Learn, then Explain

Artificial intelligence has made extraordinary progress in learning relationships from data, with increasingly capable models able to predict, classify and represent information that would previously have required substantial human expertise.

In finance, these models are applied to a distinctive scientific object. Financial econometrics studies a system whose participants learn, in which investors, firms and institutions observe outcomes, revise their expectations and adjust their behaviour, so that the regularities recorded in market data emerge from those interactions and evolve with them. Explaining a model of such a system means relating what it has learned to the behaviour that generated its data, because that relationship determines the conditions under which its estimates can be expected to hold.

Financial markets generate enormous quantities of data, but the number of independent economic experiences contained within those observations is comparatively limited. Even several decades of daily market returns provide relatively few observations of financial crises, monetary regimes, technological transitions or major changes in institutional behaviour. At the same time, modern machine-learning architectures can represent an extraordinary range of statistical relationships, many of which may explain the historical sample without providing an adequate account of the underlying economic processes.

This creates a problem familiar throughout the history of scientific inquiry. Observations alone rarely determine a unique explanation of the phenomena that generate them, particularly when the space of possible explanations is large relative to the available evidence. Scientific progress consequently depends not only on identifying empirical regularities, but on developing theories and mechanisms that constrain how those regularities can be interpreted, tested and generalised.

For machine learning in finance, this suggests a more substantive understanding of explainability. Economic interpretation is not simply an additional property imposed on a predictive model to satisfy professional or regulatory expectations. It can serve as a restriction on the space of plausible explanations, allowing prior knowledge about economic behaviour, financial contracts and institutional structures to inform what can reasonably be learned from limited observations. As machine learning becomes more expressive, this scientific role of explanation may become increasingly important.

What it means to explain

Science begins with the study of phenomena, but its ambitions extend beyond the reproduction of observations. A model of a physical system is valuable partly because it connects observable outcomes to the structures and processes that generate them, allowing researchers to understand why relationships occur and the circumstances under which they may change. A theory of fluid motion, for example, connects observed movements to pressures, material properties and physical constraints. Its explanatory power lies in describing a system of relationships that can be investigated across different conditions, rather than merely fitting the particular movements from which it was estimated.

The social sciences confront a related problem involving systems whose outcomes depend on human behaviour, institutions and interactions. Economic activity emerges from decisions made by households, firms, investors and governments, whose expectations and incentives influence both individual behaviour and collective outcomes. These interactions can generate regularities at the level of industries, financial markets and entire economies, even when the behaviour of individual participants is heterogeneous.

The distinction between physical and social sciences is therefore not one between mechanistic explanation and statistical description. Both seek to develop explanations that connect evidence to underlying processes, while recognising that the relevant mechanisms may be deterministic, stochastic or emergent. Economics differs in the extent to which its participants can learn from observations, respond to the institutions governing their behaviour and alter the system being studied.

This becomes especially important when scientific models are used to examine changes in the environment.

The Phillips curve is the textbook case taught in undergraduate macroeconomics. In 1958, A. W. Phillips documented a strikingly stable inverse relationship between wage inflation and unemployment across almost a century of UK data, and policymakers came to read it as a menu: tolerate a little more inflation, buy a little less unemployment. Edmund Phelps (1967) and Milton Friedman (1968) argued that the trade-off could not survive being exploited, because workers and firms would come to expect the higher inflation and build it into wages and prices, leading to the stagflation of the 1970s, which saw both high inflation and high unemployment together.

In his 1976 critique of econometric policy evaluation, Robert Lucas argued that statistical relationships estimated under one policy regime could change when policy altered the expectations and decisions of economic agents. The implication extended beyond a particular class of macroeconomic models: relationships observed within an existing institutional environment could reflect behavioural responses to that environment, rather than structural properties that would necessarily persist when the conditions changed. The Phillips curve had not been mismeasured; it had been measured under expectations that policy itself then changed.

For financial researchers, the problem is particularly apparent in asset markets. Interest rates, equity valuations and credit spreads are influenced by financing conditions, expectations, risk preferences and institutional constraints, while participants respond strategically to the information and opportunities available to them. A relationship that appears stable over a historical period may subsequently change because investors learn to exploit it, companies adjust their financing arrangements or new technologies alter competitive conditions.

The statistical learning problem in financial markets

The importance of scientific explanation becomes clearer when considered through statistical learning theory. Machine learning involves selecting a function or model from a hypothesis class using information contained in a finite sample. The resulting model is expected to perform not only on the observations used during training, but on previously unseen observations drawn from the relevant environment.

The difficulty depends partly on the relationship between the amount of information available and the complexity of the hypothesis class. A highly restricted model has comparatively few relationships available to fit the data, while an expressive model may accommodate an enormous variety of functions, including relationships that arise through sampling variation, or noise, rather than persistent structure.

A classical generalisation bound makes this relationship explicit. For a finite hypothesis class $\mathcal H$, bounded loss in the interval $[0,1]$, and $n$ independent observations drawn from a common distribution, Hoeffding's inequality and a union bound imply that, with probability at least $1-\delta$,

$$R(h)\leq \widehat R_n(h)+ \sqrt{\frac{\log|\mathcal H|+\log(1/\delta)}{2n}} \qquad \forall h\in\mathcal H$$

Here $R(h)$ is the expected population loss, $\widehat R_n(h)$ is the empirical loss, and $|\mathcal H|$ measures the number of candidate hypotheses. The expression illustrates how the statistical confidence associated with observed performance depends on both sample size and the complexity of the class from which the model is selected. More general learning theory replaces the cardinality of a finite hypothesis class with appropriate capacity measures, including VC dimension, covering numbers and Rademacher complexity.

The finite-class bound counts hypotheses, but most models used in finance contain infinitely many. Their capacity is measured instead by the VC dimension, or its real-valued analogue the pseudo-dimension, which takes the place of $\log|\mathcal H|$ in the corresponding bound up to logarithmic factors. For familiar models, the differences are large:

Table 1. Approximate capacity of familiar model classes, for d input features.

Model	Capacity
Fama–French five factors plus momentum (linear)	d + 1 = 7
Support vector machine	Up to d + 1 with a linear kernel; unbounded with an RBF kernel, where the margin rather than the dimension controls generalisation
Decision tree with k leaves	Grows with k, roughly k log d
ReLU network with W weights and L layers	Of order W L log W (Bartlett et al., 2019): tens of millions or more for a network with millions of weights

Against the roughly 7,500 observations in thirty years of daily returns, only the first row is comfortably smaller than the sample. These are worst-case measures, and over-parameterised networks often generalise better than they imply, but that is an empirical finding about particular data rather than a guarantee the bound provides.

Financial observations rarely satisfy the simple independent and identically distributed assumptions underlying this particular expression, but the underlying relationship between information and model complexity is central to the problem. Serial dependence, common economic factors and changing market conditions affect how much independent evidence the sample provides, while the capacity of the learning architecture determines the range of relationships that can potentially be fitted.

Consider an investment model trained on thirty years of daily equity returns. The sample contains approximately 7,500 trading-day observations, although many of these observations are affected by common economic conditions. The period might contain only a handful of major monetary transitions or financial crises, with even fewer examples of fundamental changes in market structure. A model with millions of parameters may be capable of representing relationships far more complex than the independent economic information contained within those historical experiences can distinguish.

Cross-sectional information expands the available evidence considerably, but it introduces a related problem. Thousands of securities generate many observations of firm-specific behaviour while remaining exposed to overlapping economic shocks, common financing conditions and market-wide expectations. The number of rows in a dataset consequently provides an incomplete indication of the amount of information available to distinguish competing explanations of aggregate financial behaviour.

This tension has long influenced quantitative finance. Factor models, covariance shrinkage, Bayesian estimation and regularisation methods can be understood partly as ways of controlling estimation complexity or introducing structure into an otherwise difficult statistical problem. Their contribution is not simply computational convenience. By limiting the relationships that must be independently estimated, they make portfolio and asset-pricing models more tractable under finite information.

Modern deep learning introduces greater representational flexibility, allowing models to learn nonlinear interactions, hierarchical features and relationships across data modalities. These capabilities can improve predictive performance, particularly when conventional models impose restrictive functional forms. Yet the additional flexibility increases the importance of understanding which relationships have been learned, which assumptions make them plausible and whether they are likely to remain informative outside the estimation sample.

The question is therefore not whether sufficiently complex neural networks can fit financial observations. It is how empirical evidence can distinguish among the many models capable of doing so, and how the resulting estimates should be interpreted when the underlying economic system continues to evolve.

Economic mechanisms as restrictions on learning

The statistical learning problem provides a substantive motivation for incorporating economic mechanisms into machine learning. When many models are compatible with a finite sample, established knowledge about accounting, contractual relationships, economic behaviour and financial institutions can restrict the class of admissible explanations.

Some restrictions arise from relationships that are definitional or contractual. Accounting identities govern the reconciliation of financial statements, while debt agreements determine interest obligations, repayment priorities and other contractual cash flows. Other restrictions arise from economic theories of production, demand, competition and investment, whose empirical relevance depends on the conditions being studied.

These mechanisms constrain learning by establishing relationships that candidate models must represent consistently. A model of debt-service capacity, for example, can incorporate the connection between outstanding borrowing, applicable interest rates and contractual payment obligations. The statistical task then becomes estimating uncertain quantities such as future operating cash flows, refinancing conditions and the likelihood of contractual shortfalls, rather than relearning every financial relationship independently from the observed sample. Robert Merton's (1974) model of corporate debt takes the idea furthest: by treating equity as a call option on the firm's assets, struck at the face value of its debt, it makes default risk an explicit function of leverage and asset volatility. It therefore has an answer, at combinations never seen in the sample, to the question of what happens if leverage doubles; a model fitted only to historical credit spreads has none.

The Bayesian framework provides a natural formalisation of this interaction between economic theory and empirical evidence:

$$p(M\mid D,\mathcal E) = \frac{p(D\mid M,\mathcal E)\,p(M\mid\mathcal E)}{p(D\mid\mathcal E)}$$

Here $M$ represents a candidate economic mechanism, $D$ the observed evidence and $\mathcal E$ the relevant economic knowledge. The likelihood captures how well a proposed mechanism explains the observations, while the prior expresses its plausibility given existing theory and information. The product follows from Bayes' theorem rather than any assumption of independence.

The value of this formulation is that empirical fit and mechanistic plausibility contribute different forms of evidence. Two models may explain historical asset returns equally well, but one may rely on relationships inconsistent with known contractual obligations or established accounting constraints. Another may produce an attractive historical fit by exploiting a relationship that economic theory suggests will change as market participants adapt.

A third may fit well while getting the direction of causation backwards. Regress quantity on price without allowing for prices that also respond to quantities, and the estimated demand curve can slope the wrong way, a problem Elmer Working identified in 1927; in markets, fund flows both chase past returns and move prices. No amount of additional data resolves this. Identification needs an instrument or a mechanism.

Economic interpretation therefore has a role before and during model estimation, not merely after it. Structural assumptions, admissible relationships and prior distributions influence the hypotheses that the model can entertain, while the observations provide evidence for distinguishing among them.

This also clarifies the meaning of interpretability. An interpretable economic model is not necessarily one whose complete computation can be expressed through a small collection of linear coefficients. It is one in which the relationship between the quantities being modelled and the underlying economic phenomena can be established, examined and challenged through meaningful economic relationships.

From learned representations to economic measurement

Self-supervised pretraining reshapes this problem by moving most of the learning onto a separate and far larger body of data. A large language or vision model is fitted to an enormous corpus, so that its very large hypothesis class is matched by a very large $D$, and what it hands to a financial researcher is a representation $Z$: an embedding of each document, image or firm, together with the distances $d(Z, Z')$ between them. A model built on $Z$ can then remain small, with a modest $\mathcal H$ relative to the financial sample, and so generalises on far better terms than a network trained from scratch on thirty years of daily returns.

The question becomes especially consequential when the intended economic quantity is not directly observable. Productive capacity, operating efficiency, financial vulnerability and competitive relationships may need to be inferred from several imperfect sources of evidence. A learned representation becomes a candidate economic measurement when its behaviour can be connected to those quantities through independent observations and economically meaningful relationships.

There is already a substantial empirical tradition for this approach. Hassan and co-authors (2019) construct measures of firm-level political risk from earnings-call discussions and examine their relationships with corporate behaviour, demonstrating how unstructured information can support the measurement of an economically defined construct. Related research using corporate language to infer product-market relationships illustrates how financial similarity can be measured through information about business activities rather than relying exclusively on fixed industry classifications. Modern AI extends this tradition through more flexible representations of language, imagery and other forms of economic evidence, creating the opportunity to connect observations that reveal different aspects of the same underlying activity while retaining the uncertainty associated with incomplete information.

The interpretation of these representations should follow the economic object being measured. Two companies may have similar textual representations because their disclosures use comparable language, while possessing materially different leverage, contractual structures or operating conditions. Conversely, firms in different industries may share substantial economic exposure to financing costs, energy prices or common suppliers despite appearing dissimilar under conventional classifications.

A scientifically meaningful representation therefore requires an account of the economic distinctions it preserves. Its usefulness can be investigated through relationships with independently observed quantities, changes in economic conditions and the outcomes generated by the underlying mechanisms. Its interpretation lies in the relationship between the learned representation and the economic system it is intended to describe, rather than in individual pixels, tokens or latent coordinates.

This creates a natural connection to the measurement of economic states. In dynamic economic models, a state describes the information relevant to the subsequent development of a system. In statistical learning, a representation may similarly summarise information contained within a collection of observations. The important research opportunity lies in connecting these concepts: establishing when learned representations preserve economically relevant information and when their behaviour supports an interpretation in terms of underlying economic states and mechanisms.

Conclusion

Self-supervised pre-training has expanded the capacity of financial researchers to learn from complex and heterogeneous information, but this progress also sharpens an enduring scientific problem. Finite observations can support many competing statistical explanations, particularly in economic systems whose behaviour changes through learning, competition and institutional development. Increasing model capacity makes the distinction between empirical fit and scientific understanding more important, rather than less.

Statistical learning theory establishes why the complexity of the hypothesis space matters for generalisation, while economic theory provides mechanisms through which that space can be constrained and interpreted. Together, they offer a foundation for understanding explainability as a component of scientific inference rather than a separate requirement imposed on machine-learning systems after estimation.

For AI as economic measurement, the next step is to establish how learned representations can be connected to economically meaningful states, which information their learning objectives preserve and how the resulting measurements can be interpreted across changing environments.

The central ambition is therefore not simply to explain how financial AI produces its predictions, but to understand what those models have learned about the economy, why that information is meaningful and which mechanisms make it useful beyond the observations on which they were trained.