Sparsity as to how variables and factors interact with each other. Our next inductive bias for high-level variables can be stated simply: the joint distribution between high-level concepts can be represented by a sparse factor graph.

Any joint distribution can be expressed as a factor graph, but we claim that the ones which can be conveniently described with natural language have the property that they should be sparse. A factor graph is a particular factorization of the joint distribution. A factor graph is bipartite, with variable nodes on one hand and factor nodes on the other. Factor nodes represent dependencies between the variables to which they are connected.

To illustrate the sparsity of verbalizable knowledge, consider knowledge graphs and other relational systems, in which relations between variables often involve only two arguments (i.e., two variables). In practice, we may want factors with more than two arguments, but probably not a lot more.

A factor may capture a causal mechanism between its argument variables, and thus we should introduce an additional semantic element to these factors: each argument of a causal factor should either play the role of cause or of effect, making the bipartite graph directed.

It is easy to see that linguistically expressed knowledge satisfies this sparsity property by noting that statements about the world can be expressed with a sentence and each sentence typically has only a few words, and thus relates very few concepts. When we write "If I drop the ball, it will fall on the ground", the sentence clearly involves very few variables, and yet it can make a very strong prediction about the position of the ball.

A factor in a factor graph involving a subset S of variables is simply stating a probabilistic constraint among these variables. It allows one to predict the value of one variable given the others (if we ignore other constraints or factors), or more generally it allows us to describe a preference for joint sets of values for a subset of S.

The fact that natural language allows us to make such strong predictions conditioned on so few variables should be seen as surprising: it only works because the variables are semantic ones. If we consider the space of pixel values in images, it is very difficult to find such strongly predictive rules, e.g., to predict the value of one pixel given the value of three other pixels. What this means is that pixel space does not satisfy the sparsity prior associated with the proposed inductive bias.

We claim that the proposed inductive bias is closely related to the bottleneck of the GWT of conscious processing. Our interpretation of this restriction on write access in the GWT by a very small number of specialists selected on the fly by an attention mechanism is that it stems from an assumption on the form of the joint distribution between high-level variables whose values are broadcast.

If the joint distribution factor graph is sparse, then only a few variables (those involved in one factor or a few connected factors) need to be synchronized at each step of an inference process, e.g., consider loopy belief propagation. By constraining the size of the working memory, evolution may have thus enforced the sparsity of the factor graph.

The GWT also makes a claim that the workspace is associated with the conscious contents of cognition, which can be reported verbally. One can also make links with the original von Neumann architecture of computers.

In both the GWT and the von Neumann architecture, we have a communication bottleneck with in the former the working memory and in the latter the CPU registers where operations are performed. The communication bottleneck only allows a few variables to be brought to the nexus (working memory in brains, registers in the CPU).

In addition, the operations on these variables are extremely sparse, in the sense that they take very few variables at a time as arguments (no more than the handful in working memory, in the case of brains, and generally no more than two or three in typical assembly languages). This sparsity constraint is consistent with a decomposition of computation in small chunks, each involving only a few elements.

In the case of the sparse factor graph assumption we only consider that sparsity constraint for declarative knowledge (verbalizing "how the world works", its dynamics and statistical or causal structure).

This assumption about the joint distribution between the high-level variables at the top of our deep learning hierarchy is different from the assumption commonly found in many papers on disentangling factors of variation, where the high-level variables are assumed to be marginally independent of each other, i.e., their joint distribution factorizes into independent marginals.

We think this deviates from the original goals of deep learning to learn abstract high-level representations which capture the underlying explanations for the data. Note that one can easily transform one representation (with a factorized joint) into another (with a non-factorized joint) by some transformation (think about the independent noise variables in a structural causal model, Section 4).

However, we would then lose the properties introduced up to now (that each variable is causal and corresponds to a word or phrase, that the factor graph is sparse, and that changes in distribution can be originated to one or very few variables or factors).

Instead of thinking about the high-level variables as completely independent, we propose to see them as having a very structured joint distribution, with a sparse factor graph and other characteristics (such as dependencies which can be instantiated on particular variables from generic schemas or rules, described below).

We argue that if these high-level variables have to capture semantic variables expressible with natural language, then the joint distribution of these high-level semantic variables must have sparse dependencies rather than being independent. For example, high-level concepts such as "table" and "chair" are not statistically independent, instead they come in very powerful and strong but sparse relationships.

Instead of imposing a very strong prior of complete independence at the highest level of representation, we can have this slightly weaker but very structured prior, that the joint is represented by a sparse factor graph.

Interestingly, recent studies confirm that the top-level variables in generative adversarial networks (GANs), which are independent by construction, generally do not have a semantic interpretation (as a word or short phrase), whereas many units in slightly lower layers do have a semantic interpretation.

Why not represent the causal structure with a directed graphical model? In these models, which are the basis of standard representations of causal structure (e.g., in structural causal models, described below), knowledge to be learned is stored in the conditional distribution of each variable (given its direct causal parents). However, it is not clear that this is consistent with the requirements of independent mechanisms.

For example, typical verbally expressed rules have the property that many rules could apply to the same variable. Insisting that the independent units of knowledge are conditionals would then necessarily lump the corresponding factors in the same conditional.

This issue becomes even more severe if we think of the rules as generic pieces of knowledge which can be reused to be applied to many different tuples of instances, as elaborated in the next subsection. Another reason for a formulation that is not constrained to an acyclic graph is that humans also reason about relations between variables at equilibrium (such as voltage and current), which can mutually be causes of each other (i.e., arrows can go both ways).