Consistency Around Conditional Cycles

Why Conditional Relationships Must Agree Around a Cycle

In fully conditional specification, each variable is modeled conditionally on the others. These models describe overlapping features of the same data distribution. If they are all conditional distributions from one joint distribution, their implications must agree: changing the variables in one order or another cannot give different relative probabilities for the same two states.

This requirement is especially clear for binary variables. Moving from one joint state to another by changing one variable at a time, each move has a conditional odds ratio. If a joint distribution exists, multiplying those odds along any path between the same starting and ending states must produce the same result. A discrepancy around a cycle is therefore evidence that the specified conditional models cannot all be conditionals of one joint distribution.

A cycle here means a loop through possible joint states, not a causal loop. For example, the state sequence \(000 \rightarrow 100 \rightarrow 110 \rightarrow 111\) changes one binary variable at each step. The same endpoints can be connected by changing the variables in a different order. Compatibility requires that both routes imply the same ratio of probability at \(111\) to probability at \(000\).

Deriving the Path-Agreement Requirement

Let \(X,Y,Z\) be binary variables, each taking values 0 or 1, and suppose a joint probability mass function \(p(x,y,z)\) exists and assigns positive probability to all states under consideration. Define the conditional odds for changing \(X\) from 0 to 1, while fixing \(Y=y,Z=z\), by

The second equality follows from the definition of conditional probability: the common denominator \(P(Y=y,Z=z)\) cancels. Define \(O_Y(x,z)\) and \(O_Z(x,y)\) in the same way. Thus each conditional odds value is the ratio of joint probabilities at the two states joined by that one-variable move.

For one route from \(000\) to \(111\), change \(X\), then \(Y\), then \(Z\). The odds ratios along the edges are \(O_X(0,0)\), \(O_Y(1,0)\), and \(O_Z(1,1)\). Multiplying cancels the intermediate probabilities:

If instead the route changes \(Y\), then \(Z\), then \(X\), it must give the same endpoint ratio:

Consequently, the two products must be equal. The argument is a telescoping product: every intermediate joint probability appears once in a numerator and once in a denominator. This is a necessary compatibility condition. Passing this particular path comparison does not, by itself, establish that all conditional specifications are compatible; the other relevant paths and cycles must also agree.

A Three-Variable Specification That Fails the Cycle Check

Now consider separately specified full conditionals for binary \(X,Y,Z\). Suppose their conditional odds are as follows, for all values of the variables not shown:

• \(O_X(y,z)=2\) for every \(y,z\).
• \(O_Y(x,z)=3\) when \(x=0\), and \(O_Y(x,z)=5\) when \(x=1\).
• \(O_Z(x,y)=7\) when \(y=0\), and \(O_Z(x,y)=11\) when \(y=1\).

These are individually valid binary conditional distributions: any positive odds \(o\) correspond to conditional probability \(o/(1+o)\), which is strictly between 0 and 1. But individually valid conditionals need not be compatible with one another. For the route \(000 \rightarrow 100 \rightarrow 110 \rightarrow 111\), the specified odds product is

For the route \(000 \rightarrow 010 \rightarrow 011 \rightarrow 111\), the product is

If a joint distribution generated these full conditionals, both products would have to equal the same ratio \(p(1,1,1)/p(0,0,0)\). Since \(110\ne66\), no such joint distribution can have all the specified conditional odds. This conclusion concerns the specification itself, not whether a particular chain using it happens to run or produce stable-looking summaries.

How to Use Cycle Checks and Recognize Their Limits

A useful diagnostic is to write each conditional probability as conditional odds (or log odds), select a closed loop of states, and multiply the edge odds in each direction. In log form, multiplication becomes addition, so a cycle discrepancy is the difference between two sums of log odds. This can make discrepancies easier to locate: each term is associated with one conditional update at one configuration of the other variables.

With fitted models, estimated conditionals may not satisfy the exact algebraic equalities even if a compatible population model is plausible. Separate fitting, sampling variability, sparse combinations of predictors, and model misspecification can all contribute. A discrepancy is a diagnostic signal to investigate, not an automatic measure of its practical importance. In particular, the cycle check is exact for the specified probabilities; estimated probabilities also carry uncertainty.

Cycle checks are useful for reasoning about the difference between two approaches. In a joint-model approach, one specifies a joint distribution and obtains all full conditionals from it; the cycle equalities then hold automatically. In fully conditional specification, conditional models are specified directly, so agreement is not guaranteed and must be considered. The earlier material on compatibility establishes this distinction; the cycle calculation makes one necessary implication concrete.

There are important limits to what a cycle test establishes. A mismatch on even one cycle rules out a common joint distribution for the specified conditionals. Agreement on a few selected cycles, however, is not a proof of global compatibility. Also, compatibility is not the same as correctness: even compatible conditionals may fail to represent the data-generating relationships of interest. Finally, cycle agreement alone does not establish that a particular iterative algorithm converges to a desired distribution.

• Failure mode: comparing conditional probabilities without accounting for their conditioning values. Each edge ratio must use the other variables fixed at the state on that edge.
• Failure mode: treating any set of valid probability models as a compatible set. Validity of each model separately does not enforce agreement around cycles.
• Interpretation pitfall: describing a cycle among conditional associations as a causal feedback loop. The cycle is a mathematical consistency requirement, not a causal claim.
• Overclaim: concluding global compatibility because one loop agrees. Check the relevant cycles; one failed cycle is decisive, but one successful cycle is not.

For three binary variables, the state space has eight vertices, with an edge joining states that differ in exactly one variable. Full conditionals determine the odds on those edges. If all edge odds come from one positive joint distribution, their products are path-independent between any pair of vertices. Thinking in terms of the state-space graph keeps the condition precise even when the conditional formulas are more complex than the illustrative odds above.