Auditing Conditional-Model Compatibility

Why Audit Conditional Models?

Fully conditional specification (FCS) defines an update by specifying a model for each variable with missing values, conditional on the other variables. This offers flexibility: different variables can have different conditional-model forms. But separate models do not automatically fit together as the full conditional distributions of one joint probability distribution. Auditing this issue helps clarify what probability model, if any, the updates represent.

The audit is a model-coherence check, not a test of whether an FCS chain has mixed well. A set of conditionals can be mutually compatible yet poorly estimated or computationally difficult. Conversely, an FCS chain can appear stable even when its specified conditionals cannot all be the full conditionals of one shared distribution.

The checks below assume the conditionals describe the same set of random variables, condition on the same relevant covariates, and have well-defined distributions on a common support. When support includes structural zeros, the allowable states must be treated explicitly: calculations that divide by zero or compare impossible configurations do not apply. In practice, estimated models also have uncertainty, so a small discrepancy in fitted coefficients is not automatically decisive evidence of an important incompatibility.

Pairwise Constraints: A Binary Example

For two binary variables \(X\) and \(Y\), the full conditionals must agree on the association implied by their joint distribution. Define the joint odds ratio as \(P(X=1,Y=1)P(X=0,Y=0)/[P(X=1,Y=0)P(X=0,Y=1)]\), assuming all four joint probabilities are positive. The same quantity can be recovered from either conditional model: it is the odds of \(X=1\) at \(Y=1\) divided by those odds at \(Y=0\), and also the odds of \(Y=1\) at \(X=1\) divided by those odds at \(X=0\).

This equality follows by writing each conditional odds as a ratio of joint cell probabilities. For example, the first ratio is \((p_{11}/p_{01})/(p_{10}/p_{00})=p_{11}p_{00}/(p_{01}p_{10})\), where \(p_{xy}=P(X=x,Y=y)\). The second conditional gives the same joint-cell ratio. Thus a disagreement is a direct pairwise incompatibility, not merely a difference in model notation.

For binary logistic conditionals with \(0/1\) coding, this check has a convenient coefficient form. If \(\operatorname{logit}P(X=1\mid Y)=a+bY\) and \(\operatorname{logit}P(Y=1\mid X)=c+dX\), compatibility requires \(b=d\). Both coefficients represent the log of the same joint odds ratio. This is a useful diagnostic for a simple two-variable specification, not a universal rule for arbitrary models with additional predictors, interactions, or nonbinary outcomes.

Cycle Consistency and Gaussian Conditionals

Pairwise checks can be organized as a cycle check. Consider a sequence of possible joint states in which each step changes just one variable and the sequence eventually returns to its starting state. For each step, the corresponding conditional distribution supplies the ratio of the probability density or mass at the new state to that at the old state, with other coordinates held fixed. If all conditionals come from a common joint distribution, multiplying those ratios around the closed path must give one: the joint probability ratios telescope and return to their starting value. Equivalently, the sum of their logarithms must be zero.

This gives a general way to think about cycle consistency: different routes through the conditional updates must not imply different relative probabilities for the same states. The requirement depends on the full conditional ratios, not just on a collection of fitted slopes. Pairwise and cycle checks can expose contradictions, but compatibility also requires that the resulting joint distribution be normalizable and have the proposed support.

For normal linear conditionals, the constraints can be made explicit. Suppose, conditional on any fixed covariates, two variables have proposed conditionals \(X\mid Y\sim N(a+bY,\sigma_X^2)\) and \(Y\mid X\sim N(c+dX,\sigma_Y^2)\), with positive conditional variances. If they arise from a bivariate normal joint distribution, its precision matrix (the inverse covariance matrix) must be symmetric. The \(X,Y\) off-diagonal entry derived from the first conditional is \(-b/\sigma_X^2\); from the second it is \(-d/\sigma_Y^2\). Therefore compatibility requires \(b/\sigma_X^2=d/\sigma_Y^2\). Symmetry alone is not enough: the precision matrix must also be positive definite for a proper normal joint distribution.

With three or more normal variables, the same check applies to every pair. If \(X_i\mid X_{-i}\) has a linear conditional mean with coefficient \(b_{ij}\) on \(X_j\), and conditional variance \(\sigma_i^2\), the implied precision entry is \(-b_{ij}/\sigma_i^2\). A common normal joint distribution requires \(b_{ij}/\sigma_i^2=b_{ji}/\sigma_j^2\) for every pair \(i,j\). After assembling the implied precision matrix, also check positive definiteness. These conditions ensure that the proposed Gaussian conditional means and variances correspond to a proper multivariate normal joint distribution.

What an Audit Can and Cannot Establish

A practical audit starts by writing each conditional model precisely: its outcome, predictors, covariates held fixed, support, link or distributional form, and parameter values or constraints. Then compare reciprocal relationships for each variable pair. For binary logistic models, compare the shared association implied by the two directions when the simple two-variable conditions apply. For Gaussian linear conditionals, compare the variance-scaled slopes. For more complex specifications, inspect conditional probability or density ratios around closed paths, and ask whether the implied joint distribution is proper.

A successful set of checks supports a coherent shared model only when the checks cover the relevant constraints and the joint distribution is normalizable. Passing a few pairwise comparisons is not a general proof for arbitrary nonlinear, mixed-type conditionals. Conversely, a failed necessary constraint is enough to rule out the proposed shared joint distribution under the assumptions of that check.

Compatibility is distinct from whether FCS can be run. With incompatible conditionals, the chained updates still define a transition procedure, and a stationary distribution may exist. But that stationary distribution generally cannot have every proposed conditional as its full conditional at once. Its behavior can depend on the entire update scheme, so it should not be described as sampling from a joint distribution whose full conditionals are exactly the incompatible models. Compatibility also does not guarantee good empirical performance: chain exploration and Monte Carlo error remain separate concerns.

The main alternative is to specify a joint model first and derive its full conditionals, which enforces compatibility by construction but may constrain model forms. FCS specifies the conditional models directly and can accommodate different forms by variable, but then coherence must be assessed rather than presumed. In either approach, model adequacy and the intended analysis remain important; compatibility alone does not establish that the assumptions are substantively appropriate.

Common Pitfall

• Wrong reasoning: “Each conditional model fits its variable reasonably well, so together they must describe one joint distribution.” A good fit considered one equation at a time does not ensure that reciprocal associations agree.
• Correction: Check constraints implied by a shared joint distribution, such as equality of binary conditional odds ratios or symmetry of variance-scaled Gaussian slopes. Also check properness and support where relevant.
• Wrong reasoning: “An FCS chain ran and its summaries looked stable, so the specified conditionals were compatible.” Stability of summaries is not a compatibility proof.
• Correction: Treat compatibility as a property of the conditional models. Treat chain behavior as a separate computational question.