Latent Variables and Model Mis-Specification

4Ilya Shpitser

2Charlie Steiner

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Hi Jacob. We (@JHU) read your paper on problems with ML recently!

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"On the other hand, some people take robust statistical correlation to be the *definition* of a causal relationship, and thus do consider causal and counterfactual reasoning to be the same thing."

These people would be wrong, because if A <- U -> B, A and B are robustly correlated (due to a spurious association via U), but intuitively we would not call this association causal. Example: A and B are eye color of siblings. Pretty stable, but not causal.

However, I don't see how the former part of the above sentence implies the part after "thus."

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Causal and counterfactual reasoning intersect, but neither is a subset of the other. An example of counterfactual reasoning I do that isn't causal is missing data. An example of causal reasoning that isn't counterfactual is stuff Phil Dawid does.

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If you are worried about robustness to model misspecification, you may be interested in reading about multiply robust methods based on theory of semi-parametric statistical models and influence functions. My friends and I have some papers on this. Here is the original paper (first author at JHU now) showing double ("two choose one") robustness in a missing data context:

https://pdfs.semanticscholar.org/e841/fa3834e787092e4266e9484158689405b7b0.pdf

Here is a paper on mediation analysis I was involved in that gets "three choose two" robustness:

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4710381/

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I don't know how counterfactuals get you around model misspecification. My take is, counterfactuals are something that might be of primary interest sometimes, in which case model specification is one issue you have to worry about.

Posted as part of the AI Alignment Forum sequence onValue Learning.Machine learning is very good at optimizing predictions to match an observed signal — for instance, given a dataset of input images and labels of the images (e.g. dog, cat, etc.), machine learning is very good at correctly predicting the label of a new image. However, performance can quickly break down as soon as we care about criteria other than predicting observables. There are several cases where we might care about such criteria:

In all of the above scenarios, the primary object of interest — the scientific theory, the effects of a policy, and the value function, respectively — is not part of the observed data. Instead, we can think of it as an unobserved (or “latent”) variable in the model we are using to make predictions. While we might hope that a model that makes good predictions will also place correct values on unobserved variables as well, this need not be the case in general, especially if the model is

mis-specified.I am interested in latent variable inference because I think it is a potentially important sub-problem for building AI systems that behave safely and are aligned with human values. The connection is most direct for value learning, where the value function is the latent variable of interest and the fidelity with which it is learned directly impacts the well-behavedness of the system. However, one can imagine other uses as well, such as making sure that the concepts that an AI learns sufficiently match the concepts that the human designer had in mind. It will also turn out that latent variable inference is related to

counterfactual reasoning, which has a large number of tie-ins with building safe AI systems that I will elaborate on in forthcoming posts.The goal of this post is to explain why problems show up if one cares about predicting latent variables rather than observed variables, and to point to a research direction (counterfactual reasoning) that I find promising for addressing these issues. More specifically, in the remainder of this post, I will: (1) give some formal settings where we want to infer unobserved variables and explain why we can run into problems; (2) propose a possible approach to resolving these problems, based on counterfactual reasoning.

## 1 Identifying Parameters in Regression Problems

Suppose that we have a regression model pθ(y|x), which outputs a probability distribution over y given a value for x. Also suppose we are explicitly interested in identifying the “true” value of θ rather than simply making good predictions about y given x. For instance, we might be interested in whether smoking causes cancer, and so we care not just about predicting whether a given person will get cancer (y) given information about that person (x), but specifically whether the coefficients in θ that correspond to a history of smoking are large and positive.

In a typical setting, we are given data points (x1,y1)…(xn,yn) on which to fit a model. Most methods of training machine learning systems optimize predictive performance, i.e. they will output a parameter ^θ that (approximately) maximizes ∑ni=1log pθ(yi,xi). For instance, for a linear regression problem we have log pθ(yi,xi)=−(yi−⟨θ,xi⟩)2. Various more sophisticated methods might employ some form of regularization to reduce overfitting, but they are still fundamentally trying to maximize some measure of predictive accuracy, at least in the limit of infinite data.

Call a model

well-specifiedif there is some parameter θ∗ for which pθ∗(y,x) matches the true distribution over y, and call a modelmis-specifiedif no such θ∗ exists. One can show that for well-specified models, maximizing predictive accuracy works well (modulo a number of technical conditions). In particular, maximizing ∑ni=1log pθ(yi,xi) will (asymptotically, as n→∞) lead to recovering the parameter θ∗.However, if a model is mis-specified

,then it is not even clear what it means to correctly infer θ. We could declare the θ maximizing predictive accuracy to be the “correct” value of θ, but this has issues:The two problems above also suggest a solution: we will say that we have done a good job of inferring a value for θ if θ can be used to make

good predictions in a wide variety of situations, and not just the situation we happened to train the model on. (For the latter case of predicting causal relationships, the “wide variety of situations” should include the situation in which the relevant causal intervention is applied.)Note that both of the problems above are different from the typical statistical problem of overfitting. Classically, overfitting occurs when a model is too complex relative to the amount of data at hand, but even if we have a large amount of data the problems above could occur. This is illustrated in the following graph:

Here the blue line is the data we have (x,y), and the green line is the model we fit (with slope and intercept parametrized by θ). We have more than enough data to fit a line to it. However, because the true relationship is quadratic, the best linear fit depends heavily on the distribution of the training data. If we had fit to a different part of the quadratic, we would have gotten a potentially very different result. Indeed, in this situation, there is no linear relationship that can do a good job of extrapolating to new situations, unless the domain of those new situations is restricted to the part of the quadratic that we’ve already seen.

I will refer to the type of error in the diagram above as

mis-specification error. Again, mis-specification error is different from error due to overfitting. Overfitting occurs when there is too little data and noise is driving the estimate of the model; in contrast, mis-specification error can occur even if there is plenty of data, and instead occurs because the best-performing model is different in different scenarios.## 2 Structural Equation Models

We will next consider a slightly subtler setting, which in economics is referred to as a

structural equation model. In this setting we again have an output y whose distribution depends on an input x, but now this relationship is mediated by anunobservedvariable z. A common example is a discrete choice model, where consumers make a choice among multiple goods (y) based on a consumer-specific utility function (z) that is influenced by demographic and other information about the consumer (x). Natural language processing provides another source of examples: in semantic parsing, we have an input utterance (x) and output denotation (y), mediated by a latent logical form z; in machine translation, we have input and output sentences (x and y) mediated by a latent alignment (z).Symbolically, we represent a structural equation model as a parametrized probability distribution pθ(y,z|x), where we are trying to fit the parameters θ. Of course, we can always turn a structural equation model into a regression model by using the identity pθ(y|x)=∑zpθ(y,z|x), which allows us to ignore z altogether. In economics this is called a

reduced form model. We use structural equation models if we are specifically interested in the unobserved variable z (for instance, in the examples above we are interested in the value function for each individual, or in the logical form representing the sentence’s meaning).In the regression setting where we cared about identifying θ, it was obvious that there was no meaningful “true” value of θ when the model was mis-specified. In this structural equation setting, we now care about the latent variable z, which can take on a meaningful true value (e.g. the actual utility function of a given individual) even if the overall model pθ(y,z|x) is mis-specified. It is therefore tempting to think that if we fit parameters θ and use them to impute z, we will have meaningful information about the actual utility functions of individual consumers. However, this is a notational sleight of hand — just because we call z “the utility function” does not make it so. The variable z need not correspond to the actual utility function of the consumer, nor does the consumer’s preferences even need to be representable by a utility function.

We can understand what goes wrong by consider the following procedure, which formalizes the proposal above:

As before, if the model is well-specified, one can show that such a procedure asymptotically outputs the correct probability distribution over z. However, if the model is mis-specified, things can quickly go wrong. For example, suppose that y represents what choice of drink a consumer buys, and z represents consumer utility (which might be a function of the price, attributes, and quantity of the drink). Now suppose that individuals have preferences which are influenced by unmodeled covariates: for instance, a preference for cold drinks on warm days, while the input x does not have information about the outside temperature when the drink was bought. This could cause any of several effects:

In practice we will likely have some mixture of all of these, and this will lead to systematic biases in our conclusions about the consumers’ utility functions.

The same problems as before arise: while we by design place probability mass on values of z that correctly predict the observation y, under model mis-specification this could be due to spurious correlations or other perversities of the model. Furthermore, even though predictive performance is high on the observed data (and data similar to the observed data), there is no reason for this to continue to be the case in settings very different from the observed data, which is particularly problematic if one is considering the effects of an intervention. For instance, while inferring preferences between hot and cold drinks might seem like a silly example, the design of timber auctions constitutes a much more important example with a roughly similar flavour, where it is important to correctly understand the utility functions of bidders in order to predict their behaviour under alternative auction designs (the model is also more complex, allowing even more opportunities for mis-specification to cause problems).

## 3 A Possible Solution: Counterfactual Reasoning

In general, under model mis-specification we have the following problems:

We can make sense of both of these problems by thinking in terms of

counterfactual reasoning. Without defining it too formally, counterfactual reasoning is the problem of making good predictions not just in the actual world, but in a wide variety of counterfactual worlds that “could” exist. (I recommend this paper as a good overview for machine learning researchers.)While typically machine learning models are optimized to predict well on a specific distribution, systems capable of counterfactual reasoning must make good predictions on many distributions (essentially any distribution that can be captured by a reasonable counterfactual). This stronger guarantee allows us to resolve many of the issues discussed above, while still thinking in terms of predictive performance, which historically seems to have been a successful paradigm for machine learning. In particular:

(Note that it is less clear how to actually train models to optimize counterfactual performance, since we typically won’t observe the counterfactuals! But it does at least define an end goal with good properties.)

Many people have a strong association between the concepts of “counterfactual reasoning” and “causal reasoning”. It is important to note that these are distinct ideas; causal reasoning is a type of counterfactual reasoning (where the counterfactuals are often thought of as centered around interventions), but I think of counterfactual reasoning as any type of reasoning that involves making robustly correct statistical inferences across a wide variety of distributions. On the other hand, some people take robust statistical correlation to be the

definitionof a causal relationship, and thus do consider causal and counterfactual reasoning to be the same thing.I think that building machine learning systems that can do a good job of counterfactual reasoning is likely to be an important challenge, especially in cases where reliability and safety are important, and necessitates changes in how we evaluate machine learning models. In my mind, while the Turing test has many flaws, one thing it gets very right is the ability to evaluate the accuracy of counterfactual predictions (since dialogue provides the opportunity to set up counterfactual worlds via shared hypotheticals). In contrast, most existing tasks focus on repeatedly making the same type of prediction with respect to a fixed test distribution. This latter type of benchmarking is of course easier and more clear-cut, but fails to probe important aspects of our models. I think it would be very exciting to design good benchmarks that require systems to do counterfactual reasoning, and I would even be happy to incentivize such work monetarily.

AcknowledgementsThanks to Michael Webb, Sindy Li, and Holden Karnofsky for providing feedback on drafts of this post. If any readers have additional feedback, please feel free to send it my way.