Local Positivity for Continuous Treatments

Why Exact-Dose Probability Is the Wrong Test

For a continuous treatment, asking whether a person has positive probability of receiving exactly dose $a_0$ usually gives the same answer at every dose: zero. That fact does not imply that the dose is unsupported. A continuous distribution can place substantial probability in every neighborhood of $a_0$, even though the probability of the single point $a_0$ is zero. Positivity must therefore be described by the treatment distribution near the target dose, conditional on covariates.

Consider one treatment decision, baseline covariates $L$, treatment $A$, outcome $Y$, and potential outcome $Y^a$ under dose $a$. The relevant conditional distribution is that of $A$ given $L=l$. Local positivity asks whether treatment values close to a specified target dose occur among individuals with covariate value $l$. It is separate from exchangeability: even if treatment is unconfounded given $L$, a dose cannot be learned from covariate groups in which nearby doses are absent.

Definition: The conditional support of $A$ given $L=l$ is the set $S_l$ of doses $a_0$ such that every open neighborhood of $a_0$ has positive conditional probability: $$ a_0\in S_l \quad\Longleftrightarrow\quad P\bigl(A\in(a_0-\epsilon,a_0+\epsilon)\mid L=l\bigr)>0 \text{ for every }\epsilon>0. $$ A target dose is locally supported for the covariate values of interest if it belongs to $S_l$ for almost every such $l$, with respect to the covariate distribution relevant to the target population.

When $A\mid L=l$ has a density $f(a\mid l)$ with respect to Lebesgue measure, conditional probabilities of intervals are integrals of that density. In particular, if the conditional distribution has no atom at $a_0$, then $P(A=a_0\mid L=l)=0$, while the probability of an interval around $a_0$ can be positive. The density $f(a_0\mid l)$ is not itself a probability. It has units inverse to the units of treatment and describes probability per unit of dose near $a_0$.

Worked Example: A Dose With Zero Point Probability

Suppose $A\mid L=l\sim N(0,1)$ for every $l$, and the target dose is $a_0=0.5$. The conditional density is $$ f(a\mid l)=\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{a^2}{2}\right). $$ Because this is a continuous distribution, $P(A=0.5\mid L=l)=0$. Nevertheless, the density at the target is $$ f(0.5\mid l)=\frac{1}{\sqrt{2\pi}}e^{-0.125}\approx 0.3521>0. $$ The probability of receiving a dose within $0.1$ of the target is $$ P(0.4<A<0.6\mid L=l)=\Phi(0.6)-\Phi(0.4)\approx 0.7257-0.6554=0.0703. $$ Thus about 7.03% of this conditional treatment distribution lies in that interval. Exact-dose probability is zero, but every neighborhood has positive probability; the target is locally supported.

Support, Density Positivity, and Neighborhood Strength

The support definition is deliberately local: every neighborhood must contain some conditional treatment probability. With a density, a useful stronger condition for a specified neighborhood is that $f(a\mid l)>0$ for Lebesgue-almost every $a$ in that neighborhood. Stronger still is a lower bound $f(a\mid l)\ge c_l>0$ there, almost everywhere. These conditions should not be conflated. Support does not require a density to be uniformly large nearby; density positivity almost everywhere does not supply a common positive lower bound.

A pointwise statement such as $f(a_0\mid l)>0$ needs care. Densities are defined only up to changes on sets of Lebesgue measure zero, so their value at one point can be changed without changing the conditional distribution. If the density has a continuous version near $a_0$, then a positive value of that version implies it is bounded below by a positive number on some sufficiently small neighborhood. Without such regularity, a single density value is not a stable support criterion. Neighborhood probabilities are distributional quantities and avoid this ambiguity.

Positivity must also be assessed for the covariate values relevant to the estimand. For a population-average effect, this generally means almost every $l$ under the target distribution of $L$. A treatment density that is positive for most observed covariate values may still be zero, or extremely small, for a subgroup included in the target population. An average density across covariate groups can conceal this lack of local support.

Worked Example: A Boundary Dose and One-Sided Neighborhoods

Let $A\mid L=l\sim\operatorname{Uniform}(0,1)$, and consider target dose $a_0=0$. Then $P(A=0\mid L=l)=0$. For every $\epsilon>0$, $$ P(A\in(-\epsilon,\epsilon)\mid L=l)=P(0<A<\min(\epsilon,1)\mid L=l)=\min(\epsilon,1)>0. $$ So $0$ is in the conditional support under the neighborhood definition, even though it is a boundary point and there are no observed doses below zero. For a neighborhood radius $\epsilon=0.1$, the probability is $0.1$, not zero.

This boundary case differs from an interior target: information is available only from doses above zero. A two-sided smoothness argument cannot automatically be used at the boundary. Any interpretation of the response exactly at zero must respect the one-sided support and the regularity assumptions used to approach the boundary.

How Local Support Enters Identification

Under consistency $Y=Y^A$ and conditional exchangeability $Y^a\perp A\mid L$, the observed conditional mean $m(a,l)=E[Y\mid A=a,L=l]$ corresponds to the causal mean $\mu(a,l)=E[Y^a\mid L=l]$ at treatment values where the conditional treatment distribution provides information. For continuous $A$, these conditional-mean equalities are naturally statements that hold almost everywhere under the conditional treatment distribution. Moving from that almost-everywhere statement to a particular dose requires local support and, for a pointwise dose-response value, appropriate continuity.

One way to make the role of a neighborhood explicit is to consider a stochastic intervention that assigns doses uniformly in a small interval around $a_0$. Let $q_h(a)$ be the uniform density on $(a_0-h,a_0+h)$, where $h>0$. If the observed density is positive almost everywhere on this interval for the covariate values of interest, then this intervention does not assign positive probability to treatment values absent from the observed conditional distribution. The intervention mean averages causal responses across the interval; shrinking the interval can recover the point-dose mean when that response is continuous.

Theorem: Suppose consistency and $Y^a\perp A\mid L$ hold for the doses under consideration. Suppose $A\mid L=l$ has density $f(a\mid l)$, with $f(a\mid l)>0$ for Lebesgue-almost every $a\in(a_0-h_0,a_0+h_0)$ for almost every $l$ in the target population. Assume there is a version of $\mu(a,l)=E[Y^a\mid L=l]$ continuous in $a$ at $a_0$ for almost every $l$, and a measurable bound $M(l)$ such that $|\mu(a,l)|\le M(l)$ in that interval and $E[M(L)]<\infty$. Then the mean under the uniform intervention on $(a_0-h,a_0+h)$ is identified from the observed data for $0<h\le h_0$, and it converges to $E[Y^{a_0}]$ as $h\downarrow0$.

Proof. The intervention mean is $$ \psi_h=E\left[\int_{a_0-h}^{a_0+h}\mu(a,L)\frac{1}{2h}\,da\right]. $$ For almost every $l$, exchangeability and consistency give $m(a,l)=\mu(a,l)$ for $f(a\mid l)\,da$-almost every $a$. Since $f(a\mid l)>0$ Lebesgue-almost everywhere on the interval, a set with zero $f(a\mid l)\,da$ measure there also has zero Lebesgue measure. Thus the integral defining $\psi_h$ can use the observed conditional mean $m(a,l)$ for Lebesgue-almost every dose in the interval. This proves identification of $\psi_h$.

For almost every $l$, continuity at $a_0$ implies $$ \lim_{h\downarrow0}\frac{1}{2h}\int_{a_0-h}^{a_0+h}\mu(a,l)\,da=\mu(a_0,l). $$ Indeed, the absolute difference from $\mu(a_0,l)$ is at most $\sup_{|a-a_0|<h}|\mu(a,l)-\mu(a_0,l)|$, which tends to zero. The interval average is bounded in absolute value by $M(l)$, and $\mu(a_0,l)$ is also bounded by $M(l)$. Since $E[M(L)]<\infty$, the dominated convergence theorem permits taking the limit through expectation over $L$. Therefore $\psi_h\to E[\mu(a_0,L)]=E[Y^{a_0}]$, by the law of iterated expectation. $\square$

This result makes two distinct requirements visible. Local density support allows the nearby stochastic intervention to be related to observed treatment data. Continuity of the causal mean allows nearby-dose averages to converge to the response at the exact target. Neither requirement says that the probability of the exact dose is positive.

Worked Example: A Local Dose Average Approaches a Point Response

Suppose the conditional causal mean does not depend on $L$ and is $\mu(a)=2+a+\tfrac12a^2$. Consider the uniform intervention on $(0.8,1.2)$, centered at $a_0=1$, so $h=0.2$. The point response is $$ \mu(1)=2+1+\tfrac12(1)^2=3.5. $$ Write $A^*=1+Z$, where $Z$ is uniform on $(-0.2,0.2)$. Then $E[Z]=0$ and $$ E[Z^2]=\frac{1}{0.4}\int_{-0.2}^{0.2}z^2\,dz =\frac{0.2^2}{3}=\frac{0.04}{3}. $$ Expanding the response gives $\mu(1+Z)=3.5+2Z+\tfrac12Z^2$. Hence the intervention mean is $$ E[\mu(A^*)]=3.5+2E[Z]+\tfrac12E[Z^2] =3.5+\frac{0.02}{3}\approx3.5067. $$ The local average is close to, but not exactly, the point response. As the interval shrinks, its $Z^2$ term tends to zero and the average converges to $3.5$.

Checking the Strength of Local Positivity

Local support is a qualitative condition; practical estimation also depends on how much treatment density is available near the target. If the density is positive but tiny, estimates can be unstable because relatively few observations lie near the target dose for a given covariate profile. This is often called weak or near positivity. It is not the same as a strict support failure, but it can make an estimate highly sensitive to modeling and smoothing choices.

Worked Example: Positive Density but Sparse Local Data

Let $L$ be binary. Suppose $A\mid L=0\sim N(3,1)$ and $A\mid L=1\sim N(0,1)$, with target dose $a_0=3$. For $L=0$, the density at the target is $1/\sqrt{2\pi}\approx0.3989$. For $L=1$, it is $$ f(3\mid L=1)=\frac{1}{\sqrt{2\pi}}e^{-9/2}\approx0.004432. $$ Both densities are positive, so the target is locally supported in both groups. But the density ratio is approximately $0.3989/0.004432\approx90.0$, revealing a substantial difference in local availability. For $L=1$, the probability of a dose within $0.1$ of 3 is $$ P(2.9<A<3.1\mid L=1)=\Phi(3.1)-\Phi(2.9)\approx0.00090. $$ That is about nine observations per 10,000 people in this covariate group in expectation. A study may therefore have little practical information about the target response for $L=1$, despite satisfying qualitative local support.

Useful diagnostics examine conditional treatment distributions rather than only the marginal distribution of $A$. In practice, inspect treatment histograms or density estimates within important covariate strata, estimate probabilities in prespecified neighborhoods of the target, and check whether conclusions change as the neighborhood width or outcome model changes. These checks cannot prove positivity, especially in high-dimensional $L$, but they can expose unsupported regions and sparse neighborhoods. A marginally common dose may still be locally rare after conditioning.

The main alternative to a point-dose target is a stochastic intervention that spreads treatment over a range. Its density can be chosen to stay within observed support and avoids requiring an outcome curve at a single dose. The trade-off is interpretive: it estimates an average over a dose distribution rather than the response to exactly $a_0$. A deterministic point-dose estimand is more specific, but depends more heavily on local support and on regularity for the dose-response function.

Common Pitfalls and Limits

Demanding positive probability at one exact dose. For an atomless treatment distribution that probability is zero at every dose. Check neighborhood probabilities or conditional density support instead.
Treating a positive density value as enough by itself. A density is only defined almost everywhere. A pointwise value is not a stable criterion without a regular version, such as a continuous density.
Equating local support with plentiful data. Every neighborhood may have positive probability while its probability is far too small for reliable estimation in a finite sample.
Ignoring covariates or boundaries. Marginal support does not guarantee conditional support, and a boundary target has only one-sided treatment information.

Finally, positivity is not a cure for unmeasured confounding, inconsistent treatment definitions, or a misspecified outcome model. Nor does local support alone justify extrapolating a dose-response curve outside observed neighborhoods. Identification and estimation require their own assumptions; positivity describes where the observed treatment distribution can inform the target.

Check Your Understanding

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

If $A\mid L=l$ has a continuous distribution, must $P(A=a_0\mid L=l)$ be positive for local positivity at $a_0$?
What does it mean for $a_0$ to belong to the conditional support of $A\mid L=l$?
Why does a pointwise claim $f(a_0\mid l)>0$ require care?
What additional condition allows the mean under shrinking uniform dose intervals to converge to a point-dose mean?
Can a dose be locally supported yet practically difficult to estimate? Explain briefly.

Solutions. (1) No. The exact-dose probability is typically zero; local positivity concerns positive probability in every neighborhood. (2) Every interval $(a_0-\epsilon,a_0+\epsilon)$, for $\epsilon>0$, has positive conditional probability. (3) Density versions can differ at a single point without changing the distribution, so neighborhood behavior or a regular density version is needed. (4) Continuity of the causal mean at the target, together with the integrability condition needed to pass the limit through expectation. (5) Yes. The density can be positive but extremely small near the target, leaving very few relevant observations and unstable estimates.