Research Statement

How can we build machines that can reason coherently about the real world, in all of its complexity and ambiguity? Machines that are both logically consistent, i.e., free of self-contradiction, as well as consistent with observed data. Machines that can distinguish causation from correlation, and cause from effect. Machines that can truly explain, at the appropriate level of abstraction, attuned to both the knowledge and intentions of the recipient. Machines that can introspect on their own reasoning to explain how and why they arrive at the conclusions that they do. Machines that achieve all of these tasks not only accurately and efficiently, but while adhering to the societal norms and human values that we wish to uphold.

My research agenda is to answer these questions – to understand the principles governing such a machine, and from these principles to ultimately construct one. To this end, I have taken probabilistic programming as a starting point. Its central thesis is that (i) real-world phenomena can be modelled to tremendous levels of detail using simulation programs expressed in universal programming languages¹, (ii) any uncertainty about these phenomena can be expressed with probability distributions within the simulations, and (iii) we can use inference algorithms to automatically compute answers to queries about our models, and consequently derive conclusions about the phenomena themselves.

However, probabilistic programming in its current form is unable to address all of the criteria laid out above. A primary challenge is that probabilistic inference is in general computationally intractable: the set of queries for which we can efficiently compute answers remains a small subset of those of interest. A more fundamental limitation is that there are kinds of knowledge and classes of query that are inexpressible even within universal probabilistic programming. A prime example of this is causality, which cannot, in principle, be handled within the language of probability. My research has focused on addressing both of these problems. Concretely:

1. I develop algorithms for probabilistic and causal reasoning. These algorithms are generic in the sense that they are able to compute answers to a wide variety of queries posed on a wide variety of models.

2. I formalise novel forms of reasoning, reducing the gap between what we can reason about as humans, and what is within the domain of automated reasoning.

To address these problems, my research exploits the compositional structure of programs. Programs are not black-boxes; we can do more than simply execute them. They are structured, symbolic objects that can be interrogated, manipulated, and interpreted in different ways. Moreover, programs are compositional – even extremely complex programs, millions of lines long, are no more than a small number of primitive expressions composed together. Taking advantage of this structure, I have developed methods that execute programs backwards from observations to causal factors, and methods that intervene on the causal structure of programs to automate counter-factual “what-if” reasoning. Below, I elaborate on these and other projects, followed by my plans for future work.

Inference With Declarative Knowledge

Much of human knowledge is declarative – we believe and communicate facts, such as that the stove is on, that Barack Obama was the 44th American president, or that London is the capital of the United Kingdom. Probability theory governs how to reason with declarative knowledge in exactly the same manner as with observed data, but in practice most methods of inference do not support most expressions of declarative knowledge.

I introduced a framework called Predicate Exchange [@tavares2019predicate] to support inference with declarative knowledge. In this framework, generative knowledge is expressed in the form of probabilistic simulation programs. In one example, we modelled a diabetic patient’s glucose levels over time as a simulation of a stochastic dynamical system, using probability distributions to represent uncertain causal factors that affect the transition from one time-step to the next. Declarative knowledge, in contrast, is expressed in the form of predicates: programs that compute whether a proposition about the generative process is true or false. For example, the proposition that the patient is having a hypoglycemic episode is expressible through a predicate which computes whether the glucose levels have fallen below a threshold. Predicate exchange conditions a generative model on a predicate, revising the model such that declarative propositions become fact, and updating all interdependent factors correspondingly. Continuing the example, conditional on the hypoglycemic episode occurring, the probability that the patient had recently eaten decreases, and the necessity of a medical intervention increases.

In most cases, declarative propositions that we would like to condition on are extremely unlikely to be true of the generative process a priori. As a result, sampling values of variables that are consistent with the declarative knowledge becomes an extremely challenging constraint problem. Predicate exchange aims to address this, motivated by my observation that predicates provide only a single, impoverished bit of information (true or false), even though there is often a sense in which a proposition is almost (or very far from) true, such as if the patient’s glucose levels are almost below the hypoglycemic threshold. Predicate exchange applies an automatic program transformation to predicates such that they output a number between 0 and 1 rather than just true or false, whereby a value close to but not quite 1 indicates that the predicate is close to but not quite satisfied. Informally, these soft Boolean values provide partial credit, guiding an inference procedure towards regions that are consistent with the declarative propositions. Formally, they are used as a quasi-likelihood in a Markov Chain Monte Carlo process, yielding an inference procedure that is both efficient and asymptotically exact.

Parametric Inversion of Non-Invertible Programs

I have taken this approach of inference-through-program-transformation further, motivated by the observation that every inference problem involves the inversion of a function. That is, a probabilistic model computes the values of observable variables as a function of latent causes, and inference involves the inverse process of computing latent causes given values of observable variables. One example of this I have worked on is vision as inverse rendering – given a prior distribution over three dimensional scenes (geometry, lighting, camera pose, etc.) and a rendering function that simulates light to transform a scene into a two dimensional image, a fundamental vision problem is to invert the rendering function to infer the scene causally responsible for an observed image. Virtually all methods of inference tackle inverse problems indirectly, in part by simulating the model in the forward direction and evaluating consistency with the observed output. I developed a framework called parametric inversion [@tavares2021pi] which takes the idea of inversion literally, inverting a program to compute its input from a given output.

Most functions of interest are deemed non-invertible since they transform multiple different inputs to the same output, which renders the inverse process ambiguous. As a mathematical framework, parametric inversion is an answer to the question of what it means to invert a non-invertible function. A parametric inverse is a kind of generalized inverse function that parametrically represents the set of all inputs that a function maps to a particular output. In other words, given a function \(f\) that maps \(x\) to \(y\), its parametric inverse \(f^{-1}\) maps \(y\) and some parameter \(\theta\) to \(x\), such that (i) \(f(x) = y\), and (ii) changing \(\theta\) yields a different input \(x'\) that also maps to \(y\).

To invert a complex program, parametric inversion uses a program transformation that first replaces each primitive function in the composition with a corresponding primitive parametric inverse. Then, it reverses the order in which they are executed. This method is provably sound and complete in the sense that it produces correct inverses and is applicable to any program. While several technical hurdles remain (inverting non-invertible functions is unsurprisingly very hard!), promising initial results inverting programs of rendering-level complexity provides evidence in support of radically different approaches to inference that necessarily exploit programmatic structure.

Counterfactual Generative Models

There are several kinds of query that are not conventional probabilistic inference queries. One powerful example of this is the counterfactual. Counterfactuals are statements such as “If colonial powers hadn’t invaded, the Americas would be very different”. More generally, they take the form: “Given that \(X\) is true, what if \(Y\) were the case?”.

I developed counterfactual reasoning within universal probabilistic programming [@tavares2018language]. Counterfactuals require both probabilistic conditioning (“Given that \(X\) is true”) and causal interventions (“what if \(Y\) were the case?”) Conventional probabilistic programming languages have, by definition, generic forms of conditioning, but lack operators for causal interventions. Using a programming languages concept called lazy semantics, I formulated interventions as a kind of program transformation.

I developed a probabilistic programming language called Omega, which includes causal interventions and general conditioning as primitives. The result is the ability to compute counterfactual queries in the kinds of simulation models that can only be expressed as probabilistic programs. For instance, I used Omega to emulate the kind of decision a conservationist might make: assuming a dynamical predator-prey system that models competing populations over time, could culling the prey species in the past have prevented overpopulation of the predator species now?

Omega allows us to just as easily pose queries that are not strictly counterfactuals but are causal and useful nonetheless. For instance, we could ask “Suppose we were to give a medical treatment and subsequently patient \(A\) recovers, what should we predict about the treatment’s effect on patient \(B\)?”. We can even infer the distribution over interventions that is likely to bring about a desired outcome, such as the patient surviving.

Distributional Inference

I formalised another non-standard form of inference that I call distributional inference [@tavares2019random]. Distributional inference allows us to incorporate statistics into our probabilistic models. For example, a report on near-term climate change [@kirtman2013near] places the expected increase in the range 700 ppm² to 1000 ppm over the next two decades. If a climatologist constructs a probabilistic dynamical simulation of emissions, distributional inference is a mechanism to incorporate this expectation bound directly into her model. In addition, as elaborated on below, enforcing fairness, robustness, and many of the criteria of trustworthiness onto intelligent systems are problems of distributional inference that can be addressed within the framework.

Distributional inference provides a mechanism to condition a probabilistic model such that a probability, expectation, divergence, entropy, or any other distributional property takes a desired value. To formalize this, I introduced a novel mathematical construct that I call the random conditional distribution. The random conditional distribution is an answer to the question of what it means to condition, say, a probability to take a particular value, which is non-obvious since conventionally a probability is fixed and hence cannot be conditioned. Informally, a probability or any distributional property can be viewed as uncertain – and hence becomes conditionable – if its value is contingent on other causal factors in the model. For instance, even if the probability of rain tomorrow under a weather simulation is 60%, this probability would change if clouds were observed. Uncertainty over whether or not clouds will emerge can therefore be used to induce uncertainty over the probability of rain occurring. The random conditional distribution captures this concept in a generalisation of the probability concept of a conditional expectation, relying on introspection of the program’s causal structure using the causal framework outlined above, as well as mechanisms from higher-order functional programming.

I incorporated the random conditional distribution operator into Omega as a primitive. The benefits of this are analogous to those of making conditioning a primitive construct in conventional probabilistic programming languages: (i) distributional inference problems can be expressed extremely succinctly, often exactly mirroring their mathematical form, (ii) different distributional inference queries can be posed against the same model, and (iii) different algorithms can be used to perform distributional inference, which is often even more computationally challenging than conventional inference.

Future Directions

Plainly put, the development of automatic probabilistic inference methods able to handle the diversity of problems expressible within probabilistic programming languages will be a major milestone. I believe that analysis and manipulation of models’ internal structure will be a necessary part of any solution. Two possible future areas in this vein are:

• Relational Inference: Most inference methods simulate a model only in the forward direction. With parametric inversion I have demonstrated that inverse simulation is possible. Many effective methods of non-probabilistic reasoning such as SAT and SMT do neither of these extremes, and instead more freely choose what part of a model to evaluate when. This approach can be applied to universal probabilistic programming languages in a generalization of parametric inversion. Doing so will address some of the challenges in scaling parametric inversion up to very large and complex models.

• Model Specialization: Many models have structure for which specialised inference procedures have been developed, such as Hamiltonian Monte Carlo for continuous models, and variable elimination in certain discrete models. Automatic identification of structural properties, such as independence, finiteness, and continuity, would allow automatic specialisation of inference procedures to different models or even parts of models.

Still, even in the hypothetical scenario of perfect inference, there are several remaining challenges.

Causality Beyond Counterfactuals

Humans routinely ask and answer questions of actual causality: whether some event \(A\) that actually occurred (e.g., a vaccine was administered) actually caused some other event \(B\) to occur (e.g., a patient was immune). I propose to build a system that can answer questions of actual causality. Most formalisations of actual causality rely on counterfactuals: \(A\) actually caused \(B\) means had \(A\) not occurred (the vaccine wasn’t administered) neither would have \(B\) (the patient wouldn’t have been immune). This suffers from the problem of preemption: if some other event \(C\) (e.g., the patient caught the virus) would have caused \(B\) to occur in the absence of \(A\), following this counterfactual definition leads us to erroneously conclude that \(A\) is not the actual cause. An alternative approach that I will develop will be based on analysing the primitive causal steps that occur in the execution of a simulation model of the domain, rather than on counterfactuals, hopefully sidestepping the preemption problem. This approach will look more like how a programmer debugs a program – starting at the error and working backwards, recursively using knowledge of the language to answer, “Why did this happen?”

Actual causality lies at the heart of causal explanations. Explanation has seen increased attention recently in machine learning, but much of the structure, complexity, and utility of human explanations has not been addressed. Human explanations are often communicative acts, taking into account the knowledge and intentions of the explainer and explainee. Constructing a system that generates causal explanations is, in my view, a grand challenge of artificial intelligence that is within reach. It will require probabilistic programs to capture both the knowledge that causal explanations are based on and the beliefs and intentions of its recipient, actual causality since causal explanations denote actual causes, optimal program synthesis to explore the space of causal explanations.

Trustworthiness

Machine learning models often are brittle, violate privacy, and unfairly replicate or exacerbate biases. Awareness of these failures has spurred research into developing formal definitions of privacy, robustness, and fairness. Typically, these definitions impose different kinds of constraints on distributional properties.

Causal probabilistic programming is ideally suited to not only verify whether models meet or fail these criteria, but also to synthesize models that are trustworthy by construction. A concrete project I propose to address is to synthesize models that are counterfactually fair [@kusner2017counterfactual]. A decision that a person receives is counterfactually unfair if it would have been different had some protected property such as race or gender been different. Enforcing counterfactual fairness requires both the distributional and counterfactual inference frameworks that I have developed. Moreover, counterfactual fairness relies on a counterfactual definition of actual causality, and hence suffers from the problems of preemption mentioned above. Developing better methods to compute actual causality will therefore yield better definitions of fairness.

Polystructural Models: From Reasoning to Understanding

Human knowledge is broad, spanning a multitude of different domains, and deep, describing phenomena from multiple perspectives and degrees of abstraction. Human reasoning is able to span both this breadth and depth. For example, an epidemiologist may predict how a mutation in SARS-CoV-2’s morphology will likely affect the border restrictions or long term economic growth of a country. In contrast, formal computational models, including probabilistic programs, tend to be narrow and shallow, focusing on only one domain in isolation to a single degree of abstraction. In my opinion, this is the primary foundational gap between the current paradigms of modelling and inference, and machines that could, for example, infer that the assassination of Franz Ferdinand was the probable cause of World War I; that an experiment for X should control for Y; or that an unreliable water supply is ultimately caused by chronic government underinvestment.

Rather than a single isolated model, one avenue towards this would be to construct representations of interconnected networks of models, whereby multiple models of the same thing can both coexist and interact by virtue of sharing structure. Such a “model-base” could be incrementally built, incorporating more and more types and sources of knowledge. Perhaps speculatively, I conjecture that this may take us a few steps from automated reasoning towards a form of understanding and meaning, whereby the meaning of a concept is a function of the conceptual role it plays in all the different models within the network.

Broader Vision

As is true for many, I am driven both by a basic curiosity to uncover the workings of cognition – particularly in reasoning, knowledge, and learning – but also to build technology that has a positive impact on society. Fortunately, under the program I have described, these desires have not only been compatible but mutually reinforcing. The complex demands of societal problems casts an unflattering light upon the inadequacies of even state of the art technical solutions. Still, in my view, causal probabilistic programming is the most compelling candidate for a foundation to meet these demands, enabling us to encode the knowledge that we have, to learn from data, to support various forms of reasoning, and to enforce important societal values and constraints. However, building effective, trustworthy systems is not purely a technical problem. Fairness, robustness, and privacy are elevated in current literature largely because they may be simple enough to be captured with our current mathematical formalisms, and not because they are any more important than the more nebulous concepts of power, freedom, and equality, to name just a few. Moreover, even these are fraught with difficulties; on what basis should we decide if a definition is “correct”, especially given the real disagreement among people? As summarised potently by Kalluri in [@kalluri2020don]: fairness according to whom? Answering these questions will require humility, conceptual breakthroughs, and deep collaboration across fields as diverse as philosophy and sociology, all of which I am tremendously excited to do.

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