Luke Muehlhauser: You’ve spent many years studying AI safety issues, in particular in medical contexts, e.g. in your 2000 book with Subrata Das, Safe and Sound: Artificial Intelligence in Hazardous Applications. What kinds of AI safety challenges have you focused on in the past decade or so?

John Fox: From my first research job, as a post-doc with AI founders Allen Newell and Herb Simon at CMU, I have been interested in computational theories of high level cognition. As a cognitive scientist I have been interested in theories that subsume a range of cognitive functions, from perception and reasoning to the uses of knowledge in autonomous decision-making. After I came back to the UK in 1975 I began to combine my theoretical interests with the practical goals of designing and deploying AI systems in medicine.

Since our book was published in 2000 I have been committed to testing the ideas in it by designing and deploying many kind of clinical systems, and demonstrating that AI techniques can significantly improve quality and safety of clinical decision-making and process management. Patient safety is fundamental to clinical practice so, alongside the goals of building systems that can improve on human performance, safety and ethics have always been near the top of my research agenda.

Luke Muehlhauser: Was it straightforward to address issues like safety and ethics in practice?

John Fox: While our concepts and technologies have proved to be clinically successful we have not achieved everything we hoped for. Our attempts to ensure, for example, that practical and commercial deployments of AI technologies should explicitly honor ethical principles and carry out active safety management have not yet achieved the traction that we need to achieve. I regard this as a serious cause for concern, and unfinished business in both scientific and engineering terms.

The next generation of large-scale knowledge based systems and software agents that we are now working on will be more intelligent and will have far more autonomous capabilities than current systems. The challenges for human safety and ethical use of AI that this implies are beginning to mirror those raised by the singularity hypothesis. We have much to learn from singularity researchers, and perhaps our experience in deploying autonomous agents in human healthcare will offer opportunities to ground some of the singularity debates as well.

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Luke Muehlhauser: The abstract of Ackerman, Freer, and Roy (2010) begins:

As inductive inference and machine learning methods in computer science see continued success, researchers are aiming to describe even more complex probabilistic models and inference algorithms. What are the limits of mechanizing probabilistic inference? We investigate the computability of conditional probability… and show that there are computable joint distributions with noncomputable conditional distributions, ruling out the prospect of general inference algorithms.

In what sense does your result (with Ackerman & Freer) rule out the prospect of general inference algorithms?

Daniel Roy: First, it’s important to highlight that when we say “probabilistic inference” we are referring to the problem of computing conditional probabilities, while highlighting the role of conditioning in Bayesian statistical analysis.

Bayesian inference centers around so-called posterior distributions. From a subjectivist standpoint, the posterior represents one’s updated beliefs after seeing (i.e., conditioning on) the data. Mathematically, a posterior distribution is simply a conditional distribution (and every conditional distribution can be interpreted as a posterior distribution in some statistical model), and so our study of the computability of conditioning also bears on the problem of computing posterior distributions, which is arguably one of the core computational problems in Bayesian analyses.

Second, it’s important to clarify what we mean by “general inference”. In machine learning and artificial intelligence (AI), there is a long tradition of defining formal languages in which one can specify probabilistic models over a collection of variables. Defining distributions can be difficult, but these languages can make it much more straightforward.

The goal is then to design algorithms that can use these representations to support important operations, like computing conditional distributions. Bayesian networks can be thought of as such a language: You specify a distribution over a collection of variables by specifying a graph over these variables, which breaks down the entire distribution into “local” conditional distributions corresponding with each node, which are themselves often represented as tables of probabilities (at least in the case where all variables take on only a finite set of values). Together, the graph and the local conditional distributions determine a unique distribution over all the variables.

An inference algorithms that support the entire class of all finite, discrete, Bayesian networks might be called general, but as a class of distributions, those having finite, discrete Bayesian networks is a rather small one.

In this work, we are interested in the prospect of algorithms that work on very large classes of distributions. Namely, we are considering the class of samplable distributions, i.e., the class of distributions for which there exists a probabilistic program that can generate a sample using, e.g., uniformly distributed random numbers or independent coin flips as a source of randomness. The class of samplable distributions is a natural one: indeed it is equivalent to the class of computable distributions, i.e., those for which we can devise algorithms to compute lower bounds on probabilities from descriptions of open sets. The class of samplable distributions is also equivalent to the class of distributions for which we can compute expectations from descriptions of bounded continuous functions.

The class of samplable distributions is, in a sense, the richest class you might hope to deal with. The question we asked was: is there an algorithm that, given a samplable distribution on two variables X and Y, represented by a program that samples values for both variables, can compute the conditional distribution of, say, Y given X=x, for almost all values for X? When X takes values in a finite, discrete set, e.g., when X is binary valued, there is a general algorithm, although it is inefficient. But when X is continuous, e.g., when it can take on every value in the unit interval [0,1], then problems can arise. In particular, there exists a distribution on a pair of numbers in [0,1] from which one can generate perfect samples, but for which it is impossible to compute conditional probabilities for one of the variables given the other. As one might expect, the proof reduces the halting problem to that of conditioning a specially crafted distribution.

This pathological distribution rules out the possibility of a general algorithm for conditioning (equivalently, for probabilistic inference). The paper ends by giving some further conditions that, when present, allow one to devise general inference algorithms. Those familiar with computing conditional distributions for finite-dimensional statistical models will not be surprised that conditions necessary for Bayes’ theorem are one example.
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