Inverse Decision Modeling:
Learning Interpretable Representations of Behavior

Consider the standard setup for sequential decision-making, where an agent interacts with a (potentially partially-obser- vable) environment. First, let $\psi$$\doteq$$(\mathcal{S},\mathcal{X},\mathcal{Z},\mathcal{U},\mathcal{T},\mathcal{O})$ give the problem setting, where $\mathcal{S}$ denotes the space of (external) environment states, $\mathcal{X}$ of environment observables, $\mathcal{Z}$ of (internal) agent states, $\mathcal{U}$ of agent actions, $\mathcal{T}$$\doteq$ $\Delta(\mathcal{S})^{\mathcal{S}\times\mathcal{U}}$ of environment transitions, and $\mathcal{O}$$\doteq$$\Delta(\mathcal{X})^{\mathcal{U}\times\mathcal{S}}$ of environment emissions. Second, denote with $\theta$ the planning parameter: the parameterization of (subjective) factors that a planning algorithm uses to produce behavior, e.g. utility functions $\upsilon$$\in$$\mathbb{R}^{\mathcal{S}\times\mathcal{U}}$, discount factors $\gamma$$\in$$[0,1)$, or any other biases that an agent might be subject to, such as imperfect knowledge $\tau,\omega$ of true environment dynamics $\tau_{\text{env}},\omega_{\text{env}}$$\in$$\mathcal{T}$$\times$$\mathcal{O}$. Note that access to the true dynamics is only (indirectly) possible via such knowledge, or by sampling online/from batch data. Now, a planner is a mapping producing observable behavior:

(Note: Our notation may not be immediately familiar as we seek to unify terminology across multiple fields. For reference, a summary of notation is provided in Appendix E).

This definition is very general: It encapsulates a wide range of standard algorithms in the literature (see Table 2), including decision-rule policies and neural-network planners. Importantly, however, observe that in most contexts, a global optimizer for $\rho$ is (trivially) either an identity function, or perfect Bayesian inference (with the practical caveat, of course, that in model-free contexts actually reaching such an optimum may be difficult, such as with a deep recurrent network). Therefore in addition to just $\pi$, what Definition 2 makes explicit is the potential for $\rho$ to be biased—that is, to deviate from (perfect) Bayes updates; this will be one of the important developments made in our subsequent example.

Note that by equating a planner with such a mapping, we are implicitly assuming that the embedded optimization (Equation 4) is well-defined—that is, that there exists a single global optimum. In general if the optimization is non-trivial, this requires that the spaces of policies $\pi,\rho\in\mathcal{P}$$\times$$\mathcal{R}$ be suitably restricted: This is satisfied by the usual (hard-/ soft-$Q$) Boltzmann-rationality for decision policies, and by uniquely fixing the semantics of internal states as (subjective) beliefs, i.e. probability distributions over states, with recognition policies being (possibly-biased) Bayes updates.

A more practical question is whether this optimum is reachable. While this may seem more difficult (at least in the most general case), for our interpretative purposes it is rarely a problem, because (simple) human-understandable models are what we desire to be working with in the first instance. In healthcare, for example, diseases are often modeled in terms of discrete states, and subjective beliefs over those states are eminently transparent factors that medical practitioners can readily comprehend and reason about (sonnenberg1983markov, ; jackson2003multistate, ). This is prevalent in research and practice, e.g. two-to-four states in progressive dementia (obryant2008staging, ; jarrett2018match, ; jarrett2019dynamic, ), cancer screening (petousis2019using, ; cardoso2019early, ), cystic fibrosis (alaa2019attentive, ), as well as pulmonary disease (wang2014unsupervised, ). Of course, this is not to say our exposition is incompatible with model-free, online settings with complex spaces and black-box approximators. But our focus here is to establish an interpretative paradigm—for which simple state-based models are most amenable to human reasoning.

Given any setting and appropriate planner, $\theta$ gives a complete account of $\phi^{*}=F(\psi,\theta)$: This deals with generation —that is, of behavior from its parameterization. In the opposite, given observed behavior $\phi_{\text{demo}}$ produced by some planner, we can ask what its $\theta$ appears to be: This now deals with inference—that is, of parameterizations from behavior.

First, note that absent any restrictions, this endeavor immediately falls prey to the celebrated “no free lunch” result: It is in general impossible to infer anything of use from $\phi_{\text{demo}}$ alone, if we posit nothing about $\theta$ (or $F$) to begin with (armstrong2018occam, ; christiano2015medium, ). The only close attempt has recruited inductive biases requiring multiple environments, and is not interpretable due to the use of differentiable planners (shah2018inferring, ; shah2019feasibility, ).

On the other extreme, the vast literature on IRL has largely restricted attention to perfectly optimal agents—that is, with full visibility of states, certain knowledge of dynamics, and perfect ability to optimize $\upsilon$. While this indeed fends off the impossibility result, it is overly restrictive for understanding behavior: Summarizing $\phi_{\text{demo}}$ using $\upsilon$ alone is not informative as to specific types of biases we may be interested in. How aggressive does this clinician seem? How flexible do their actions appear? It is difficult to tease out such nuances from just $\upsilon$—let alone comparing between agents (michaud2020understanding, ; gleave2021quantifying, ).

We take a generalized approach to allow any middle ground of choice. While some normative specifications are required to fend off the impossibility result (armstrong2018occam, ; shah2019feasibility, ), they need not be so strong as to restrict us to perfect optimality. Formally:

As with the forward case, this definition is broad: It encapsulates a wide range of inverse optimization techniques in the literature (see Table 3). Although not all techniques entail learning imitating policies in the process, by far the most dominant paradigms do (i.e. maximum margin, soft policy matching, and distribution matching). Moreover, it is normatively flexible in the sense of the middle ground we wanted: $\theta_{\text{norm}}$ can encode precisely the information we desire.⁴⁴4We can verify that $\theta_{\text{desc}}$$=$$\upsilon$ alone recovers the usual IRL paradigm. This opens up new possibilities for interpretative research. For instance, contrary to IRL for imitation or apprenticeship, we may often not wish to recover $\upsilon$ at all. Suppose—as an investigator—we believe that a certain $\upsilon$ we defined is the “ought-to-be” ideal. By allowing $\upsilon$ to be encoded in $\theta_{\text{norm}}$ (instead of $\theta_{\text{desc}}$), we may now ask questions of the form: How “consistently” does $\phi_{\text{demo}}$ appear to be in pursuing $\upsilon$? Does it seem “optimistic” or “pessimistic” relative to neutral beliefs about the world? All that is required is for appropriate measures of such notions (and any others) to be represented in $\theta_{\text{desc}}$. (Section 4 shall provide one such exemplar).

Note that parameter identifiability depends on the degrees of freedom in the target $\theta_{\text{desc}}$ and the nature of the identifi- cation strategy $G$. From our generalized standpoint, we sim- ply note that—beyond the usual restrictions (e.g. on scaling, shifting, reward shaping) in conjunction with $G$—Bayesian inference remains a valid option to address ambiguities, as in (jeon2018bayesian, ) for distribution matching,(ramachandran2007bayesian, ; choi2011map, ; dimitrakakis2011bayesian, ; rothkopf2011preference, ; balakrishnan2020efficient, ; makino2012apprenticeship, ; jarrett2020inverse, ) for soft policy matching, and (brown2019deep, ; brown2020safe, ) for preference extrapolation.

Now we have the ingredients to formally define the business of inverse decision modeling. Compacting notation, denote $F_{\theta_{\text{norm}}}(\kern 1.00006pt\kern 1.00006pt\cdot\kern 1.00006pt\kern 1.00006pt)$$\doteq$$F(\psi,(\theta_{\text{norm}},\kern 1.00006pt\cdot\kern 1.00006pt\kern 1.00006pt))$, and $G_{\theta_{\text{norm}}}(\kern 1.00006pt\kern 1.00006pt\cdot\kern 1.00006pt\kern 1.00006pt)$$\doteq$$G(\kern 1.00006pt\kern 1.00006pt\kern 1.00006pt\cdot\kern 1.00006pt\kern 1.00006pt\kern 1.00006pt,\theta_{\text{norm}})$. First, we require a projection operator that maps onto the sp- ace of behaviors that are parameterizable by $\theta$ given $F_{\theta_{\text{norm}}}$:

Interpretability  What dictates our choices? For pure imitation (i.e. replicating expert actions), a black-box decision-rule fitted by soft policy matching may do well. For apprenticeship (i.e. matching expert returns), a perfectly optimal planner inversed by distribution matching may do well. But for understanding, however, we wish to place appropriate structure on $\Theta$ depending on the question of interest: Precisely, the mission here is to choose some (interpretable) $F_{\theta_{\text{norm}}},G_{\theta_{\text{norm}}}$ such that $\phi_{\text{imit}}^{*}$ is amenable to human reasoning.

Note that these are not passive assumptions: We are not making the (factual) claim that $\theta$ gives a scientific explanation of the complex neurobiological processes in a clinician’s head. Instead, these are active specifications: We are making the (effective) claim that the learned $\theta$ is a parameterized “as-if” interpretation of the observed behavior. For instance, while there exist a multitude of commonly studied human biases in psychology, it is difficult to measure their magnitudes—much less compare them among agents. Section 4 shows an example of how inverse decision modeling can tackle this. (Figure 1 visualizes inverse decision modeling in a nutshell).

Consider that an agent has uncertain knowledge of the environment, captured by a prior over dynamics $\tilde{\sigma}$$\in$$\Delta(\mathcal{T}$$\times$$\mathcal{O})$. As a normative baseline, let this be given by some (unbiased) posterior $\tilde{\sigma}\doteq p(\tau,\omega|\mathcal{E})$, where $\mathcal{E}$ refers to any manner of experience (e.g. observed data about environment dynamics) with which we may come to form such a neutral belief.

Now, an agent may deviate from $\tilde{\sigma}$ depending on the situation, relying instead on $\tau,\omega$$\sim$$\sigma(\cdot|z,u)$—where $z,u$ allows the (biased) $\sigma$$\in$$\Delta(\mathcal{T}$$\times$$\mathcal{O})^{\mathcal{Z}\times\mathcal{U}}$ to be context-dependent. Consider recognition policies thereby parameterized by $\sigma$:

where $\rho_{\tau,\omega}$ denotes the policy for adapting $z$ to $x^{\prime}$ given (a point value for) $\tau,\omega$. For interpretability, we let $\rho_{\tau,\omega}$ be the usual Bayes belief-update. Importantly, however, $\rho$ can now effectively be biased (i.e. by $\sigma$) even while $\rho_{\tau,\omega}$ is Bayesian.

Forward Process  The forward (“inference”) process yields the occupancy measure. First, the stepwise conditional is:

Define Markov operator $\mathbb{M}_{\pi,\rho}$$\in$$\Delta(\mathcal{Z})^{\Delta(\mathcal{Z})}$ such that for any distribution $\mu$$\in$$\Delta(\mathcal{Z})$ : $(\mathbb{M}_{\pi,\rho}\mu)(z^{\prime})\doteq\mathbb{E}_{z\sim\mu}p(z^{\prime}|z)$. Then

defines the occupancy measure $\mu_{\pi,\rho}$$\in$$\Delta(\mathcal{Z})$ for any initial (internal-state) distribution $\rho_{0}$, and discount rate $\gamma\in[0,1)$.

Backward Process  The backward (“control”) process yie- lds the value function. We want that $\mu_{\pi,\rho}$ maximize utility:

Using $V$$\in$$\mathbb{R}^{\mathcal{Z}}$ to denote the multiplier, the Lagrangian is giv- en by $\mathcal{L}_{\pi,\rho}(\mu,V)\doteq J_{\pi,\rho}-\langle V,\mu-\gamma\mathbb{M}_{\pi,\rho}\mu-(1-\gamma)\rho_{0}\rangle$.

For perfectly rational agents, the best decision policy given any $z$ simply maximizes $V^{\phi_{\pi,\rho}}(z)$, thus it selects actions according to $\text{argmax}_{u}Q^{\phi_{\pi,\rho}}(z,u)$. And the best recognition policy simply corresponds to their unbiased knowledge of the world, thus it sets $\sigma(\cdot|z,u)=\tilde{\sigma},\forall z,u$ (in Equation 9).

Information Constraints  But control is resource-intensive. We formalize an agent’s boundedness in terms of capacities for processing information. First, decision complexity captures the informational effort in determining actions $\pi(\cdot|z)$, relative to some prior $\tilde{\pi}$ (e.g. baseline clinical guidelines):

Second, specification complexity captures the average regret of their internal model $\sigma(\cdot|z,u)$ deviating from their prior (i.e. unbiased knowledge $\tilde{\sigma}$) about environment dynamics:

Finally, recognition complexity captures the statistical surprise in adapting to successive beliefs about the partially-ob- servable states of the world (again, relative to some prior $\tilde{\varrho}$):

where $\varrho_{\tau,\omega}(\cdot|z,u)\doteq\mathbb{E}_{s\sim p(\cdot|z),s^{\prime}\sim\tau(\cdot|s,u),x^{\prime}\sim\omega(\cdot|u,s^{\prime})}\rho_{\tau,\omega}(\cdot$ $z,u,x^{\prime})$ gives the internal-state update. We shall see, these measures generalize information-theoretic ideas in control.

Backward Process  With capacity constraints, the maximi- zation in Equation 13 now becomes subject to $\mathbb{I}_{\pi,\rho}[\pi;\tilde{\pi}]\leq$ $A$, $\mathbb{I}_{\pi,\rho}[\sigma;\tilde{\sigma}]\leq B$, and $\mathbb{I}_{\pi,\rho}[\varrho;\tilde{\varrho}]\leq C$. So the Lagrangian (now with the additional multipliers $\alpha,\beta,\eta$$\in$$\mathbb{R}$) is given by $\mathcal{L}_{\pi,\rho}(\mu,\alpha,\beta,\eta,V)$$\doteq$$J_{\pi,\rho}$$-$$\langle V,\mu$$-$$\gamma\mathbb{M}_{\pi,\rho}\mu$$-$$(1$$-$$\gamma)\rho_{0}\rangle$$-$ $\alpha$$\cdot$$(\mathbb{I}_{\pi,\rho}[\pi;\tilde{\pi}]$$-$$A)-\beta$$\cdot$$(\mathbb{I}_{\pi,\rho}[\sigma;\tilde{\sigma}]$$-$$B)-\eta$$\cdot$$(\mathbb{I}_{\pi,\rho}[\varrho;\tilde{\varrho}]$$-$$C)$.

Policies and Values  The (dis-)/utility-seeking decision policy (min-)/maximizes $V^{\phi_{\pi,\rho}}(z)$, and a pessimistic/optimis- tic recognition policy min-/maximizes $Q^{\phi_{\pi,\rho}}(z,u)$ via $\sigma$.⁵⁵5In general, flipping the direction of optimization for $\pi$ or $\rho$ corre- sponds to the signs of $\alpha$ or $\beta$, but does not change Theorems 4–5. These optimal policies depend on optimal value functions: