1  The Machines Talk Back: Modeling Turn-Taking Text as a Stochastic Process

AJ Alvero Cornell University

Anna Seo Gyeong Choi Cornell University

Abstract: The primary way most people interact with generative AI is in turn-taking interactions with online chat interfaces. These chats are powerful influences on many different user behaviors, perceptions, and actions; in the near future these text based interactions will become important artifacts for social scientists. In this chapter, we present an analytical framework which models turn-taking text as a stochastic process with socially relevant, well-defined end-points as determined by the researcher. The stochastic model incorporates a regression framework to make it useful for quantitative social scientists analyzing this type of data. The model is flexible by design. Its text-measurement layer is modular, allowing the analyst to substitute representations ranging from word counts to topic mixtures, embeddings, or LLM-coded discourse moves.

AI usage statement: AI was used to edit the writing and assist with mathematical notation.

1.1 Introduction

Computational text analysis has become commonplace and popular across the social sciences, though oftentimes the methods are designed to study a piece of text independent of its communicative context. For example, part of the rationale behind topic modeling (according to the authors of the original Latent Dirichlet allocation paper) was to help organize vast swaths of text into coherent groupings, like organizing a library (Blei et al. 2003). Many social science applications of these methods operate well in this paradigm, as the text available to researchers likewise resembles the static documents and/or books one might find in a library, such as open-ended responses in surveys (Roberts et al. 2014), social media posts (Regla-Vargas et al. 2026), scientific articles (Kusumegi et al. 2025), and other forms of writing, like narrative writing (Moon et al. 2025). These will continue to be important sources of data for social inference, but the methods designed for them do not always transfer naturally to interaction and sequence data. Political scientists have grappled with similar methodological tensions and have likewise proposed stochastic models to analyze sequential data, though specifically for audio data (Knox and Lucas 2021). Our method contributes to this literature as a means to better analyze sequential text-based interactions while also leveraging some of the unique qualities of LLMs as part of the method (if the analyst so chooses to use one). We next describe our generative model.

1.2 A Generative Model of Multi-Turn Text with Absorbing Outcomes

This section develops a model that treats the interaction, understood here as multi-speaker linguistic exchange encoded into text, as the object of inquiry: a sequence of latent discursive states that can continue, shift, or terminate with a specified outcome. For example, at the end of a conversation, one speaker might end up agreeing or disagreeing with the other, or an interaction with an LLM might end with someone making one decision rather than another. We adapt well-known tools and devise a covariate-conditioned hidden Markov model with absorbing terminal states. The contribution is the set of social-scientific quantities routed through it: an outcome-probability trajectory per interaction, a decomposition of its movement into expectation and surprise, and a five-component accounting of group differences in outcomes that separates what is discussed, how discourse moves, when interactions end, and how the same discourse is decided.

1.2.1 Setup and notation

We observe a corpus of \(N\) interactions, indexed \(i = 1, \dots, N\). Interaction \(i\) consists of a sequence of turns \(t = 1, \dots, T_i\), where each turn is a span of text \(w_{it}\). By default this is represented as a vector of token counts \(c_{it} \in \mathbb{N}^V\) over a vocabulary of size \(V\), though any per-turn representation (e.g., topic mixtures, sentence embeddings, dictionary derived feature vectors) can be substituted, with a corresponding change to the emission family. Each transition between turns carries a covariate vector \(x_{it} \in \mathbb{R}^P\), which may include time-varying features (e.g., the role of the upcoming speaker) and fixed interaction-level features (e.g., speaker characteristics). The absorption layer accommodates \(M\) terminal categories with ordered or nested structure where outcomes have them, though by default we assume the binary case and denote as \(o_i \in \{G, D\}\).

The latent structure is a Markov chain over \(K\) transient discursive states \(\mathcal{S} = \{1, \dots, K\}\) augmented with two absorbing states \(\{G, D\}\). Let \(s_{it} \in \mathcal{S}\) denote the latent state of turn \(t\). The generative model can be described as:

  1. \(s_{i1} \sim \text{Categorical}(\pi)\), \(\pi \in \Delta^{K-1}\).
  2. Conditional on \(s_{it} = k\), the turn’s tokens are drawn \(c_{it} \sim \text{Multinomial}(\varphi_k)\), where \(\varphi_k \in \Delta^{V-1}\) is a state-specific lexical distribution. Emissions are shared across groups by design (Section 1.2.5).
  3. From state \(k\), the chain moves to a destination \(d \in \mathcal{S} \cup \{G, D\}\) via a covariate-conditioned multinomial logit: \[ P(s_{i,t+1} = d \mid s_{it} = k, x_{it}) \;=\; \frac{\exp(\eta_{kd} + x_{it}^\top \beta_{kd})}{\sum_{d'} \exp(\eta_{kd'} + x_{it}^\top \beta_{kd'})}, \] with one destination per row fixed as reference for identification. Because absorbing destinations sit inside the same softmax as transient ones, continuation and termination compete directly; survival to turn \(t+1\) is implicit in the transient entries.
  4. The interaction ends when the chain enters \(G\) or \(D\); the observed data record absorption after turn \(T_i\) into \(o_i\).

Write \(T_x \in \mathbb{R}^{K \times K}\) for the transient-to-transient block of the transition matrix at covariates \(x\), and \(a^{(m)}_x \in \mathbb{R}^K\), \(m \in \{G, D\}\), for the columns of absorption probabilities.

The latent state space is the framework’s standardization device: since interactions differ in lengths and turns taken, the \(K\) states are the common coordinates in which all of them are described. Every estimand below presupposes this: “the same discursive state” is a well-defined event across interactions only because the state space is shared. The two extremes clarify the design. At \(K = 1\) the filtered state distribution is degenerate, text has no beliefs to revise, and the model collapses to a covariate-only logit and the trajectory goes flat by construction. At the other extreme, one state per turn gives each interaction its own state space (i.e., no shared transition matrix, no comparable splits, nothing to pool; this is the analog of one dummy per observation), and even the position-indexed repair aligns interactions by when rather than by what is happening, which Section 1.2.2’s structural-alignment principle rejects. \(K\) is therefore a compression resolution, bounded below by the requirement that text discriminate states (the evidence floor of Section 1.2.5) and above by pooling and comparability; its selection is treated in Section 1.2.5.

When the outcome is genuinely the terminal event of the interaction, absorption is part of the sequence and the stopping hazards below are substantive. When the outcome is instead a label attached to a completed sequence, such as a decision issued long after the interaction ends or any externally clock-limited exchange, the same likelihood applies. But the model then functions as a terminal-outcome HMM: the hazard component describes sequence length, not decision timing, and should be interpreted (or ignored) accordingly. For terminal-label applications the cleaner variant conditions on observed length. Using the reparameterization of Section 1.2.4, the likelihood is built from the continuation kernel \(\tilde T\) for \(t < T_i\) and the decision split \(q\) at \(T_i\). The hazard, and with it any pressure of the implied geometric length distribution on the transient dynamics, drops out of estimation entirely.

Speaker information can enter along either of two design axes. As a transition covariate, the identity of the upcoming speaker conditions the dynamics but turn-taking is treated as exogenous. This is appropriate when turn order is institutionally scripted or follows some kind of internally coherent conventional structure. Alternatively, speaker identity can be moved into the emission: each latent state emits a (speaker, tokens) pair, \(P(r_{it}, c_{it} \mid s_{it} = k) = \rho_{k}(r_{it})\, b_k(c_{it})\), so that who speaks is generated by the latent state and turn-taking becomes endogenous. The choice is substantive. Exogenous covariates treat turn-taking as structure; emitted speaker labels treat turn-taking as part of the phenomenon. A given variable should enter on one axis, not both. Conditioning transitions on a quantity the emission model elsewhere generates produces a conditionally specified hybrid rather than the generative model above. When the dynamic reading is wanted on both axes, a coherent repair is to condition transitions on the variable’s lagged value.

1.2.2 Likelihood

The likelihood for interaction \(i\) marginalizes over latent state paths: \[ \mathcal{L}_i(\theta) \;=\; \sum_{s_{1:T_i} \in \mathcal{S}^{T_i}} \pi_{s_1} b_{s_1}(c_{i1}) \prod_{t=2}^{T_i} T_{x_{i,t-1}}(s_t \mid s_{t-1})\, b_{s_t}(c_{it}) \;\cdot\; a_{x_{i,T_i}}^{(o_i)}(s_{T_i}), \] where \(b_k(c) = \prod_v \varphi_{kv}^{c_v}\) up to a multinomial constant. The marginal is computed exactly by the forward algorithm: with \(\alpha_1(k) = \pi_k b_k(c_{i1})\) and \[ \alpha_t(k) \;=\; b_k(c_{it}) \sum_{j=1}^K \alpha_{t-1}(j)\, T_{x_{i,t-1}}(k \mid j), \] the likelihood is \(\mathcal{L}_i = \sum_k \alpha_{T_i}(k)\, a^{(o_i)}_{x_{i,T_i}}(k)\), at cost \(O(T_i K^2)\) per interaction.

Two structural facts deserve emphasis. First, under the first-order Markov assumption, transition parameters pool information by state, not by turn position. A transition out of state \(k\) at turn 3 and a transition out of state \(k\) at turn 40 contribute to the same parameters. Sequence enters through state assignment (the forward–backward posteriors condition on the full sequence) and through the trajectory object below. The framework is structurally aligned, not temporally aligned. Second, the outcome enters the likelihood as the terminal absorption event, not as a label predicted from features. This is what makes the trajectory in Section 1.2.3 a coherent model quantity rather than a classifier score.

1.2.3 The outcome-probability trajectory

The primary analytic object is the per-turn probability of the eventual outcome given the history so far, \[ p_{it} \;=\; P(o_i = G \mid c_{i,1:t}, x_{i,1:t}), \] a model-derived quantity, not a literal observable. It factors into a filtering term and an eventual-absorption term. Let \(\bar\alpha_t(k) = \alpha_t(k) / \sum_j \alpha_t(j)\) be the filtered state distribution. For a covariate profile \(x\) held fixed going forward, the probability of eventual absorption into \(G\) from state \(k\), \(u_k(x)\), solves \[ u(x) \;=\; a^{(G)}_x + T_x\, u(x) \quad\Longrightarrow\quad u(x) = (I - T_x)^{-1} a^{(G)}_x, \] which exists whenever the interaction is certain to eventually end under profile \(x\). (In estimation, hazards driven toward zero can make this inversion unstable, so a numerical guard is prudent.) Then \(p_{it} = \sum_k \bar\alpha_t(k)\, u_k(x)\).

Time-varying covariates require a choice about the future. If the relevant covariates are fixed for the interaction, or evaluated at a chosen reference profile, the formula above is exact. If they vary, one can use the local computation \(u(x_{it})\), which holds the current covariate profile fixed going forward and uses no future information. The realized future covariate path should never be used: it is not knowable at turn \(t\), and using it leaks the future into \(p_{it}\). Empirical displays should state which version they use.

Trajectory increments decompose exactly. Let \(\tilde T_x\) denote the transient block renormalized on observed continuation, \(\hat\alpha_{t|t-1} = \bar\alpha_{t-1} \tilde T_{x_{t-1}}\) the one-step-ahead predicted state distribution, and \(u_{t-1}, u_t\) the absorption maps at the two profiles. Then \[ p_t - p_{t-1} \;=\; \underbrace{\big(\hat\alpha_{t|t-1} - \bar\alpha_{t-1}\big) u_{t-1}}_{\text{drift}} \;+\; \underbrace{\big(\bar\alpha_t - \hat\alpha_{t|t-1}\big) u_{t-1}}_{\text{innovation}} \;+\; \underbrace{\bar\alpha_t\,\big(u_t - u_{t-1}\big)}_{\text{covariate update}}, \] with the third term vanishing under a fixed or reference profile, in which case the two-term decomposition is exact. Under the model, the innovation averages to zero given everything observed so far: surprises are unpredictable by construction, which motivates the diagnostic in Section 1.2.5. Two cautions apply. The drift term is not a failure of this property; it comes from conditioning on the interaction having continued, which \(\tilde T\) encodes. And because the local computation \(u(x_{it})\) assumes the current covariates persist, strongly autocorrelated covariates bias the expected drift, so empirical innovations mix genuine discursive surprise with covariate changes the approximation could not anticipate. The diagnostic therefore tests the model and the approximation together.

This makes moments of divergence measurable: turns at which an interaction headed toward one outcome shifted toward the other appear as turns with large innovation opposing the prevailing drift. Innovations can be ranked corpus-wide and the extreme cases read qualitatively. Derived interaction-level quantities include near-misses (\(\max_t p_t\) high, terminal \(D\)) and reversal timing (last crossing of \(p_t = 0.5\)); both feed decomposition component 5. Because the trajectory \(p_{it} = \sum_k \bar\alpha_t(k) u_k(x)\) is a mixture that averages eventual-absorption prospects over the filtered state distribution, it is approximately invariant to the choice of \(K\). Provided \(K\) sits above the evidence floor, splitting a broad discursive state into functionally similar micro-states merely partitions the filtered probability mass \(\bar\alpha_t(k)\) across states with nearly identical expected outcomes \(u_k(x)\); this local re-labeling changes the individual summands but preserves the overall sum. Instability of the trajectories across adjacent values of \(K\) is therefore itself a diagnostic. The decomposition of Section 1.2.4, however, evaluates disparities conditionally state-by-state, and is therefore intrinsically dependent on the exact state space. One caution attaches throughout: \(p_t\) is evolving evidence, so a reversal is agnostic between the interaction genuinely turning and the model learning what kind of interaction it is observing.

The drift–innovation contrast is one of four comparisons supported by the model to describe situations where a given sequence of interaction appeared to be on one trajectory (i.e., toward \(G\) or \(D\)) but then ended up in another direction. Each of these comparisons reside between two posteriors of the same fitted model rather than between a text representation and an external outcome column. (i) Content versus context: what a turn looks like read in isolation versus what it is doing in sequence. (ii) Expectation versus realization: drift versus innovation. (iii) Real time versus hindsight: the divergence \(D_t = \mathrm{KL}(\gamma_t \,\|\, \bar\alpha_t)\) between the hindsight posterior \(\gamma_t = P(s_t \mid w_{1:T}, o)\), which knows the whole interaction and its outcome, and the real-time posterior \(\bar\alpha_t\) locates turns that read as routine when they occurred but were, retrospectively, where the outcome was being determined. (iv) Belief versus outcome: near-misses and reversal timing. These nominate moments for close reading; they are not estimands. The estimands are in Section 1.2.4.

Fixed covariates are known from the first turn, so they are priced into \(p_{i1}\): trajectories open at covariate-conditional starting points, and a group gap in opening levels is the fitted marginal disparity, not a dynamic finding. Group comparisons of dynamics should therefore use baseline-centered curves \(p_{it} - p_{i1}\), whose group differences measure differences in movement (signal for components 2 and 5, not misspecification; a group mean hovering at zero is a finding of no dynamic difference). We recommend pairing a levels plot with a centered-movement plot as standard practice.

1.2.4 A reparameterization, and the five-component decomposition

Direct comparison of absorption columns across groups conflates the timing of interaction termination with how outcomes are assigned. For state \(k\) at covariates \(x\) with \(0 < h_k(x) < 1\), define

  • \(h_k(x) = a^{(G)}_x(k) + a^{(D)}_x(k)\);
  • \(q_k(x) = a^{(G)}_x(k) / h_k(x)\);
  • \(\tilde T_x(\cdot \mid k) = T_x(\cdot \mid k) / (1 - h_k(x))\).

The triple \((\tilde T, h, q)\) carries exactly the same information as the original transition row whenever \(0 < h_k < 1\) (\(q\) is undefined at \(h_k = 0\) and \(\tilde T\) at \(h_k = 1\); the softmax keeps estimates off the boundary, but reported quantities should note the restriction). With group membership entering the transition model as a covariate (emissions shared; Section 1.2.5), group differences in aggregate outcome rates decompose into five comparisons:

  1. Differences in expected time spent in each discursive state. For the decomposition we compute occupancy from filtered posteriors \(\bar\alpha_{it}(k)\), which do not condition on the realized terminal outcome (though they are computed under a model whose parameters include the outcome process). Smoothed occupancy \(\gamma_{it}(k)\) conditions on \(o_i\) directly and would mechanically absorb part of the very outcome gap being decomposed. Smoothed occupancy remains the better description of where time was spent; the two serve different purposes and should be labeled.
  2. Differences in the continuation kernels \(\tilde T^{(g)}\).
  3. Differences \(q_k^{(g_1)} - q_k^{(g_2)}\), holding state \(k\) fixed. We frame this as a descriptive conditional disparity: among interactions terminating in the same discursive state, do groups receive different decisions? Its causal upgrade, “differential treatment holding discourse constant,” faces a structural obstacle beyond latent-space quality. The terminal state is plausibly caused by group membership (components 1–2 expect exactly this), and conditioning on a group-affected intermediate distorts the conditional comparison even with a perfect latent space. This is the same logic as the critique by Knox et al. (2020) of conditioning on police stops: here the stopping hazard is “being stopped” and the split is “outcome conditional on stop.” The chapter adopts their conclusion. The conditional disparity is a well-defined and valuable descriptive estimand whose causal interpretation requires assumptions the design itself does not supply. The audit analogy is motivation, not license: audits randomize the signal; we condition on it.
  4. Zero by construction under shared emissions; relaxing to group-specific \(\varphi_k^{(g)}\) and testing whether the shared version predicts held-out interactions as well provides a check on whether “the same state” means the same thing across groups.
  5. Functional differences in centered trajectories (slope, volatility, reversal timing, near-miss rates) compared across groups conditional on outcome. The same fence as component 3 applies: the outcome is itself group-affected, so these are descriptive conditional comparisons, not causal contrasts.

Components (1)–(3) admit counterfactual accounting in the Kitagawa–Oaxaca–Blinder style (Rowold et al. 2025). We define as follows: \(\Pi(\tilde T, h, q)\) is the model-implied aggregate probability of outcome \(G\), computed by simulating interactions from the fitted initial distribution under a stated covariate distribution (pooled, or a designated reference group’s) held fixed across all swaps; for fixed profiles it is available in closed form. The choice of that covariate distribution is itself a reference point of the decomposition, familiar from this literature, and is reported alongside the ordering choice. The gap \(\Pi^{(g_1)} - \Pi^{(g_2)}\) is decomposed by swapping one component at a time. Because the result depends on the order of swaps, we report all orderings or their average. Component swaps evaluate one group’s dynamics under another’s splits, which presupposes that the groups actually overlap in the covariate–state space; where they do not, the corresponding terms should be flagged rather than extrapolated. Nothing here is specific to demographic groups: the same machinery compares policy periods, jurisdictions, or interaction formats, decomposing a policy effect into changes in what is discussed, how long interactions run, and how the same discourse is decided. This is a mechanism-level description unavailable to endpoint analysis. The decomposition defines the estimand vector; for policy and period comparisons, causal identification comes from the surrounding design (comparison groups, timing). For component 3 we state the stronger conclusion plainly: absent randomization of the discursive intermediate, no observational design identifies the causal contrast it gestures at, because the unobserved factors steering an interaction into state \(k\) are entangled with the outcome by construction. What remains available are sensitivity analyses in the Knox–Lowe–Mummolo style, and experimental companions that randomize the group signal upstream, such as vignette or simulated-interaction designs in which perceived group membership is assigned, for which \(\Delta q_k\) is the natural target quantity.

1.2.5 Identification and design choices

All groups are fit jointly: one set of emissions, one initial distribution, with group entering only through transition covariates. This is a measurement invariance assumption, not a computational convenience. Component 3 is meaningful only if “the same state” is defined identically across groups, which shared emissions enforce and the component-4 test probes. Separate per-group fits would yield latent spaces that cannot be compared.

The emission density \(b_k(w_t)\) is the framework’s single modular port: every quantity above survives any substitution of the per-turn representation. The options form a ladder of increasing semantic resolution and decreasing self-evidence: raw token counts; a background mixture \(\varphi_{\text{eff}} = \lambda\varphi_k + (1-\lambda)\,\text{bg}_i\) that absorbs interaction-specific vocabulary into a per-interaction background; topic or embedding emissions; and, at the top, LLM-based measurement, in two forms. In the first, an LLM codes each turn into discourse-functional categories drawn from an existing qualitative codebook (e.g., concession, challenge, accountability, probing, deflection), and those codes become the emission alphabet. This makes the emission space theoretically meaningful by construction and invariant to interaction-specific topics by design. In the second, the \(K\) states are fixed a priori from the codebook and the LLM supplies \(b_k(w_t)\) directly as scored compatibility of each turn with each named state, leaving the model to estimate only dynamics and absorption. This eliminates label switching (states are named before fitting) and enforces measurement invariance by construction (identical prompts for all groups). Both forms carry a non-negotiable obligation: an LLM inside the measurement layer of a disparity analysis is itself an evaluator that must be audited. The protocol is counterfactual perturbation testing (demographic and case markers swapped within turns, label invariance required), agreement against a human-coded subset, cross-model agreement, and pinned model versions with released prompts in the pre-registration. The instrument is held to the same standard as the process it measures.

Models of this type are identifiable under generic conditions rather than automatically; the relevant line of results is Allman et al. (2009) and its extensions to covariate-dependent transitions. We rely on those results plus the empirical stability checks below, and note the assumption explicitly.

Within a fit, labels are arbitrary but consistent. Across random restarts, and across bootstrap replicates (each of which refits the model), states are aligned by matching emission distributions before any quantity is aggregated. Unaligned replicates would inflate every interval.

\(K\) answers two different questions: the measurement resolution (how many kinds of talk can the emissions distinguish?) and the process resolution (how many states do the dynamics and the decision actually use?). Because the text term dominates the likelihood by orders of magnitude, unguided selection answers the first while appearing to answer the second. We therefore select in two stages. The measurement stage runs a dashboard over \(K \in \{3, \dots, 8\}\), with criteria in two tiers. The first tier applies to any emission family, since its criteria are computed from the fitted process and state assignments rather than from the emission parameters: how well the fit predicts the text of held-out interactions (held out whole, never split within an interaction, which the dynamics make leaky); whether the same states reappear across random restarts and split halves, with instability at \(K{+}1\) the classic sign of one state too many; and degeneracy checks, namely states that are nearly empty, states whose typical visit collapses to a single turn, and pairs of states with nearly identical transition behavior. The second tier asks whether each state admits a coherent characterization in whatever space its emission lives in: coherence and exclusivity of top words for word-based emissions (Roberts et al. 2014; Mimno et al. 2011), separation and readable exemplar turns for embedding emissions, and annotator validity for LLM-coded emissions, whose categories are interpretable by construction. The process stage then asks which of the selected states the process distinguishes. How well the fit predicts held-out outcomes, \(P(o_i \mid w_{1:T_i})\), is the one criterion that belongs to the outcome process rather than the text, and the only one comparable across emission families; it plateaus at the resolution the decision requires. Merge tests, asking whether candidate pairs of states share the same continuation rows, hazards, and splits, identify states the process treats identically. Decomposition components are reported at this coarser, process-justified resolution \(K_{\text{proc}} \le K_{\text{text}}\); the finer text resolution is retained for description and labeling. Because \(K\) and the sequence unit jointly determine the evidence floor below (more states means smaller separations between them), the two are tuned together, not sequentially. Finally, the posture is stated rather than implied. Latent states are a resolution at which the process is described, not a count of objects in it. The data bound \(K\) from below, robustness bounds the claims from above, and every reported state-level estimand is required to be stable across \(K \pm 2\). A finding that exists at only one resolution is an artifact of the coordinate system and is not reported.

Sequences simulated from the fitted model are compared with the observed corpus on distributions the likelihood does not directly target: sequence lengths, time spent in each state, speaker-change counts, outcome rates by group, and trajectory volatility. These checks probe the generative claim itself rather than the fit of any single component.

The granularity of a “step” (utterance, speaker-merged turn, or block of \(b\) turns) is an estimable hyperparameter bounded by two inequalities: the evidence floor and the contamination ceiling. For the evidence floor, a step must carry enough text to discriminate states. For token-count emissions, the discriminating evidence in a step of \(n\) tokens grows as \(n \cdot D(\varphi_a \,\|\, \varphi_b)\), where \(D\) measures how separated two state vocabularies are; requiring a modest fixed amount of evidence \(c\) against the nearest competing state gives \(\bar n(b) \ge c / D_{\min}\), where \(D_{\min}\) is the smallest such separation. Because \(D_{\min}\) can be driven by one rare, poorly estimated state, an occupancy-weighted version should be reported alongside it. Auxiliary emission factors (e.g., a speaker factor) contribute additional evidence per step.

For the contamination ceiling, pooling \(b\) turns treats the combined text as coming from a single state, valid only if the state did not change within the block. If a state persists from one turn to the next with probability \(\rho\) (state-specific in reality, so a conservative choice is the smallest or an occupancy-weighted value), a \(b\)-turn block straddles a state change with probability \(1 - \rho^{\,b-1}\), and a tolerance \(\alpha\) gives \(b \le 1 + \log(1-\alpha)/\log\rho\).

Both bounds depend on quantities estimated at some granularity, and they are not independent. The evidence-floor calculation assumes single-state steps, which is precisely what the ceiling polices, so the floor is trustworthy only where contamination is low. Selection is therefore a fixed-point procedure: fit at the finest feasible unit, estimate \(D_{\min}\) and \(\rho\), compute the interval, refit inside it, confirm stability. Mixing within blocks shrinks the estimated separations, so coarse pilot fits yield conservative floors. An empty interval (\(b_{\min} > b_{\max}\)) is itself diagnostic: by the time steps carry discriminating text they already straddle changes. This is a failure of the emission representation, remedied by enriching emissions, not by aggregation.

Under correct granularity, the innovations of Section 1.2.3 should carry no leftover correlation from one step to the next, so testing for such correlation diagnoses a step too fine. The innovations should be standardized first, because their variability moves with how uncertain the model is about the state. The test does not detect over-aggregation, which will generally pass. The contamination ceiling guards against coarse, the correlation test against fine, and neither substitutes for the other. The test is a diagnostic for one failure mode, not a validation of the specification.

First-order dynamics make the trajectory and decomposition computable in closed form and keep parameters linear in covariates. The cost is that duration dependence and long-range dependencies are absorbed into state definitions. If turn position matters net of state, it enters as a transition covariate, which can serve as a robustness check.

1.2.6 Estimation

The marginal log-likelihood is maximized directly by stochastic gradient ascent (Adam) with automatic differentiation. This avoids the weighted regression updates an EM treatment would require; both procedures climb the same objective. Parameters are kept unconstrained through standard transformations. Uncertainty for all reported decomposition components comes from a bootstrap that resamples whole interactions, not turns, with per-replicate refitting and label alignment before aggregation. Because the objective has many local optima, point-estimate stability precedes uncertainty. Fits use multiple informed initializations (\(k\)-means on emissions), with the dispersion of decomposition quantities across restarts reported as a stability statistic. Bootstrap replicates are warm-started from the full-data estimate, which keeps replicates near the same solution, makes refits cheap, and reduces label switching before alignment.

1.3 Discussion

Many studies have described the outcomes of using chatbots, and our method could provide novel insight into how the conversational (i.e., turn-taking) structure is shaping these processes. Further, experimental designs which involve participants using a chatbot could further probe the discursive moves into the interactions to dive deeper into data or even bolster their claims (Lin et al. 2025). Beyond using our method to examine AI interactions, researchers can also incorporate AI into the method itself in two key ways. First, AI can assist in assessing whether candidate latent states are interpretable, one of the criteria for selecting \(K\), the number of latent states in the model. Second, AI could be used to encode each interaction in the sequence. Finally, the covariate structure built into the stochastic process allows social scientists to more precisely describe the mechanisms of how and why chatbot interactions lead to disparate results for users from different social backgrounds.