Tensions in Practice: A small observable state in tension with a sufficient predictive state¶
A deterministic cycle observed through three labels
The actual cycle repeats A, B, A, C. Start at a uniformly chosen position in that four-step cycle. Seeing A alone gives equal one-step chances of B and C. But A after B must be followed by C, while A after C must be followed by B. A three-label summary is enough for that restricted one-step marginal question; splitting A into its two phases retains the history needed for correct multi-step predictions.
Keep a compact one-step summary
Track only the three visible labels.
Predict valid sequences
Retain enough phase information to determine the next step.
Why these aims pull against each other
Merging the two A phases saves state but loses a predictive distinction. Recovering it requires a richer state and access to the phase, not merely labeling the coarse process memoryless.
Choose an arrangement to see what changes and what remains difficult.
The coarse picture merges two occurrences of A; the enriched picture separates them into different cycle positions. Read the coarse arrows only as one-step marginals, not as repeatable sequence rules.
What this choice protects
What it costs
When it fits
Compare the arrangements
Keep the visible label
Use the stationary one-step frequencies for A, B and C under the uniform starting phase.
- What it protects
- Only the visible three-label state is needed for that limited conditional forecast.
- What it costs
- The summary cannot be iterated as a Markov transition model: it would allow B, A, B, which this cycle never produces.
- When it fits
- Fits one-step questions when only the current label is retained and the declared phase mixture applies.
Illustration note: This is a valid marginal summary, explicitly not a Markov representation of the full observed sequence.
Split the two A phases
Track A-after-C and A-after-B separately, alongside B and C.
- What it protects
- Every displayed transition reproduces the actual cycle, including multi-step paths.
- What it costs
- Four states and reliable phase tracking are required; seeing A in isolation does not identify its phase.
- When it fits
- Fits sequence prediction when the previous label or another phase signal is available.
Illustration note: The enriched process is deterministic and Markov. This does not make arbitrary systems observable or easy to predict.
What this illustration does—and does not—establish
The source supplies the structural tension; the invented example makes one relation inspectable. Costs and conditions are part of each arrangement, not exceptions to a universal recommendation.
- The four-step cycle and uniform starting phase are invented and exact. They are not an estimated transition model.
- The coarse arrows are one-step conditional frequencies; the fine arrows are deterministic state transitions. Their scopes are explicitly different.
- No claim is made that adding any variable always restores the Markov property; this selected phase variable does so in this finite example.
Source entries
Markov Process
The canonical tension motivates this comparison. The setting, finite values and arrangements are declared editorial illustrations, not measured findings.
Enriching the state to restore the Markov property trades memory for dimensionality
The practitioner must navigate between a state too thin to be Markov and a state so fat that the memorylessness buys no tractability.
The source operation
A Markov process embodies the *memorylessness* (Markov) property: the future evolution of a system is conditionally independent of its entire past history given its present state, an idea first made rigorous by Markov (1906) in his study of dependent sequences of random variables. The current state "screens off" the past, so that once you know where the system is now, knowing how it got there adds nothing to your prediction of where it goes next. The structural claim is sharp and almost paradoxically strong: a single, sufficiently-rich present state is a *complete summary* of all history relevant to the future, so prediction requires only the state now and a transition rule, not the trajectory that produced it.