Choosing Proposal Distributions for SMC-FCS

Why the Proposal Distribution Matters

In SMC-FCS, an incomplete covariate is drawn from a conditional distribution that is compatible with the substantive model. The desired distribution combines information from the covariate model and the observed outcome. Rejection sampling can generate draws from that target, but its practical performance depends on the distribution used to propose candidates. A proposal that often generates plausible target values can be efficient; a proposal that rarely does so can require many rejected candidates.

Proposal choice is not merely a computational detail. A proposal must cover every value that has positive probability under the target, and its relationship to the target must permit a valid rejection bound. A convenient distribution that fails either condition does not produce valid SMC-FCS draws, regardless of how many candidates are generated.

Definition: Let $X$ be an incomplete covariate, $x_{\mathrm{obs}}$ its observed values, and $z$ the other conditioning variables. For a fixed observed outcome $y$, the SMC-FCS target density or mass function for a missing value $x$ is proportional to $$ g(x)=f_X(x\mid x_{\mathrm{obs}},z)f_Y(y\mid x,x_{\mathrm{obs}},z). $$ Its normalized version is $ \pi(x)=g(x)/Z $, where $ Z=\int g(u)\,du $ for a continuous covariate, or $ Z=\sum_u g(u) $ for a discrete covariate. A proposal $q(x)$ is a normalized density or mass function used to generate candidates.

The factor $f_X$ describes the covariate distribution conditional on the available covariate information; the factor $f_Y$ is the likelihood contribution from the substantive model. The variables conditioned on may be more extensive in a particular analysis, but the principle is the same: the substantive likelihood modifies the covariate-model distribution. The proposal is an algorithmic device, not a replacement target. Changing $q$ can change efficiency, but a correctly implemented rejection sampler continues to target $\pi$.

For a proposal $q$, rejection sampling requires a finite constant $C$ such that $g(x)\leq Cq(x)$ everywhere the target is positive. A proposed candidate $x$ is accepted with probability

a(x)=\frac{g(x)}{Cq(x)}.

For continuous covariates, the inequality must hold over the relevant continuous support; for discrete covariates, it must hold at every target-supported value. In either case, $q(x)>0$ wherever $g(x)>0$. The accepted draws then have density or mass function proportional to $q(x)a(x)=g(x)/C$, which is proportional to the target. This gives the central design goal: choose a proposal with adequate support and a ratio $g(x)/q(x)$ that can be bounded tightly.

Choosing a Proposal for the Covariate Type

For a continuous incomplete covariate, a natural starting point is the conditional distribution from its covariate model, such as a normal regression distribution given observed covariates. If $q(x)=f_X(x\mid x_{\mathrm{obs}},z)$, then the covariate factor cancels in the ratio:

\frac{g(x)}{q(x)}=f_Y(y\mid x,x_{\mathrm{obs}},z).

When the substantive likelihood is bounded as a function of $x$, a bound on that likelihood supplies $C$. This proposal is often simple and correctly reflects the covariate distribution, but it can be inefficient if the outcome likelihood concentrates the target in a small region that the covariate proposal visits infrequently. For a continuous covariate, also check that the proposal respects the variable's domain. A normal proposal on an unrestricted real line, for example, is not automatically appropriate for a strictly positive variable. A truncated, transformed, or otherwise support-compatible proposal may be needed.

For a binary or categorical covariate, the proposal is a set of probabilities over its allowed categories. A categorical proposal can assign probabilities based on the covariate model; the substantive likelihood then changes the relative target weights. The proposal must assign positive probability to every category with positive target mass. Structural restrictions should be built into the set of allowed categories rather than handled by proposing impossible values and hoping they are rejected.

Another option for either covariate type is a proposal closer to the target itself. This may be obtained from an approximation to the target or, in a tractable special case, from its exact conditional distribution. Such proposals can increase acceptance substantially, but they may be harder to derive and must still permit a valid bound. An approximation is not automatically valid just because it resembles the target; the acceptance ratio must correct for the difference.

Worked Example: A Continuous Covariate With a Logistic Outcome

Suppose a hypothetical binary outcome $Y$ and continuous covariate $X$ have substantive model $P(Y=1\mid X=x)=\operatorname{logit}^{-1}(-1+2x)$. For an individual with observed $Y=1$, take the covariate-model conditional distribution as the proposal $q(x)=f_X(x\mid x_{\mathrm{obs}},z)$. Assume this proposal has support on the real line and the substantive model is correctly specified.

Because $Y$ is binary, its likelihood contribution for the observed value $Y=1$ is $f_Y(1\mid x)=\operatorname{logit}^{-1}(-1+2x)$, which is at most 1. Thus $g(x)/q(x)=f_Y(1\mid x)\leq 1$, so $C=1$ is a valid bound. For a proposed value $x=0.5$, the linear predictor is $-1+2(0.5)=0$, and the acceptance probability is

a(0.5)=\frac{f_Y(1\mid 0.5)}{1} =\frac{1}{1+\exp(0)} =0.5000.

This candidate is accepted with probability 0.5000. At $x=1$, the likelihood is $\operatorname{logit}^{-1}(1)=0.7311$, so the acceptance probability is 0.7311. At $x=-1$, it is $\operatorname{logit}^{-1}(-3)=0.04743$, so the proposal is much more likely to be rejected. This variation is appropriate: given $Y=1$, larger values of $X$ have higher target weight under the specified model. The calculation assumes a valid covariate proposal and a logistic likelihood as stated; if the covariate has a restricted domain, the proposal support must be restricted accordingly.

Discrete Proposals and Target Reweighting

For a discrete covariate, the acceptance calculation is especially transparent because it can be checked category by category. If the covariate-model probabilities form $q$, the substantive likelihood reweights those probabilities. A category with a larger likelihood for the observed outcome receives greater target mass relative to its proposal mass. The acceptance step carries out this reweighting without needing to calculate the normalizing constant $Z$.

Worked Example: A Binary Covariate Proposal

Suppose a hypothetical binary covariate has proposal probabilities $q(0)=0.70$ and $q(1)=0.30$. For an observed binary outcome $Y=1$, suppose the substantive model gives likelihoods $f_Y(1\mid X=0)=0.20$ and $f_Y(1\mid X=1)=0.80$. These probabilities are valid binary likelihoods, and both categories have positive proposal mass.

The unnormalized target masses are $g(0)=0.70(0.20)=0.14$ and $g(1)=0.30(0.80)=0.24$. Since $g(x)/q(x)$ is the likelihood, its maximum is $0.80$; take $C=0.80$. The acceptance probabilities are

a(0)=\frac{0.14}{0.80(0.70)}=0.2500, \qquad a(1)=\frac{0.24}{0.80(0.30)}=1.000.

The normalized target probability of category 1 is $0.24/(0.14+0.24)=0.6316$, while its proposal probability is only 0.30. Acceptance preferentially retains category 1, producing the intended target reweighting. The overall acceptance probability is

P(\text{accept})=\sum_{x\in\{0,1\}}q(x)a(x) =0.70(0.2500)+0.30(1.000)=0.4750.

This equals $Z/C=(0.14+0.24)/0.80=0.4750$. The equality is a useful check on the arithmetic. The example assumes both categories are allowed under the covariate model. If category 1 had positive target mass but $q(1)=0$, no finite bound could repair the missing proposal support.

Tailoring a Continuous Proposal to the Target

A proposal that closely follows the target can reduce rejections. The most favorable case is $q(x)=\pi(x)$: then $g(x)/q(x)=Z$ is constant, $C=Z$, and every candidate is accepted. Usually $Z$ or $\pi$ is not known directly, which is why this ideal is not generally available. Nevertheless, a useful approximation can be constructed when the model structure is simple.

Worked Example: Prior Proposal Versus an Exact Normal Proposal

Consider a hypothetical model with $X\sim N(0,1)$ and $Y\mid X=x\sim N(x,1)$, and suppose the observed outcome is $y=1$. The covariate-model proposal is $q_0(x)=f_X(x)$. The target is proportional to $f_X(x)f_Y(1\mid x)$. Completing the square gives

x^2+(1-x)^2 =2\left(x-\frac{1}{2}\right)^2+\frac{1}{2}, \qquad X\mid Y=1\sim N\left(\frac{1}{2},\frac{1}{2}\right).

Here the second parameter of the normal distribution is the variance. With the prior proposal $q_0$, the ratio $g(x)/q_0(x)=f_Y(1\mid x)$ is a normal density with variance 1, so it is bounded above by $C=1/\sqrt{2\pi}=0.3989$. For a proposed value $x=0$, the acceptance probability is

a_0(0)=\frac{f_Y(1\mid 0)}{C} =\frac{(1/\sqrt{2\pi})\exp(-1/2)}{1/\sqrt{2\pi}} =\exp(-1/2)=0.6065.

Instead, use $q_1(x)=N(1/2,1/2)$, which is the exact target in this special case. Then $g(x)/q_1(x)=Z$ everywhere, so $C=Z$ and $a_1(x)=1$ for every candidate. The improved acceptance is not because the target changed: both valid procedures generate $N(1/2,1/2)$ draws. It is because the second proposal already has the target shape. This example assumes the specified normal models and known parameters; in a more complex analysis, an approximate target proposal still requires a valid envelope and the correct ratio $g(x)/q(x)$.

Justifying the Acceptance Rate and Checking a Proposal

For a valid envelope $g(x)\leq Cq(x)$, the probability of accepting one proposal is the proposal-average acceptance probability. In the continuous case it is an integral; in the discrete case it is a sum:

P(\text{accept}) =\int q(x)\frac{g(x)}{Cq(x)}\,dx =\frac{1}{C}\int g(x)\,dx =\frac{Z}{C}.

The same argument with summation gives $P(\text{accept})=\sum_x q(x)g(x)/(Cq(x))=Z/C$ for a discrete covariate. Thus a smaller valid envelope constant generally means higher acceptance. If a candidate-specific acceptance probability exceeds 1, the proposed $C$ is too small or the ratio has been calculated incorrectly; it must not be silently capped at 1, because doing so changes the target.

Before using a proposal, check its support, the bound, and the model conditions that make the target meaningful. In particular, ensure that the covariate model assigns positive density or mass wherever the target does, and that the substantive likelihood and covariate conditional are evaluated for the same conditioning information used to define $g$. For constrained variables, verify that proposed values obey the constraint. In practical runs, a very low acceptance rate, repeated extreme candidates, or computational failures in evaluating a likelihood can indicate a poorly aligned proposal, an inappropriate envelope, or numerical instability. These signals call for checking the calculation and model support, not for changing the target informally.

Specify the target.
Write the covariate conditional multiplied by the substantive likelihood for the observed outcome.

Choose a support-compatible proposal.
Use a continuous density or discrete mass function that covers every target-supported value and respects known constraints.

Find and verify an envelope.
Establish a finite $C$ with $g(x)\leq Cq(x)$ over the full support; then compute $a(x)=g(x)/(Cq(x))$.

Assess efficiency without changing the target.
Use the acceptance rate and proposal-target alignment to consider alternatives, while retaining the exact correction ratio.

Comparison, Limitations, and Common Pitfalls

Proposal choice	Strength	Main concern
Covariate conditional	Often directly available from the covariate model; simple likelihood correction	May be inefficient when the outcome strongly concentrates the target
Truncated or transformed continuous proposal	Can respect positivity or other domain restrictions	Its density must include the correct normalization or transformation adjustment
Categorical proposal over allowed levels	Easy to inspect category by category	Zero proposal probability for a target-supported category invalidates the sampler
Target-informed approximation	May yield higher acceptance by matching the target's location and spread	Requires a valid envelope and careful evaluation of both target and proposal

A common mistake is to assume that the proposal must itself equal the target, or, conversely, that any convenient proposal is acceptable. Neither is correct. The proposal can differ from the target, but the rejection ratio must account for that difference and the proposal must cover the target support. Another mistake is to compare proposals only by their raw spread: acceptance depends on the ratio $g/q$, not on whether a proposal looks broad or narrow in isolation. A narrow proposal can be excellent if centered on the target, but invalid if it excludes target-supported values.

Rejection sampling also has practical limits. A low acceptance rate means many candidate draws are needed, and an envelope that is difficult to establish can make a seemingly attractive proposal unusable. A proposal that improves acceptance does not correct misspecification of the covariate model or substantive model. Nor does it by itself resolve parameter uncertainty or other issues in an imputation procedure. The essential comparison is between valid proposals targeting the same conditional distribution, not between methods that implicitly change that distribution.

Key takeaway: Choose a proposal with support at least as broad as the SMC-FCS target, a tractable density or mass function, and a tight valid envelope. Covariate-model proposals are convenient baselines; target-informed proposals can improve acceptance, provided the acceptance ratio corrects exactly for the proposal.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A proposal assigns zero mass to one category, but the substantive likelihood and covariate model give that category positive target mass. Can a larger finite envelope constant make the rejection sampler valid? Explain.
A binary covariate proposal has probabilities 0.40 and 0.60. The likelihoods for the observed outcome are 0.25 and 0.75, respectively. Find a valid $C$, both acceptance probabilities, and the overall acceptance rate.
Why can a proposal closer to the target improve acceptance without changing the target distribution?
For $g(x)\leq Cq(x)$, derive the overall acceptance probability in terms of $Z$ and $C$.

Solutions. (1) No. At the unsupported category, $q(x)=0$ while $g(x)>0$, so no finite $C$ satisfies $g(x)\leq Cq(x)$. The proposal must be changed to cover that category. (2) The ratios $g(x)/q(x)$ are the likelihoods, so $C=0.75$. The acceptance probabilities are $0.25/0.75=0.3333$ and $0.75/0.75=1.000$. The overall rate is $0.40(0.3333)+0.60(1)=0.7333$. (3) Better alignment can reduce variation in $g(x)/q(x)$, allowing a smaller valid envelope. The acceptance correction still makes accepted draws proportional to $g$, so the target remains unchanged. (4) Average $a(x)=g(x)/(Cq(x))$ over proposals: $\int q(x)a(x)\,dx=(1/C)\int g(x)\,dx=Z/C$; replace the integral with a sum for a discrete covariate.