The simple conceptual logic of Normative models of behavior

Bayes, Optimal Control, and Neural Networks

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I remember how I first saw the beauty of Bayesian models. I was a PhD student in Zurich, building models of learning in primary visual cortex. My models worked and described some data, but I lacked a coherent framework for thinking about the logic behind them. Then, I attended a fun meeting (I think it was at Banbury) and essentially invited myself to visit Bruno Olshausen’s lab. He very generously hosted me, and there, on a simple blackboard (with no slides), the logic of Bayesian thinking suddenly clicked. Later, hanging out with one of my heroes, the late David Knill, that logic became even clearer. But let me talk about the logic of Bayesian models of perception — which is the logic of normative models¹.

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The logic of normative models

1. Canonical problems in a niche
   We assume there are some canonical problems that organisms must solve: for example, estimating the distance to an object or gauging the relative position of our hand to a target. We also assume that by examining the niche closely, we can identify these problems fairly well.

2. Constraints and Performance Norms
   The world (and our bodies) imposes natural constraints. For instance, we cannot see what is behind an object; our eyes have limited resolution; our muscles and sensors are noisy. Such noise necessitates statistical (often Bayesian) approaches. There are also natural goals or norms, e.g., minimizing mean squared error when estimating distance. Both constraints and norms are (we hope) discoverable by scientists studying that niche.

3. Evolution and learning produce good solutions
   Because animals have evolved (and also learn) to handle these canonical tasks, we assume that their behavior is generally close to optimal under typical niche conditions. So, the behavior we observe should roughly match the behavior predicted by a corresponding normative (optimal) model.

4. We can derive optimal behavior
   If we have accurate information about the constraints and norms, we can mathematically (or algorithmically) derive what optimal behavior should be. In simpler settings (e.g., linear relations, Gaussian noise, and quadratic cost functions), it’s often straightforward. In more complex settings, we approximate. Crucially, this derivation yields a predictive model of behavior.

5. Comparing predictions with actual behavior
   We then compare the model’s predictions to real data from humans or animals. Sometimes, we fit a few free parameters; sometimes, we can set them from known properties of the niche. This comparison produces a similarity score or measure of how well the normative model accounts for observed behavior.

6. Insights from a good fit
   When the model accounts for behavior well, we gain multiple insights:

   • We can predict behavior in new situations.

   • It serves as a sanity check that we really understand how certain tasks relate to the niche.

   • It demarcates where “optimal” assumptions seem justified and where they don’t.

   • It clarifies which aspects of the world are critical for particular behaviors.

For those interested in such models, Weiji Ma, Dan Goldreich, and I co-wrote a textbook detailing the logic, the mathematical approaches, and a range of applications in behavioral science. There are other models (e.g. here with Trommershauser and Landy, here by Knill and Richards) that list countless cases where behavior is well predicted by these ideas.

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Mechanism matters (but not here)

Neuroscientists often criticize Bayesian models for not explaining mechanisms. Early in my career, I was inclined to push back on that critique, especially since some work (e.g., from Alexandre Pouget and colleagues) tries to link Bayesian ideas to neural circuits. But David Knill convinced me that we should be precise in what normative models do and don’t offer. Normative models focus on predicting and understanding behavior. An infinite variety of brains, with widely differing properties, could all produce (approximately) optimal behaviors. Normative modeling can still guide which variables matter for a task, but it remains mostly silent on which neurons or circuits are specifically responsible for generating the behavior.

To take a concrete example: one brain might explicitly represent “beliefs” (priors) and “sensory data” (likelihoods) and then do an actual Bayesian update. Another might rely on a neural network that simply learned the approximate mapping from inputs to outputs. If both exhibit Bayesian-like behavior, normative modeling alone cannot distinguish their internal mechanisms.

Normative models are the best we can currently do at describing behavior

In normative/ Bayesian models of behavior, the link with the world basically comes from evolutionary/ learning considerations. As such, the link between mathematical theory and the real world is relatively straightforward, and is experimentally accessible. All links between model components and the world are, from the get go, spelled out. This clarity makes it hard for me to swallow other approaches to brain modeling like the free energy principle (FEP) or integrated information theory (IIT). Instead of deriving from the properties of the niche, they begin with broad theoretical insights and formalize them. However, the link of their internal variables and structures with the outside world is complicated and rarely the focus of attention (while Bayesian modeling tends to deeply discusses this point). And similarly, because the logic of the link is not the focus of attention, it is generally unclear how behavioral predictions can be obtained. Normative theories, at their core, are fundamentally about tasks and behavior. They specialize in telling us what the behavior should look like if it is good for a given environment, not how the underlying neural machinery actually gets it done.

1. I want to just briefly mention some history of the naming. These models used to be called “models of optimality”. However, in practice, behavior is never optimal - merely good and well predicted by the models. So the term models of optimality was sucking the air out of the important discussion of how good behavior is predicted. Hence, I changed it to normative models (models relating to a norm of behavior), with apologies to the field in law, and economics where the word has a somewhat different connotation.