Why View Stabilized Weights as Likelihood Ratios?
An inverse probability treatment weight compares the treatment probability actually observed for a person with the probability assigned to that person under a reference treatment mechanism. Stabilization makes this comparison explicit: it uses a probability conditional on a reduced history in the numerator and a probability conditional on the full measured history in the denominator. The resulting ratio is not just a rescaled inverse probability. It is a likelihood ratio between two conditional treatment distributions, and that interpretation tells us precisely what the weights do.
Consider one binary treatment decision. Let \(A\in\{0,1\}\) be treatment, \(H\) the full measured history available before treatment, and \(R=r(H)\) a reduced history determined by \(H\). Write \(g(a\mid H)=P(A=a\mid H)\) for the observed treatment mechanism. Let \(g_0(a\mid R)\) be a normalized numerator treatment mechanism: for every \(R\), \(g_0(a\mid R)\geq 0\) and \(\sum_{a=0}^1 g_0(a\mid R)=1\). A common choice is \(g_0(a\mid R)=P(A=a\mid R)\).
For a ratio to be defined, require \(g(A\mid H)>0\) for every treatment value that can occur under the numerator mechanism at that history: \(g_0(a\mid R)>0\) implies \(g(a\mid H)>0\), for almost every \(H\). This is the relevant positivity condition. If the numerator assigns positive probability to a treatment that the observed mechanism rules out for some full histories, no finite likelihood ratio can create that reference mechanism there.
The word “likelihood ratio” refers to conditional assignment probabilities. For each possible \(a\) and \(H\), the ratio compares the probability of assigning \(a\) under the reduced-history mechanism with its probability under the full-history mechanism. It is not a ratio of outcome likelihoods, and it does not mean that a realized weight is itself a probability.
What Distribution Does the Ratio Create?
Let \(P\) denote the observed-data distribution of \((H,A)\). Define a new distribution \(Q\) by weighting observations according to \(SW\). This construction is valid when the positivity condition above holds and the numerator probabilities sum to one. Under \(P\), the joint distribution factors into the history distribution and the observed treatment mechanism. Multiplying that joint distribution by the ratio replaces the treatment mechanism while leaving the history distribution intact.
For almost every \(H\), sum over the two possible treatment values using the observed conditional treatment probabilities.
Multiply the observed joint probability of history and treatment by the numerator-to-denominator ratio.
The denominator cancels, leaving the original history distribution multiplied by the numerator treatment mechanism.
Here is the complete derivation. Conditional on \(H\), the observed treatment probabilities are \(g(a\mid H)\). Therefore,
Taking expectations gives \(E_P[SW]=1\), so \(SW\) can define a probability reweighting. For any measurable set \(B\) of histories and any \(a\in\{0,1\}\),
Summing the final expression over \(a\) gives \(Q(H\in B)=\int_B p_H(h)\,dh=P(H\in B)\), because \(g_0\) sums to one. Dividing the joint expression by this unchanged history distribution gives \(Q(A=a\mid H)=g_0(a\mid R)\). The same calculation applies with sums for discrete histories or integrals with respect to the appropriate history measure. \(\square\)
A useful special case follows when \(g_0(a\mid R)=P(A=a\mid R)\). Since \(R\) is determined by \(H\), the reweighted joint distribution of \(R\) and \(A\) is then unchanged:
Thus this choice changes the treatment distribution conditional on the full history, but preserves the observed joint distribution of the reduced history and treatment. If a different normalized numerator is chosen, the history distribution remains unchanged, but the joint distribution of \(R\) and \(A\) need not be preserved.
Worked Examples
Worked Example: A Constant Numerator Removes History Dependence
Let \(X\) be a binary full history with \(P(X=0)=P(X=1)=0.5\). Suppose \(R\) contains no history, and the observed treatment probabilities are \(P(A=1\mid X=0)=0.2\) and \(P(A=1\mid X=1)=0.8\). First calculate the numerator: \(P(A=1)=0.5(0.2)+0.5(0.8)=0.5\), so \(P(A=0)=0.5\).
For a treated person with \(X=0\), \(SW=0.5/0.2=2.5\). For a treated person with \(X=1\), \(SW=0.5/0.8=0.625\). For a control with \(X=0\), the denominator is \(P(A=0\mid X=0)=0.8\), so \(SW=0.5/0.8=0.625\). For a control with \(X=1\), the denominator is \(0.2\), so \(SW=0.5/0.2=2.5\).
Check the reweighted joint cells. The observed probability of \((X=0,A=1)\) is \(0.5(0.2)=0.1\); multiplying by its weight gives \(0.1(2.5)=0.25\). For \((X=1,A=1)\), the result is \(0.5(0.8)(0.625)=0.25\). The two control cells also each have reweighted probability \(0.25\). The new distribution therefore has \(P_Q(X=x,A=a)=0.25\) in all four cells: \(X\) and \(A\) are independent under \(Q\), while each retains its marginal distribution.
Worked Example: Preserving a Reduced History
Let \(R\) and \(Z\) be binary, with \(P(R=0)=0.6\), \(P(R=1)=0.4\), and \(P(Z=0\mid R)=P(Z=1\mid R)=0.5\). Take \(H=(R,Z)\). The full-history treatment probabilities are \(P(A=1\mid R=0,Z=0)=0.1\), \(P(A=1\mid R=0,Z=1)=0.4\), \(P(A=1\mid R=1,Z=0)=0.6\), and \(P(A=1\mid R=1,Z=1)=0.9\).
Average over \(Z\) to obtain the numerator probabilities: \(P(A=1\mid R=0)=0.5(0.1)+0.5(0.4)=0.25\), and \(P(A=1\mid R=1)=0.5(0.6)+0.5(0.9)=0.75\). The corresponding control probabilities are \(0.75\) and \(0.25\).
For \(R=0,Z=0\), the treated weight is \(0.25/0.1=2.5\) and the control weight is \(0.75/0.9=5/6\). For \(R=0,Z=1\), they are \(0.25/0.4=0.625\) and \(0.75/0.6=1.25\). For \(R=1,Z=0\), they are \(0.75/0.6=1.25\) and \(0.25/0.4=0.625\). For \(R=1,Z=1\), they are \(0.75/0.9=5/6\) and \(0.25/0.1=2.5\).
For example, conditional on \(R=0,Z=0\), the weighted treated mass is \(0.1(2.5)=0.25\), and the weighted control mass is \(0.9(5/6)=0.75\). These sum to one, and the treatment probability in this history stratum becomes \(0.25=P(A=1\mid R=0)\). The same calculation in every \((R,Z)\) stratum gives treatment probability \(0.25\) when \(R=0\) and \(0.75\) when \(R=1\). The full history \(H\) has not been discarded from the denominator; rather, the reweighted assignment mechanism depends on \(R\) and not on the additional variable \(Z\).
Worked Example: A Weighted Treatment-Group Mean
Use the first example’s distribution and weights. Suppose the outcome is \(Y=10\) for treated people with \(X=0\), \(Y=0\) for treated people with \(X=1\), and \(Y=4\) for controls at either value of \(X\). In the observed data, the treated group has probabilities \(0.1\) and \(0.4\) in the \(X=0\) and \(X=1\) cells. Its unweighted mean is \(\{0.1(10)+0.4(0)\}/0.5=2\).
After weighting, the treated cell masses are \(0.1(2.5)=0.25\) and \(0.4(0.625)=0.25\). Their total is \(0.5\), and their weighted outcome total is \(0.25(10)+0.25(0)=2.5\). The weighted treated mean is therefore \(2.5/0.5=5\), the average of the two outcome values under the unchanged marginal distribution of \(X\). For controls, each \(X\) cell has weighted mass \(0.25\); the weighted control mean is \(\{0.25(4)+0.25(4)\}/0.5=4\). This calculation illustrates the new assignment distribution; interpreting a weighted outcome contrast causally requires the relevant causal assumptions as well.
Assumptions, Diagnostics, and Alternatives
The likelihood-ratio derivation is a statement about the true assignment probabilities. In applications, these probabilities are usually estimated. The fitted numerator and denominator must each be valid conditional probability models, and the denominator must not be zero for treatment values that occur. The more general requirement is support inclusion: whenever the numerator mechanism gives a treatment positive probability at a history, the observed mechanism must also give it positive probability there. A small denominator can produce a very large weight even when it is not exactly zero.
Useful checks include examining the range and distribution of the fitted denominator probabilities, identifying observations with especially large ratios, and comparing weighted treatment frequencies across the histories the numerator is intended to retain or remove. In the simple constant-numerator example, treatment should no longer depend on \(X\) in the reweighted distribution. In the reduced-history example, the reweighted treatment probability should agree across values of \(Z\) within each value of \(R\). Failure of these checks can indicate a coding error, an incorrect probability calculation, or that estimated models do not represent the intended mechanisms. With estimated probabilities, finite-sample deviations from the exact identities are expected.
Stabilization should not be treated as a guarantee of smaller variance. Multiplying the ordinary inverse probability weight \(1/g(A\mid H)\) by \(g_0(A\mid R)\) can reduce the scale of weights in a given application, but the numerator can also vary across observations; there is no universal guarantee that every tail or variance measure improves. Nor does stabilization solve a positivity violation: ratios can still be extreme when the denominator is close to zero.
| Approach | What it uses | Key distinction |
|---|---|---|
| Unstabilized inverse probability weight | \(1/g(A\mid H)\) | Uses only the inverse observed treatment probability; it does not specify a numerator assignment mechanism. |
| Stabilized likelihood-ratio weight | \(g_0(A\mid R)/g(A\mid H)\) | Replaces the full-history mechanism by a stated reference mechanism while retaining the history distribution. |
| Normalization of observed weights | A rescaling of a set of weights | Can set their sample mean to a chosen value, but does not by itself create the conditional treatment mechanism \(g_0\). |
A common mistake is to say that stabilized weights are “the inverse probability weights divided by their mean.” That is not generally correct. The numerator is a treatment probability conditional on \(R\), and can differ between treatment values and reduced-history strata. Normalizing weights by a single constant may alter their overall scale, but it does not generally reproduce the likelihood ratio or its change-of-measure properties. Another mistake is to infer that the reweighted data are randomized in every sense. The result proved here concerns treatment conditional on the measured history \(H\); it does not establish causal identification, correct measurement, or balance on unmeasured variables.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- If \(g_0(1\mid R)=0.3\) and \(g(1\mid H)=0.1\), what is the stabilized weight for an individual with \(A=1\)?
- Why must the numerator probabilities sum to one over the possible treatment values?
- If \(g_0(a\mid R)=P(A=a\mid R)\), which joint distribution is preserved by reweighting?
- Does the likelihood-ratio construction guarantee that stabilized weights have less variability than unstabilized weights? Explain.
Solutions. (1) The weight is \(0.3/0.1=3\), because the numerator and denominator both refer to the observed treatment value \(A=1\). (2) Normalization makes \(E_P[SW\mid H]=\sum_a g_0(a\mid R)=1\), so the weights define a probability reweighting; otherwise they would not define the stated conditional treatment distribution. (3) It preserves the joint distribution of \(R\) and \(A\), since the numerator is the observed conditional treatment probability given \(R\). The marginal distribution of \(H\) is also preserved under any normalized numerator. (4) No. The numerator changes the scale observation by observation, so it may reduce variability in some settings but does not guarantee lower variance or eliminate extreme ratios.