Why Compare Independent Chains?
A trace plot follows a chain summary across FCS iterations. Looking at several independently initialized chains adds an important comparison: do they appear to explore similar values and settle into similar behavior, or does the behavior depend on where the algorithm started? This matters because one chain can look steady while remaining in a restricted part of the state space.
In FCS, a complete iteration updates the variables with missing values in turn from their specified conditional models. A chain summary might be the mean of an imputed variable, the proportion assigned a category, or a fitted conditional-model coefficient. Compare the same summary, computed in the same way, across chains. For example, comparing one chain's imputed-variable mean with another chain's regression coefficient would not be informative.
The comparison is meaningful only when the chains are otherwise comparable. Different conditional models, update schedules, or data make a difference between chains hard to attribute to initialization or exploration. Record the settings and use the same iteration range and summary definitions for each chain. The trace plots should be examined alongside numerical summaries, as in the earlier material on trace plots for FCS chain summaries.
What Agreement Would Mean
The idealized reason for comparing starts can be stated using a transition kernel. Suppose the FCS updates define a Markov chain with a unique invariant distribution \(\pi\), and the chain converges to \(\pi\) from each starting state being considered. An invariant distribution is one that the transition leaves unchanged. Convergence here means that after many iterations the distribution of the chain's state is close to \(\pi\), regardless of its initial state.
Here, \(K^t(x,\cdot)\) is the distribution after \(t\) iterations when the chain starts at \(x\). If \(g\) is a bounded summary of the state, convergence implies that its expected value approaches the same limit from different starts:
Thus, if chains started in distinct states have similar distributions of summaries over a suitable retained segment, that is evidence consistent with them reaching similar behavior. The argument relies on the stated convergence conditions; it does not show those conditions hold just because two plots look alike. In particular, FCS conditionals need not be compatible with a single joint model. A chain can appear stable without establishing compatibility or correct specification.
Comparing Plots and Summaries
First compare trace plots on the same axes and iteration scale. Look for persistent separation in level, spread, or pattern; different apparent transitions; and a chain that continues to move while another appears confined. Then compare a small set of relevant numerical summaries for each chain. Useful choices include the mean, standard deviation, median, selected quantiles, range, and—for categorical variables—the proportion in each category. These are descriptive comparisons, not pass/fail tests.
Confirm that data, conditional models, update schedule, and summary definitions are the same. Use distinct starting states and independent random-number streams.
Include quantities relevant to the imputations or analyses, and compare them chain by chain rather than relying only on one pooled number.
Check whether chains occupy similar ranges and show similar behavior over the same iterations. Pay attention to sustained differences, not just isolated extremes.
Review chain-specific centers and spreads, and use quantiles or category proportions where a mean could conceal important differences.
Check implementation and model settings, then consider whether chains may be exploring different regions or changing slowly. Agreement is reassuring evidence, not proof.
A range is easy to compare but depends strongly on the number of iterations and on extreme observations. Overlapping ranges do not establish that two chains have the same distribution, and non-overlapping ranges are a warning to investigate rather than a formal test result. Likewise, similar means can hide differences in spread, shape, or transitions. Interpret summaries together with the plots and the substantive meaning of the quantity.
Worked Example: Reversed Traces With Matching Summaries
Suppose two separately initialized chains produce the following eight values of the same scalar summary, measured in score units:
Chain A: 4.0, 4.2, 4.4, 4.6, 4.8, 5.0, 5.2, 5.4.
Chain B: 5.4, 5.2, 5.0, 4.8, 4.6, 4.4, 4.2, 4.0.
For Chain A, the sum is \(4.0+4.2+4.4+4.6+4.8+5.0+5.2+5.4=37.6\), so the mean is \(37.6/8=4.7\) score units. Chain B contains the same values in reverse order, so its sum is also \(37.6\) and its mean is also \(4.7\) score units. Both ranges are \(5.4-4.0=1.4\) score units.
The empirical distributions match exactly, but the traces have opposite trends. The summaries alone therefore give an incomplete account of behavior: the plot reveals an ordered pattern that the matching means and ranges conceal. These short sequences do not establish that either chain has converged. The appropriate conclusion is that their values agree as an unordered set, while their iteration-by-iteration behavior calls for further investigation.
Worked Example: Chains Occupying Separated Ranges
Two independent FCS runs produce six values each for the imputed-variable mean, measured in years. Chain A gives 2.0, 2.2, 2.4, 2.3, 2.5, 2.4. Chain B gives 2.8, 3.0, 3.1, 2.9, 3.2, 3.0. Assume the runs used the same data, conditionals, schedule, and summary definition, and that these iterations are the common comparison segment.
For Chain A, the sum is \(2.0+2.2+2.4+2.3+2.5+2.4=13.8\), giving a chain mean of \(13.8/6=2.30\) years. Its range is \(2.5-2.0=0.50\) years. For Chain B, the sum is \(2.8+3.0+3.1+2.9+3.2+3.0=18.0\), giving a chain mean of \(18.0/6=3.00\) years. Its range is \(3.2-2.8=0.40\) years.
The observed ranges do not overlap, and the chain means differ by \(3.00-2.30=0.70\) years. This is a clear descriptive discrepancy for the displayed iterations. It is not a p-value and does not identify the cause. Check that the settings and calculations match; if they do, the separation suggests that the chains have not demonstrated similar behavior. Do not average the chain means and report \(2.65\) years as if pooling erased the discrepancy.
Worked Example: Equal Means, Different Spread
For one summary measured in arbitrary units, Chain A has five values: 0.8, 0.9, 1.0, 1.1, 1.2. Chain B has five values: \(-1\), 0, 1, 2, 3. Each chain mean is 1.0: Chain A sums to 5.0, and Chain B sums to 5.0.
For Chain A, deviations from 1.0 are \(-0.2,-0.1,0,0.1,0.2\). The sum of squared deviations is \(0.04+0.01+0+0.01+0.04=0.10\), so its sample standard deviation is \(s=\sqrt{0.10/(5-1)}=0.158\) units. For Chain B, the deviations are \(-2,-1,0,1,2\), with squared deviations summing to 10. Its sample standard deviation is \(s=\sqrt{10/(5-1)}=1.58\) units.
Equal means have concealed a large difference in spread: Chain B's range is \(3-(-1)=4\) units, compared with Chain A's \(1.2-0.8=0.4\) units. This pattern could reflect genuinely different exploration, unusual short-run variation, or a problem with the runs; five values cannot distinguish these explanations. The example shows why comparing only means is inadequate, not that these particular values diagnose a specific failure.
Interpretation, Alternatives, and Limitations
Independent-chain comparison is a diagnostic, not a formal convergence certificate. Separate starts are valuable because similar behavior from different initial states is harder to explain by a single shared starting point. But a shared conditional-model error, an unvisited region, or a limitation of the chosen summaries can affect all chains. Even close agreement cannot rule these out.
| Approach | What it contributes | What it can miss |
|---|---|---|
| One chain's trace and summaries | Shows how that run behaves over iterations. | Cannot reveal a discrepancy that depends on starting state. |
| Several aligned traces | Makes chain-specific separation and differing patterns visible. | Can be hard to interpret if traces are crowded or the summary is uninformative. |
| Chain-specific numerical summaries | Provides a compact comparison of centers, spreads, and selected quantiles. | A small set of numbers can hide distributional or temporal differences. |
| A single pooled summary | Condenses values across runs. | Can conceal disagreement between chains by averaging it away. |
If chains disagree, first verify that they used the same data and settings and that the summary was computed consistently. Then inspect the traces and other summaries before deciding what to investigate. Possible concerns include slow movement, dependence on initialization, and distinct regions that are not being traversed. A stable-looking trace or a chain-to-chain match does not establish that the conditional models are correctly specified or compatible with a joint distribution.
If chains agree, report the extent of the agreement rather than claiming proof of convergence. Describe which quantities and iterations were compared, and whether centers, spreads, ranges, and traces appeared similar. The comparison has limited power when chains are short, the chosen summaries miss relevant features, or all starts enter the same restricted region. It also does not, by itself, quantify the precision of chain summaries.
Common Mistakes
- Comparing unlike runs. Different conditionals or update schedules can produce different behavior for reasons other than starting state. Match the analysis settings before interpreting discrepancies.
- Treating range overlap as a decision rule. Overlap is descriptive; it is neither a necessary nor a sufficient condition for similar chain distributions.
- Checking means only. Equal means can coexist with different spreads, tails, or iteration patterns. Include plots and summaries suited to the quantity.
- Pooling away disagreement. A combined average can look reasonable even when chain-specific averages are separated. Inspect each chain before pooling.
- Claiming correctness from agreement. Chains may agree while sharing model misspecification or failing to explore a relevant region. Agreement supports a limited diagnostic conclusion only.
Check Your Understanding
Answer each question, then compare with the solutions below.
- Two chains use different conditional models but have nearly identical traces for one summary. Can their similarity be attributed to independent initialization? Explain.
- Chain A has six values with sum 24.0, and Chain B has six values with sum 27.0. Calculate their means and the difference in means, with units of years.
- Why can matching chain means be insufficient evidence that chains behave similarly?
- What does a persistent gap between matched chain traces suggest, and what does it not prove?
- Name two kinds of FCS diagnostics or model properties that chain agreement alone does not establish.
Solutions. (1) No. Different conditional models are a competing explanation for any similarities or differences, so the runs are not matched apart from their starts. (2) Chain A's mean is \(24.0/6=4.00\) years and Chain B's mean is \(27.0/6=4.50\) years; the difference is \(0.50\) years. (3) Means do not describe spread, shape, or iteration order, so compare traces and additional distributional summaries. (4) It suggests that the chains have not demonstrated similar behavior and warrants checking settings and exploration; it does not prove the cause or provide a formal test of convergence. (5) Agreement alone does not establish compatibility of the conditionals or correct specification, and it cannot rule out shared failure to explore a relevant region.