Vertical Weighted Strips (VWS)

• Vertical Weighted Strips (VWS) is a rejection-sampling framework for weighted densities f(x) ∝ w(x)g(x), enabling stratified proposals with explicit rejection probability bounds.
• The method partitions the support into regions, upper-bounds the weight function on each strip, and constructs a finite mixture proposal that is both sampleable and efficient.
• VWS extends classical vertical strip methods by incorporating nonuniform base densities and adaptive refinement strategies, including self-tuning for repeated sampling applications.

Searching arXiv for the VWS papers and related follow-up work. Vertical Weighted Strips (VWS) is a rejection-sampling framework for target distributions that can be written as weighted densities, that is, densities of the form $f(x)\propto w(x)g(x)$ where $g$ is a normalized base density and $w$ is a nonnegative weight function. The method constructs a finite mixture proposal by partitioning the support into regions and majorizing the weight function on each region, so that the proposal remains convenient to sample from while admitting an explicit upper bound on the rejection probability before sampling begins. In the formulation introduced in “Rejection Sampling with Vertical Weighted Strips” (Raim et al., 2024), VWS is particularly relevant when the normalizing constant $\psi$ is intractable and when assumptions required by adaptive rejection sampling and related methods are not met.

1. Weighted-density formulation and problem setting

VWS is built for targets whose possibly unnormalized density has the weighted form

$$f(x) = \frac{f_0(x)}{\psi},\qquad f_0(x) = w(x)\,g(x),\qquad \psi = \int_\Omega f_0(x)\,d\nu(x),$$

where $\Omega$ is the support, $\nu$ is a dominating measure, $g(x)$ is a normalized base density, $w(x)\ge 0$ is a weight function, and $\psi$ is the normalizing constant (Raim et al., 2024). In this setting, $g$0 is assumed evaluable and sampleable, while $g$1 is nonnegative and finite on the relevant support. For the univariate analysis, $g$2 is partitioned into intervals $g$3 (Raim et al., 2024).

The central computational difficulty is that generating variates from $g$4 may be nontrivial even when $g$5 is simple, because the weight function may be complicated and $g$6 may be intractable. Classical rejection sampling remains applicable in principle, but it requires a proposal that both dominates the target and is practically sampleable. VWS exploits the factorization $g$7 so that proposal design focuses on bounding the weight function rather than the full target density (Raim et al., 2024).

This formulation also clarifies the scope of the method. Some VWS constructions use only boundedness of $g$8, whereas others use stripwise log-convexity or log-concavity of $g$9 to obtain sharper majorizers (Raim et al., 2024). A plausible implication is that the method separates structural information about the target into a tractable base $w$0 and a localized envelope problem for $w$1.

2. Proposal construction by stripwise majorization

The defining step in VWS is the choice of a majorizing function $w$2 and the induced proposal

$w$3

Because $w$4, the rejection-sampling constant can be taken as $w$5 (Raim et al., 2024). This is the basic simplification that drives the framework.

To improve efficiency, the support is partitioned into disjoint regions

$w$6

and each region is assigned a strip-specific majorizer $w$7 satisfying

$w$8

The global majorizer is then

$w$9

with

$\psi$0

If $\psi$1, the strip integrals are

$\psi$2

and

$\psi$3

(Raim et al., 2024).

The proposal density therefore admits the finite-mixture representation

$\psi$4

with mixture weights

$\psi$5

and strip components

$\psi$6

Each component is the base distribution $\psi$7, truncated to $\psi$8 and reweighted by $\psi$9 (Raim et al., 2024).

The terminology “vertical weighted strips” is explained geometrically in the original paper. In classical vertical strip methods one partitions the horizontal axis and uses vertical bounds on the density; here one partitions the support into horizontal regions $$f(x) = \frac{f_0(x)}{\psi},\qquad f_0(x) = w(x)\,g(x),\qquad \psi = \int_\Omega f_0(x)\,d\nu(x),$$0 and bounds the weight $$f(x) = \frac{f_0(x)}{\psi},\qquad f_0(x) = w(x)\,g(x),\qquad \psi = \int_\Omega f_0(x)\,d\nu(x),$$1 vertically by a constant or a linear function in $$f(x) = \frac{f_0(x)}{\psi},\qquad f_0(x) = w(x)\,g(x),\qquad \psi = \int_\Omega f_0(x)\,d\nu(x),$$2, while retaining the base density $$f(x) = \frac{f_0(x)}{\psi},\qquad f_0(x) = w(x)\,g(x),\qquad \psi = \int_\Omega f_0(x)\,d\nu(x),$$3 (Raim et al., 2024).

3. Rejection-sampling algorithm and efficiency criteria

Under the VWS envelope $$f(x) = \frac{f_0(x)}{\psi},\qquad f_0(x) = w(x)\,g(x),\qquad \psi = \int_\Omega f_0(x)\,d\nu(x),$$4, the rejection sampler proceeds by drawing $$f(x) = \frac{f_0(x)}{\psi},\qquad f_0(x) = w(x)\,g(x),\qquad \psi = \int_\Omega f_0(x)\,d\nu(x),$$5, then $$f(x) = \frac{f_0(x)}{\psi},\qquad f_0(x) = w(x)\,g(x),\qquad \psi = \int_\Omega f_0(x)\,d\nu(x),$$6, and accepting if

$$f(x) = \frac{f_0(x)}{\psi},\qquad f_0(x) = w(x)\,g(x),\qquad \psi = \int_\Omega f_0(x)\,d\nu(x),$$7

Since $$f(x) = \frac{f_0(x)}{\psi},\qquad f_0(x) = w(x)\,g(x),\qquad \psi = \int_\Omega f_0(x)\,d\nu(x),$$8, the acceptance probability is

$$f(x) = \frac{f_0(x)}{\psi},\qquad f_0(x) = w(x)\,g(x),\qquad \psi = \int_\Omega f_0(x)\,d\nu(x),$$9

and the expected number of proposals per accepted draw is $\Omega$0 (Raim et al., 2024).

Sampling from the proposal requires sampling from the finite mixture. One first draws a strip index $\Omega$1 with $\Omega$2, then samples

$\Omega$3

The paper identifies three implementation routes: drawing from $\Omega$4 and conditioning when that is easy, exploiting analytic forms such as exponential-family structure, or using inverse CDF within each strip through

$\Omega$5

and the mixture CDF $\Omega$6 (Raim et al., 2024). In many examples, including exponential-family and uniform bases, the necessary integrals and inverses are available in closed form (Raim et al., 2024).

Because $\Omega$7 is often unavailable, VWS emphasizes computable efficiency diagnostics. For each strip one may define a minorizer $\Omega$8 with

$\Omega$9

and

$\nu$0

Then the rejection probability satisfies

$\nu$1

If the trivial minorizer $\nu$2 is used, the bound equals the true rejection probability (Raim et al., 2024).

The same quantity controls approximation error. Proposition 2 in the original paper shows that if $\nu$3 and $\nu$4, then for any measurable set $\nu$5,

$\nu$6

so a small rejection bound implies that the proposal is also close to the target in total variation (Raim et al., 2024). This suggests that VWS serves both as an exact-sampling device and as a controlled approximation framework.

4. Partitioning, majorizers, and adaptive bifurcation

The efficiency of VWS depends strongly on the partition $\nu$7. The original paper proposes an adaptive bifurcation scheme based on each region’s contribution to the rejection bound. For a partition with $\nu$8 regions, the contribution of region $\nu$9 is

$g(x)$0

and $g(x)$1 equals the computable upper bound on the rejection probability (Raim et al., 2024).

The bifurcation strategy starts from a coarse partition, often a single region. At each refinement step, it draws a region index with probability proportional to $g(x)$2, splits that region at a midpoint, and recomputes $g(x)$3, $g(x)$4, and $g(x)$5. The process continues until a desired number of regions is reached or the rejection bound falls below a target threshold (Raim et al., 2024). The continuous-case midpoint rules in the paper explicitly cover finite endpoints, doubly infinite intervals, and one-sided infinite intervals (Raim et al., 2024).

Two stripwise envelope families are emphasized. In constant VWS, one uses

$g(x)$6

which requires only boundedness of $g(x)$7 on each strip (Raim et al., 2024). In linear VWS, one works on the log scale and uses a majorizer of the form

$g(x)$8

with a corresponding minorizer based on an endpoint chord or another tangent, depending on convexity (Raim et al., 2024). The linear construction is available when $g(x)$9 is log-convex or log-concave on each strip and is especially effective when $w(x)\ge 0$0 belongs to an exponential family, because the resulting components remain analytically sampleable (Raim et al., 2024).

The paper treats the refinement problem as heuristic rather than globally optimal. It states that minimizing $w(x)\ge 0$1 over partitions and majorizers leads to more efficient samplers, but exact optimal partitions are not derived (Raim et al., 2024). A plausible implication is that VWS trades formal optimality for envelope constructions that remain explicit and implementable.

5. Relation to rejection-sampling literature

VWS is positioned relative to several established proposal-construction paradigms. Standard rejection sampling often selects simple envelopes such as uniform or exponential distributions without exploiting the weighted factorization $w(x)\ge 0$2. VWS instead focuses on majorizing the weight $w(x)\ge 0$3 while retaining the tractable base $w(x)\ge 0$4, which can be more efficient when $w(x)\ge 0$5 is already close to the target (Raim et al., 2024).

The method also generalizes classical vertical strips. In the bounded-support case, standard vertical strips correspond to taking $w(x)\ge 0$6 uniform on $w(x)\ge 0$7, $w(x)\ge 0$8, and stripwise majorizers and minorizers given by the maximum and minimum of $w(x)\ge 0$9 on each strip. VWS extends this by allowing nonuniform bases and by working with $\psi$0 rather than the full target density (Raim et al., 2024).

Compared with Adaptive Rejection Sampling (ARS) and its variants, the distinction is structural. ARS requires the log of the target to be concave, while ARMS relaxes this at the cost of a Metropolis step and Markov dependence. VWS does not require log-concavity of the target $\psi$1 itself; it only needs log-convexity or log-concavity of the weight function $\psi$2 on strips when linear majorizers are used, and constant majorizers require only boundedness. It produces i.i.d. exact samples rather than a Markov chain and provides a pre-sampling bound on rejection probability (Raim et al., 2024).

The original paper also compares VWS with direct sampling methods based on auxiliary variables. Raim’s 2023 direct-sampling formulation is described there as viewable as a special case of VWS where the auxiliary density is majorized by a step function. The comparison given is that VWS works directly on $\psi$3, whereas direct sampling reformulates the problem through an auxiliary variable on $\psi$4, with possible numerical difficulties when the auxiliary distribution is extremely concentrated (Raim et al., 2024).

A common misconception is that VWS is merely a restatement of generic rejection sampling. The distinctive feature is not the acceptance-rejection step itself, but the stripwise construction of a finite-mixture proposal through majorization of $\psi$5, together with the computable bound on rejection probability before sampling (Raim et al., 2024).

6. von Mises–Fisher examples and practical implementation

The main applications in the original paper are based on the von Mises–Fisher (VMF) distribution. After transformation to a univariate coordinate $\psi$6, the target density becomes

$\psi$7

and the paper uses the decomposition

$\psi$8

so that $\psi$9 is a doubly truncated exponential (Raim et al., 2024). On $g$00, $g$01, and its second derivative shows that $g$02 is log-concave for $g$03, log-convex for $g$04, and constant for $g$05 (Raim et al., 2024). This makes the example particularly suitable for comparing constant and linear VWS.

For constant VWS, the strip majorizer is $g$06 and the minorizer is $g$07, so each strip component is simply the truncated base $g$08 on that interval (Raim et al., 2024). For linear VWS, the log-weight is bounded by a tangent line, and because $g$09 is exponential-family form, each mixture component becomes a truncated exponential with parameter $g$10, from which sampling is analytic via inverse CDF (Raim et al., 2024).

The paper reports that for moderately fine partitions such as $g$11, linear VWS achieves rejection rates substantially lower than vertical strips and constant VWS, and that with $g$12 up to 100, linear VWS often yields extremely low rejection probabilities, for example well below $g$13, even at high $g$14 and dimension $g$15, outperforming the Ulrich–Wood VMF sampler in some regimes (Raim et al., 2024). It recommends constant VWS when ease of implementation is primary and linear VWS when $g$16 is log-convex or log-concave and $g$17 is exponential-family-like (Raim et al., 2024).

The implementation guidance given in the paper is procedural. One chooses a factorization $g$18, picks $g$19 close to $g$20 and convenient to sample from, partitions the support, constructs constant or linear majorizers and minorizers, computes $g$21 and $g$22, evaluates the rejection bound, and then samples from the mixture proposal and accepts with probability $g$23 (Raim et al., 2024). The paper also lists practical pitfalls: infinite or unbounded weights near support edges, numerical instability in $g$24 or Bessel functions, and excessive region counts. Suggested remedies include slight support truncation, working on the log scale, using scaled special functions, and moderating refinement when VWS is embedded in Gibbs samplers (Raim et al., 2024).

7. Subsequent development: self-tuned VWS within Gibbs

A later development revisits the method for repeated conditional simulation in Gibbs samplers. “Self-Tuned Rejection Sampling within Gibbs and a Case Study in Small Area Estimation” (Raim et al., 21 Sep 2025) considers univariate conditionals that can be written as weighted densities and maintains persistent VWS proposals that tune themselves gradually instead of rebuilding a new proposal from scratch for each conditional draw.

In that follow-up formulation, constant VWS is used with stripwise piecewise-constant majorizers and minorizers, and the same rejection-bound quantity

$g$25

governs refinement (Raim et al., 21 Sep 2025). The paper introduces a self-tuning rule in which rejected points can be added as new knots when the bound is too crude, while strips with sufficiently small contribution can be removed when coarsening does not push the bound above a threshold (Raim et al., 21 Sep 2025). This extends VWS from a single-target proposal-construction method to a persistent proposal mechanism inside Gibbs.

The case study concerns a Bayesian small area estimation model in which the unfamiliar conditional for $g$26 is proportional to an inverse-gamma density times a lognormal density, so that

$g$27

There, self-tuned VWS is used to obtain exact draws within Gibbs while controlling proposal-maintenance cost (Raim et al., 21 Sep 2025). The paper reports that self-tuned VWS is much cheaper than rebuilding proposals repeatedly, yet markedly improves mixing relative to an independent Metropolis–Hastings step for the same conditional family (Raim et al., 21 Sep 2025).

This follow-up work suggests an important interpretation of VWS. The original method is a finite-mixture rejection-sampling design for weighted densities (Raim et al., 2024); the later extension shows that the same design can be made persistent and adaptive in repeated-sampling contexts without changing its exact-draw character (Raim et al., 21 Sep 2025).