If perception corresponds to hypothesis testing (Gregory, 1980); then visual searches might be construed as experiments that generate sensory data. In this work, we explore the idea that saccadic eye movements are optimal experiments, in which data are gathered to test hypotheses or beliefs about how those data are caused. This provides a plausible model of visual search that can be motivated from the basic principles of self-organized behavior: namely, the imperative to minimize the entropy of hidden states of the world and their sensory consequences. This imperative is met if agents sample hidden states of the world efficiently. This efficient sampling of salient information can be derived in a fairly straightforward way, using approximate Bayesian inference and variational free-energy minimization. Simulations of the resulting active inference scheme reproduce sequential eye movements that are reminiscent of empirically observed saccades and provide some counterintuitive insights into the way that sensory evidence is accumulated or assimilated into beliefs about the world.
In this paper Friston et al. introduce the notion that an agent (such as the brain) minimizes uncertainty about its state in the world by actively sampling those states which minimise the uncertainty of the agent’s posterior beliefs, when visited some time in the future. The presented ideas can also be seen as reply to the commonly formulated dark-room-critique of Friston’s free energy principle which states that under the free energy principle an agent would try to find a dark, stimulus-free room in which sensory input can be perfectly predicted. Here, I review these ideas together with the technical background (see also a related post about Friston et al., 2011). Although I find the presented theoretical argument very interesting and sound (and compatible with other proposals for the origin of autonomous behaviour), I do not think that the presented simulations conclusively show that the extended free energy principle as instantiated by the particular model chosen in the paper leads to the desired exploratory behaviour.
Introduction: free energy principle and the dark room
Friston’s free energy principle has gained considerable momentum in the field of cognitive neuroscience as a unifying framework under which many cognitive phenomena may be understood. Its main axiom is that an agent tries to minimise the long-term uncertainty about its state in the world by executing actions which make prediction of changes in the agent’s world more precise, i.e., which minimise surprises. In other words, the agent tries to maintain a sort of homeostasis with its environment.
While homeostasis is a concept which most people happily associate with bodily functions, it is harder to reconcile with cognitive functions which produce behaviour. Typically, the counter-argument for the free energy principle is the dark-room-problem: changes in a dark room can be perfectly predicted (= no changes), so shouldn’t we all just try to lock ourselves into dark rooms instead of frequently exploring our environment for new things?
The shortcoming of the dark-room-problem is that an agent cannot maintain homeostasis in a dark room, because, for example, its bodily functions will stop working properly after some time without water. There may be many more environmental factors which may disturb the agent’s dark room pleasure. An experienced agent knows this and has developed a corresponding model about its world which tells it that the state of its world becomes increasingly uncertain as long as the agent only samples a small fraction of the state space of the world, as it is the case when you are in a dark room and don’t notice what happens outside of the room.
The present paper formalises this idea. It assumes that an agent only observes a small part of the world in its local surroundings, but also maintains a more comprehensive model of its world. To decrease uncertainty about the global state of the world, the agent then explores other parts of the state space which it beliefs to be informative according to its current estimate of the global world state. In the remainder I will present the technical argument in more detail, discuss the supporting experiments and conclude with my opinion about the presented approach.
Review of theoretical argument
In previous publications Friston postulated that agents try to minimise the entropy of the world states which they encounter in their life and that this minimisation is equivalent to minimising the entropy of their sensory observations (by essentially assuming that the state-observation mapping is linear). The sensory entropy can be estimated by the average of sensory surprise (negative model evidence) across (a very long) time. So the argument goes that an agent should minimise sensory surprise at all times. Because sensory surprise cannot usually be computed directly, Friston suggests a variational approximation in which the posterior distribution over world states (posterior beliefs) and model parameters is separated. Further, the posterior distributions are approximated with Gaussian distributions (Laplace approximation). Then, minimisation of surprise is approximated by minimisation of Friston’s free energy. This minimisation is done with respect to the posterior over world states and with respect to action. The former corresponds to perception and ensures that the agent maintains a good estimate of the state of the world and the latter implements how the agent manipulates its environment, i.e., produces behaviour. While the former is a particular instantiation of the Bayesian brain hypothesis, and hence not necessarily a new idea, the latter had not previously been proposed and subsequently spurred some controversy (cf. above).
At this point it is important to note that the action variables are defined on the level of primitive reflex arcs, i.e., they directly control muscles in response to unexpected basic sensations. Yet, the agent can produce arbitrary complex actions by suitably setting sensory expectations which can be done via priors in the model of the agent. In comparison with reinforcement learning, the priors of the agent about states of the world (the probability mass attributed by the prior to the states), therefore, replace values or costs. But how does the agent choose its priors? This is the main question addressed by the present paper, however, only in the context of a freely exploring (i.e., task-free) agent.
In this paper, Friston et al. postulate that an agent minimises the joint entropy of world states and sensory observations instead of only the entropy of world states. Because the joint entropy is the sum of sensory entropy and conditional entropy (world states conditioned on sensory observations), the agent needs to implement two minimisations. The minimisation of sensory entropy is exactly the same as before implementing perception and action. However, conditional entropy is minimised with respect to the priors of the agent’s model, implementing higher-level action selection.
In Friston’s dynamic free energy framework (and other filters) priors correspond to predictive distributions, i.e., distributions over the world states some time in the future given their current estimate. Friston also assumes that the prior densities are Gaussian. Hence, priors are parameterised by their mean and covariance. To manipulate the probability mass attributed by the prior to the states he, thus, has to change prior mean or covariance of the world states. In the present paper the authors use a fixed covariance (as far as I can tell) and implement changes in the prior by manipulating its mean. They do this indicrectly by introducing new, independent control variables (“controls” from here on) which parameterise the dynamics of the world states without having a dynamics associated with themselves. The controls are treated like the other hidden variables in the agent model and their values are inferred from the sensory observations via free energy minimisation. However, I guess, that the idea is to more or less fix the controls to their prior means, because the second entropy minimisation, i.e., minimisation of the conditional entropy, is with respect to these prior means. Note that the controls are pretty arbitrary and can only be interpreted once a particular model is considered (as is the case for the remaining variables mentioned so far).
As with the sensory entropy, the agent has no direct access to the conditional entropy. However, it can use the posterior over world states given by the variational approximation to approximate the conditional entropy, too. In particular, Friston et al. suggest to approximate the conditional entropy using a predictive density which looks ahead in time from the current posterior and which they call counterfactual density. The entropy of this counterfactual density tells the agent how much uncertainty about the global state of the world it can expect in the future based on its current estimate of the world state. The authors do not specify how far in the future the counterfactual density looks. They here use the denotational trick to call negative conditional entropy ‘saliency’ to make the correspondence between the suggested framework and experimental variables in their example more intuitive, i.e., minimisation of conditional entropy becomes maximisation of saliency. The actual implementation of this nonlinear optimisation is computationally demanding. In particular, it will be very hard to find global optima using gradient-based approaches. In this paper Friston et al. bypass this problem by discretising the space spanned by the controls (which are the variables with respect to which they optimise), computing conditional entropy at each discrete location and simply selecting the location with minimal entropy, i.e., they do grid search.
In summary, the present paper extends previous versions of Friston’s free energy principle by adding prior selection, or, say, high-level action, to perception and action. This is done by adding new control variables representing high-level actions and setting these variables using a new optimisation which minimises future uncertainty about the state of the world. The descriptions in the paper implicitly suggest that the three processes happen sequentially: first the agent perceives to get the best estimate of the current world state, then it produces action to take the world state closer to its expectations and then it reevaluates expectations and thus sets high-level actions (goals). However, Friston’s formulations are in continuous time such that all these processes supposedly happen in parallel. For perception and action alone this leads to unexpected interactions. (Do you rather perceive the true state of the world as it is, or change it such that it corresponds to your expectations?) Adding control variables certainly doesn’t reduce this problem, if their values are inferred (perceived), too, but if perception cannot change them, only action can reduce the part of free energy contributed by them, thereby disentangling perception and action again. Therefore, the new control variables may be a necessary extension, if used properly. To me, it does not seem plausible that high-level actions are reevaluated continuously. Shouldn’t you wait until, e.g., a goal is reached? Such a mechanism is still missing in the present proposal. Instead the authors simply reevaluate high-level actions (minimise conditional entropy with respect to control variable priors) at fixed, ad-hoc intervals spanning sufficiently large amounts of time.
Review of presented experiments (saccade model)
To illustrate the theoretical points, Friston et al. present a model for saccadic eye movements. This model is very basic and is only supposed to show in principle that the new minimisation of conditional entropy can provide sensible high-level action. The model consists of two main parts: 1) the world, which defines how sensory input changes based on the true underlying state of the world and 2) the agent, which defines how the agent believes the world behaves. In this case, the state of the world is the position in a viewed image which is currently fixated by the eye of the agent. This position, hence, determines what input the visual sensors of the agent currently get (the field of view around the fixation position is restricted), but additionally there are proprioceptive sensors which give direct feedback about the position. Action changes the fixation position. The agent has a similar, but extended model of the world. In it, the fixation position depends on the hidden controls. Additionally, the model of the agent contains several images such that the agent has to infer what image it sees based on its sensory input.
In Friston’s framework, inference results heavily depend on the setting of prior uncertainties of the agent. Here, the agent is assumed to have certain proprioception, but uncertain vision such that it tends to update its beliefs of what it sees (which image) rather than trying to update its beliefs of where it looks. [I guess, this refers to the uncertainties of the hidden states and not the uncertainties of the actual sensory input which was probably chosen to be quite certain. The text does not differentiate between these and, unfortunately, the code was not yet available within the SPM toolbox at the time of writing (08.09.2012).]
As mentioned above, every 16 time steps the prior for the hidden controls of the agent is recomputed by minimising the conditional entropy of the hidden states given sensory input (minimising the uncertainty over future states given the sensory observations up to that time point). This is implemented by defining a grid of fixation positions and computing the entropy of the counterfactual density (uncertainty of future states) while setting the mean of the prior to one of the positions. In effect, this translates for the agent into: ‘Use your internal model of the world to simulate how your estimate of the world will change when you execute a particular high-level action. (What will be your beliefs about what image you see, when fixating a particular position?) Then choose the high-level action which reduces your uncertainty about the world most. (Which position gives you most information about what image you see?)’ Up to here, the theoretical ideas were self-contained and derived from first principles, but then Friston et al. introduce inhibition of return to make their results ‘more realistic’. In particular, they introduce an inhibition of return map which is a kind of fading memory of which positions were previously chosen as saccade targets and which is subtracted from the computed conditional entropy values. [The particular form of the inhibition of return computations, especially the initial substraction of the minimal conditional entropy value, is not motivated by the authors.]
For the presented experiments the authors use an agent model which contains three images as hypotheses of what the agent observes: a face and its 90° and 180° rotated versions. The first experiment is supposed to show that the agent can correctly infer which image it observes by making saccades to low conditional entropy (‘salient’) positions. The second experiment is supposed to show that, when an image is observed which is unknown to the agent, the agent cannot be certain of which of the three images it observes. The third experiment is supposed to show that the uncertainty of the agent increases when high entropy high-level actions are chosen instead of low entropy ones (when the agent chooses positions which contain very little information). I’ll discuss them in turn.
In the first experiment, the presented posterior beliefs of the agent about the identity of the observed image show that the agent indeed identifies the correct image and becomes more certain about it. Figure 5 of the paper also shows us the fixated positions and inhibition of return adapted conditional entropy maps. The presented ‘saccadic eye movements’ are misleading: the points only show the stabilised fixated positions and the lines only connect these without showing the large overshoots which occur according to the plot of ‘hidden (oculomotor) states’. Most critically, however, it appears that the agent already had identified the right image with relative certainty before any saccade was made (time until about 200ms). The results, therefore, do not clearly show that the saccade selection is beneficial for identifying the observed image, also because the presented example is only a single trial with a particular initial fixation point and with a noiseless observed image. Also, because the image was clearly identified very quickly, my guess is that the conditional entropy maps would be very similar after each saccade without inhibition of return, i.e., always the same fixation position would be chosen and no exploratory behaviour (saccades) would be seen, but this needs to be confirmed by running the experiment without inhibition of return. My overall impression of this experiment is that it presents a single, trivial example which does not allow me to draw general conclusions about the suggested theoretical framework.
The second experiment acts like a sanity check: the agent shouldn’t be able to identify one of its three images, when it observes a fourth one. Whether the experiment shows that, depends on the interpretation of the inferred hidden states. The way these states were defined their values can be directly interpreted as the probability of observing one of the three images. If only these are considered the agent appears to be very certain at times (it doesn’t help that the scale of the posterior belief plot in Figure 6 is 4 times larger than that of the same plot in Figure 5). However, the posterior uncertainty directly associated with the hidden states appears to be indeed considerably larger than in experiment 1, but, again, this is only a single example. Something that is rather strange: the sequence of fixation positions is almost exactly the same as in experiment 1 even though the observed image and the resulting posterior beliefs were completely different. Why?
Finally, experiment three is more like a thought experiment: what would happen, if an agent chooses high-level actions which maximise future uncertainty instead of minimising it. Well, the uncertainty of the agent’s posterior beliefs increases as shown in Figure 7, which is the expected behaviour. One thing that I wonder, though, and it applies to the presented results of all experiments: In Friston’s Bayesian filtering framework the uncertainty of the posterior hidden states is a direct function of their mean values. Hence, as long as the mean values do not change, the posterior uncertainty should stay constant, too. However, we see in Figure 7 that the posterior uncertainty increases even though the posterior means stay more or less constant. So there must be an additional (unexplained) mechanism at work, or we are not shown the distribution of posterior hidden states, but something slightly different. In both cases, it would be important to know what exactly resulted in the presented plots to be able to interpret the experiments in the correct way.
The paper presents an important theoretical extension to Friston’s free energy framework. This extension consists of adding a new layer of computations which can be interpreted as a mechanism for how an agent (autonomously) chooses its high-level actions. These high-level actions are defined in terms of desired future states encoded by the probability mass which is assigned to these states by the prior state distribution. Conceptually, these ideas translate into choosing maximally informative actions given the agent’s model of the world and its current state estimate. As discussed by Friston et al. such approaches to action selection are not new (see also Tishby and Polani, 2011). So, the author’s contribution is to show that these ideas are compatible with Friston’s free energy framework. Hence, on the abstract, theoretical level this paper makes sense. It also provides a sound theoretical argument for why an agent would not seek sensory deprivation in a dark room, as feared by critics of the free energy principle. However, the presented framework heavily relies on the agent’s model of the world and it leaves open how the agent has attained this model. Although the free energy principle also provides a way for the agent to learn parameters of its model, I still, for example, haven’t seen a convincing application in which the agent actually learnt the dynamics of an unknown process in the world. Probably Friston would here also refer to evolution as providing a good initialisation for process dynamics, but I find that a too cheap way out.
From a technical point of view the paper leaves a few questions open, for example: How far does the counterfactual distribution look into the future? What does it mean for high-level actions to change how far the agent looks into his subjective future? How well does the presented approach scale? Is it important to choose the global minimum of the conditional entropy (this would be bad, as it’s probably extremely hard to find in a general setting)? When, or how often, does the agent minimise conditional entropy to set high-level actions? What happens with more than one control variables (several possible high-level actions)? How can you model discrete high-level actions in Friston’s continuous Gaussian framework? How do results depend on the setting of prior covariances / uncertainties. And many more.
Finally, I have to say that I find the presented experiments quite poor. Although providing the agent with a limited field of view such that it has to explore different regions of a presented image is a suitable setting to test the proposed ideas, the trivial example and introduction of ad-hoc inhibition of return make it impossible to judge whether the underlying principle is successfully at work, or the simulations have been engineered to work in this particular case.