Author: Matthew J. Heiner, David B. Dahl, and Samuel B. Johnson
Date: 2024-05-29

# Qslice package

#### Matthew J. Heiner, David B. Dahl, and Samuel B. Johnson

#### 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