. For this reason, we recommend the Student-$$t$$ distribution.

We visualize $$g(x)$$, $$\hat{g}(x)$$ and $$h(\psi)$$ below.

utility_pseudo(pseudo = pseu, log_target = ltarget,
type = "function",
utility_type = "AUC", plot = TRUE)

#> [1] 0.4907685

The function utility_pseudo measures how close $$h(\cdot)$$ is to a constant function. Here we use AUC $$\in (0, 1]$$, defined as the area under the curve $$h(\psi) / \max_z(h(z))$$.

The function slice_quantile performs a single draw using the quantile slice sampler. It also outputs draws of $$\psi$$ (called u in the output), which can be used as a diagnostic for pseudo-target fitness.

## Set up MCMC
n_iter <- 1e3
x_sample <- psi_sample <- numeric(n_iter + 1)
x_sample[1] <- psi_sample[1] <- 0.5  # initialize
n_eval <- 0  # count target evaluations

## Run step-and-shrink slice sampler
for (i in 2:(n_iter+1)) {
state <- slice_quantile(x = x_sample[i-1],
log_target = ltarget,
pseudo = pseu)
n_eval <- n_eval + state$nEvaluations x_sample[i] <- state$x
psi_sample[i] <- state$u } ## Check samples n_eval / n_iter # target evaluations per iteration of MCMC #> [1] 2.669  hist(x_sample, freq = FALSE, n = 30) curve(dgamma(x, shape = alpha), col = "blue", lwd = 2, add = TRUE) hist(psi_sample, freq = FALSE, n = 30) auc(u = psi_sample) # calculate AUC from transformed samples #> [1] 0.4634259  ## Null hypothesis: Samples are iid Gamma(alpha, 1) ks.test(x_sample[seq(1, n_iter, by = 10)], pgamma, shape = alpha) #> #> Asymptotic one-sample Kolmogorov-Smirnov test #> #> data: x_sample[seq(1, n_iter, by = 10)] #> D = 0.085335, p-value = 0.4602 #> alternative hypothesis: two-sided ## Optimal pseudo-targets and tuning The quantile slice sampler is effective even with crude approximations to the target. We can also find a pseudo-target within the Student-$$t$$ family that maximizes AUC. pseu_opt <- pseudo_opt(log_target = ltarget, type = "function", family = "t", degf = c(1, 5, 20), lb = 0, utility_type = "AUC", plot = TRUE) Alternatively, we can use initial samples from the target to specify a pseudo-target. This gives a simple procedure for “tuning” the sampler. pseu_opt <- pseudo_opt(samples = x_sample, type = "samples", family = "t", degf = c(1, 5, 20), lb = 0, utility_type = "AUC", plot = TRUE, nbins = 20)  names(pseu_opt) #> [1] "pseudo" "utility" "utility_type" "opt" "nbins" #> [6] "tol_int" "tol_opt" names(pseu_opt$pseudo)
#> [1] "d"      "ld"     "q"      "p"      "txt"    "params" "lb"     "ub"
#> [9] "family"
pseu_opt$pseudo$txt
#> [1] "t(loc = 1.51, sc = 1.94, degf = 20) I(0 < x < Inf)"

We conclude by running the quantile slice sampler again with an optimized pseudo-target.

## Set up MCMC
n_iter <- 1e3
x_sample <- psi_sample <- numeric(n_iter + 1)
x_sample[1] <- psi_sample[1] <- 0.5  # initialize
n_eval <- 0  # count target evaluations

## Run step-and-shrink slice sampler
for (i in 2:(n_iter+1)) {
state <- slice_quantile(x = x_sample[i-1],
log_target = ltarget,
pseudo = pseu_opt$pseudo) n_eval <- n_eval + state$nEvaluations
x_sample[i] <- state$x psi_sample[i] <- state$u
}

## Check samples
n_eval / n_iter  # target evaluations per iteration of MCMC
#> [1] 2.254

hist(x_sample, freq = FALSE, n = 30)
curve(dgamma(x, shape = alpha), col = "blue", lwd = 2, add = TRUE)

hist(psi_sample, freq = FALSE, n = 30)

auc(u = psi_sample) # calculate AUC from transformed samples
#> [1] 0.5853801

## Null hypothesis: Samples are iid Gamma(alpha, 1)
ks.test(x_sample[seq(1, n_iter, by = 10)], pgamma, shape = alpha)
#>
#>  Asymptotic one-sample Kolmogorov-Smirnov test
#>
#> data:  x_sample[seq(1, n_iter, by = 10)]
#> D = 0.073568, p-value = 0.6513
#> alternative hypothesis: two-sided

# References

• Heiner, M. J., Johnson, S. B., Christensen, J. R., and Dahl, D. B. (2024+), “Quantile Slice Sampling,” arXiv preprint arXiv:###.

• Li, Y. and Walker, S. G. (2023), “A latent slice sampling algorithm,” Computational Statistics and Data Analysis, 179, 107652. https://doi.org/10.1016/j.csda.2022.107652

• Neal, R. M. (2003), “Slice sampling,” The Annals of Statistics, 31, 705-767. https://doi.org/10.1214/aos/1056562461

• Nishihara, R., Murray, I., and Adams, R. P. (2014), “Parallel MCMC with Generalized Elliptical Slice Sampling,” Journal of Machine Learning Research, 15, 2087-2112. https://jmlr.org/papers/v15/nishihara14a.html