Discrete Latent Types in Dynamic Choice

Why Persistent Latent Types Matter

Two decision makers can face the same observed state and still make different choices because their preferences or other persistent characteristics differ. If those differences are unobserved, a model with one common parameter vector may attribute systematic behavioral differences to state dependence or random variation. A finite mixture offers one way to represent persistent heterogeneity: each decision maker belongs to one of a fixed number of latent types, and the type affects that person’s choice behavior over time.

The key implication is that a person’s type is shared across the whole observed history. Choices should therefore not be treated as independent draws from a new mixture at each date. Instead, each possible type generates a complete type-specific likelihood for the person’s history; the individual likelihood averages those likelihoods using the prior type probabilities.

Let \(a_{it}\) denote individual \(i\)’s choice at date \(t\), and let \(s_{it}\) denote the observed state. The model’s dynamic structure determines the probability of a choice conditional on the state, the observed characteristics, and the type. Write this probability as \(p_{ikt}(a\mid s,x_i)\). It may depend on the remaining horizon in a finite-horizon model. The type-specific choice probabilities are obtained from the corresponding dynamic choice model; the mixture construction below does not replace that model.

The analysis can be conditional on the observed state path or can model the joint history of choices and states. That distinction matters. If the probability of observing a state path differs by type, the state history itself contains information about type and must not be silently omitted when calculating posterior type probabilities.

From Type-Specific Histories to a Mixture Likelihood

Suppose first that the observed state path is treated as uninformative about type conditional on \(x_i\). For example, this holds if the state path is fixed or generated by a process that has the same probability under every type. For the observed choices \(a_{i1},\ldots,a_{iT}\), define the type-specific choice-history likelihood

Conditional independence of choices across dates given the type and the relevant states justifies the product. The states can affect the probability at each date; this formula does not claim that choices are independent without conditioning on type and the dynamic information they face.

Because the type is unobserved, the observed-data likelihood is obtained by summing over its possible values. The law of total probability gives the individual likelihood

Each term is the probability that the individual is type \(k\), multiplied by the likelihood of the observed choices under that type. The sum—not an average of per-period mixtures—keeps the same type attached to the individual at every date. If individuals are independent conditional on model parameters, the sample likelihood is \(\prod_i L_i\), and the log-likelihood is \(\sum_i\log L_i\).

When the State History Is Informative

If initial states or transitions depend on type, a likelihood based only on choices does not describe the joint observed history. Let \(f_k(s_{i1}\mid x_i)\) be the type-\(k\) probability of the initial state. Let \(g_{ikt}(s'\mid s,a,x_i)\) be the type-\(k\) probability of the next state, conditional on the current state and choice. Under a first-order Markov state model and the specified conditional choice model, the joint type-specific likelihood for states and choices is

This expression assumes the displayed factorization: the choice probability is conditional on the current state and type, and the next-state probability is conditional on the current state, choice, and type. If the observed data do not include the next state after the final choice, omit that final transition factor. The joint individual likelihood is \(\sum_k\pi_k(x_i)L_{ik}^{\mathrm{joint}}\). If states are conditioned on rather than modeled jointly, the appropriate type probabilities at the start of the choice likelihood must instead account for any information in those states.

In particular, “conditioning on the observed states” does not by itself mean those states are irrelevant to type. If the initial state is type-dependent, the type probability conditional on that initial state differs from \(\pi_k(x_i)\). Likewise, if transitions are type-dependent, later observed states can further change type probabilities. Type-independent state probabilities cancel from posterior odds across types, but type dependence cannot be ignored merely because states are observed.

Updating Type Probabilities with a History

After observing a history \(h_i\), the posterior probability that individual \(i\) is type \(k\) is the probability of that type conditional on the observed history and the characteristics used in the type probabilities. For a general joint history, Bayes’ rule gives

where \(L_{ik}(h_i)\) is the likelihood of all components of \(h_i\) included in the conditioning information under type \(k\). If \(h_i\) includes the state path, \(L_{ik}(h_i)\) must include its initial-state and transition likelihoods whenever they depend on type. Alternatively, if the state path is conditioned on, use a type probability already conditional on that path as the prior for the choice-history likelihood. The formula is only as appropriate as the history included in its likelihood.

Posterior weights can be updated sequentially. Start with type probabilities conditional on any observed initial state. After observing a choice, multiply each current weight by its type-specific choice probability and normalize. If the next state is also observed and its transition probability is type-dependent, multiply by that probability as well, then normalize again. This is Bayes’ rule applied repeatedly, not a new assumption about the type changing over time.

Derivation of the Mixture and Posterior Formulas

The basic result follows directly by partitioning over mutually exclusive latent types. For a history \(h_i\), the events \(z_i=1,\ldots,z_i=K\) exhaust the possibilities. Therefore,

For the posterior, divide the joint probability of the history and type \(k\) by the probability of the history. The numerator is \(P(h_i\mid z_i=k,x_i)P(z_i=k\mid x_i)=L_{ik}(h_i)\pi_k(x_i)\), and the denominator is the mixture likelihood just derived. Hence

This derivation also explains the state-history requirement: \(h_i\) must mean the same observed information in both the type-specific likelihood and the posterior conditioning event. Leaving an informative state out of \(L_{ik}\) while conditioning on it in the posterior breaks the equality.

Practical Interpretation and a Common Pitfall

Latent types let a dynamic choice model represent persistent differences with a finite number of behavioral profiles. They can also make estimation more flexible: the model can allow distinct type-specific parameters, type probabilities that vary with observed characteristics, or both. But the resulting types are model-based summaries, not necessarily literal categories in the population. The posterior weights quantify uncertainty about an individual’s type given the model and observed history.

A common mistake is to calculate a posterior using the choice likelihood alone while defining the history to include observed states, even though the initial-state distribution or transition probabilities vary by type. For instance, the formula \(\pi_k(x_i)L_{ik}^{a}/\sum_j\pi_j(x_i)L_{ij}^{a}\) is valid for a choice history conditional on a state path only when the prior is already conditional on that path, or when the state path is uninformative about type given \(x_i\). Otherwise, include the state-history likelihood in each type-specific term. Another mistake is to assign each date a fresh type draw. That describes a different model: persistent latent types require one type-specific likelihood for the individual’s complete history.