Abstract
A psychological model is a simplified, explicit representation of a mental or behavioural process, built so that its consequences can be derived and checked against data. Models range from verbal and diagrammatic sketches to fully specified mathematical and computational systems that generate quantitative predictions. Their value lies in precision: casting a theory as a model exposes its assumptions, forces its predictions to be explicit, and makes it possible to compare rival accounts by how well and how economically each fits behaviour. This article distinguishes the major classes of psychological model, sets out the three levels at which any process can be described, and explains how models are fitted to data and compared while guarding against overfitting. Three interactive demonstrations explore fitting a learning curve, the cost of excess flexibility, and an evidence-accumulation model of choice.
Keywords: psychological models, computational modeling, model comparison
A psychological model is a deliberately simplified account of how some part of the mind or behaviour works, stated precisely enough that its consequences can be worked out and tested. Where an informal theory says that memory fades with time or that people weigh risks against rewards, a model commits to how — a specific equation, mechanism, or computation from which predictions follow of their own accord (Lewandowsky & Farrell, 2018). The model is never the whole truth; it is a tractable stand-in that captures the features thought to matter and sets the rest aside.
What earns a model its place is that this discipline pays off. A verbal theory can absorb almost any result after the fact, because its terms are loose enough to be read either way; a formal model cannot, because its predictions are fixed before the data arrive and can be shown to be wrong (Guest & Martin, 2021). Making a theory explicit in this way is often where the real theoretical work happens: assumptions that seemed harmless in prose turn out, once written as a mechanism, to be doing much of the explaining, or to be incoherent (van Rooij & Baggio, 2021). The sections below set out what a psychological model is, the kinds that MeSH distinguishes, the major theoretical classes, the levels at which a process can be modelled, and how models are fitted and compared.
- A psychological model is a simplified, explicit representation of a mental or behavioural process, built so its predictions can be derived and tested against data (Lewandowsky & Farrell, 2018).
- Formal models discipline theory: they force assumptions into the open and can be falsified in a way loose verbal accounts cannot (Guest & Martin, 2021; Palminteri et al., 2017).
- The major classes include statistical and associative-learning models, symbolic information-processing architectures, connectionist networks, and probabilistic (Bayesian) models (Estes, 1950; Newell & Simon, 1972; McClelland & Rumelhart, 1981; Tenenbaum et al., 2011).
- A process can be modelled at three levels — the problem it solves, the algorithm it uses, and its physical implementation — and confusion between them is a common source of sterile dispute (Marr, 1982).
- A model is judged not by fit alone but by fit weighed against complexity, because an over-flexible model fits noise and predicts new data poorly (Roberts & Pashler, 2000; Pitt & Myung, 2002).
What a Psychological Model Is
MeSH defines a psychological model as a theoretical representation that simulates psychological or social processes. The word model is used loosely across psychology, but at its core it names the same thing in every case: a representation that is simpler than what it represents, chosen so that reasoning about the representation stands in for reasoning about the real system. A model of forgetting is not a claim about every detail of memory; it is a compact rule — say, that the probability of recall declines as a power function of elapsed time — whose behaviour can be compared with what people actually do (Lewandowsky & Farrell, 2018).
Models differ in how explicit they are, and the difference matters. A verbal model states its claims in ordinary language; it is easy to build and easy to communicate, but its predictions are only as sharp as the words allow, and slippery terms let it survive results that should have refuted it. A mathematical or computational model states its claims as equations or as a program, so that its predictions are entailed rather than argued for: once the assumptions are fixed, the model’s behaviour is whatever the mathematics or the code produces, and the modeller cannot quietly adjust it to fit (Guest & Martin, 2021). This is why the move from a verbal to a formal model is so often clarifying: it removes the modeller’s discretion after the fact.
A model is always a compromise between fidelity and tractability. Adding detail can bring a model closer to the target system, but a model complex enough to reproduce every feature of behaviour is no easier to understand than the behaviour itself, and it typically buys its fit by absorbing noise rather than capturing structure. The art of modelling lies in idealisation — deciding what to leave out — and a good model is valued as much for what it deliberately ignores as for what it includes (van Rooij & Baggio, 2021).
Types of Psychological Models
In the Medical Subject Headings vocabulary, Models, Psychological sits under the broader heading Models, Theoretical and is subdivided into several narrower descriptors. These MeSH children are a useful reminder that “psychological model” is an indexing category, not a single formalism: the terms below are specific, named models drawn largely from health and social psychology, where explicit models of behaviour change are used to organise research and design interventions. They are one slice through the space of psychological models, orthogonal to the broad theoretical classes — statistical, symbolic, connectionist, Bayesian — described in the next section; a given model can belong to a MeSH category here and to a formal class there at the same time. MeSH is a classification built for literature retrieval, so its subdivisions reflect how the published literature is organised rather than a deductive taxonomy of mechanisms.
MeSH Narrower Descriptors of “Models, Psychological”
| Model | What it represents |
|---|---|
| Health Belief Model | A model of health behaviour in which the likelihood of acting depends on perceived susceptibility, severity, benefits, and barriers. |
| Information Motivation Behavioral Skills Model | A model holding that health behaviour follows from being informed, motivated, and equipped with the necessary behavioural skills. |
| Neurolinguistic Programming | An approach claiming a link between neurological processes, language, and learned behaviour; classified here despite weak empirical support. |
| Theory of Planned Behavior | A model deriving intention, and hence behaviour, from attitudes, subjective norms, and perceived behavioural control. |
Major Classes of Psychological Model
Cutting across the indexing categories above, formal psychological models fall into a handful of broad theoretical families, distinguished by the kind of machinery they use to turn assumptions into predictions.
Statistical and associative-learning models describe behaviour with probability and simple update rules. Estes’s stimulus-sampling theory recast learning as the gradual conditioning of a population of stimulus elements, giving learning curves a statistical derivation rather than a merely descriptive one (Estes, 1950). The Rescorla–Wagner model of classical conditioning, which makes the change in associative strength on a trial proportional to the discrepancy between what was expected and what occurred, remains one of the most influential formal models in psychology precisely because that one error-correcting equation predicts a wide range of conditioning phenomena (Rescorla & Wagner, 1972). Symbolic information-processing models treat cognition as computation over discrete symbols, in the tradition of Newell and Simon’s analysis of human problem solving as search through a space of states using production rules (Newell & Simon, 1972). Their modern descendants are cognitive architectures — unified theories, such as ACT-R, that specify a fixed set of memory and control mechanisms intended to hold across many tasks, so that a model of a particular task is built from the same standing components rather than invented anew (Anderson et al., 2004). Connectionist models replace discrete symbols with networks of simple units whose weighted connections carry the computation, so that behaviour emerges from parallel interaction rather than sequential rule-following. The interactive-activation model of letter perception showed how such a network reproduces subtle context effects — a letter identified faster in a word than in isolation — without any rule stating the effect (McClelland & Rumelhart, 1981). Probabilistic, or Bayesian, models instead cast cognition as inference under uncertainty, asking what an ideal reasoner should conclude given noisy evidence and prior knowledge, and comparing human behaviour with that standard (Tenenbaum et al., 2011). A related strand builds explicit process models of memory as probabilistic inference, as in the retrieving-effectively-from-memory account of recognition (Shiffrin & Steyvers, 1997). These families are not mutually exclusive — a model can be connectionist in mechanism and Bayesian in the computation it approximates — which is exactly what the levels of analysis in the next section make precise.Levels of Analysis
A recurring source of confusion in modelling is that two theorists can describe the same process at different levels and mistake their difference for a disagreement. Marr’s influential proposal was that any information-processing system should be understood at three distinct levels (Marr, 1982). The computational level asks what problem the system solves and why — what is being computed, and what makes that the right thing to compute given the organism’s goals and the structure of its environment. The algorithmic level asks how the problem is solved — what representations the system uses and what procedure transforms input to output. The implementational level asks how that algorithm is physically realised in neural tissue.
Figure 1
Marr’s Three Levels of Analysis
The levels are loosely coupled, and this is the point. The same computational goal can be met by different algorithms, and the same algorithm can be implemented in different physical substrates, so a claim at one level does not fix the others. A Bayesian model is usually a computational-level account — a statement of the inference the system ought to perform — and it is fully compatible with a connectionist algorithm that approximates that inference and with a neural implementation of that network (Kriegeskorte & Douglas, 2018). Keeping the levels apart clarifies what a given model is and is not claiming, and it explains why a good fit at one level neither guarantees nor requires correctness at another. It also sets a research agenda: a mature account of a cognitive function connects all three, showing how a computational goal is met by a specific algorithm realised in specific circuitry.
Fitting and Comparing Models
A model with free parameters is fitted to data by finding the parameter values that make the model’s predictions come closest to the observations — minimising a measure of error such as the sum of squared deviations, or maximising the likelihood of the data under the model. Fitting a learning curve, for instance, means choosing the rate and asymptote that best reproduce the observed improvement. The first demonstration lets the reader fit a power-law model of practice to a fixed dataset by hand and watch the error fall as the parameters approach their best-fitting values.
Fitting a power law of practice
Practice times fall as a power function of the number of trials, T(n) = asymptote + 600 · n−rate. Adjust the two parameters to bring the curve onto the fixed data and watch the fitting error fall.
The error surface has a single minimum here, so hand-fitting converges; real fitting is done numerically, but the logic — choose parameters that minimise error — is the same.
Goodness of fit alone, however, is a treacherous guide, and this is one of the most important lessons in the practice of modelling. A model with enough free parameters can fit almost any dataset, including the noise in it, and a model that fits the noise will predict fresh data worse than a simpler model that does not (Pitt & Myung, 2002). A close fit is therefore weak evidence for a theory unless the model was at genuine risk of fitting badly — unless it could, in principle, have failed (Roberts & Pashler, 2000). The concern is not merely the number of parameters but the model’s flexibility: its capacity to accommodate patterns it will never actually be asked to explain.
The response is to weigh fit against complexity. Model-selection criteria such as the Akaike and Bayesian information criteria add an explicit penalty for the number of parameters, so that an extra parameter must earn its place by improving fit more than the penalty costs. The aim is not the model that fits the data in hand best, but the one that will predict new data best — a balance between capturing structure and ignoring noise (Navarro, 2019). The second demonstration shows this directly: as a fitted polynomial is given more terms, its fit to the sample keeps improving while its ability to predict new points first improves, then sharply worsens.
Overfitting: fit versus prediction
A polynomial is fitted to nine sample points (navy) drawn from a smooth curve plus noise. As its degree rises, its fit to the sample keeps improving, but its error on five held-out test points (gold) first falls, then climbs as it starts chasing noise.
The sample comes from a quadratic, so a degree-2 or degree-3 fit generalises best. Higher degrees hug the sample more tightly while predicting the held-out points worse — the signature of overfitting.
A model that only fits after the fact explains little; the stronger test is whether it predicts, and whether it can be broken. Good modelling practice therefore treats a model as a hypothesis to be falsified rather than confirmed: fitting it to data is a necessary step, but the decisive question is whether it survives attempts to make it fail and whether it out-predicts genuine rivals on data it has not seen (Palminteri et al., 2017). A shared set of methodological conventions — simulating the model to check that its parameters can be recovered, comparing it against alternatives, and validating on held-out data — has grown up to keep the enterprise honest (Wilson & Collins, 2019).
The clearest illustration of a model earning its keep is one that predicts more than it was built to. Sequential-sampling models of choice are the standard example: they assume that on each decision noisy evidence accumulates over time toward one of two boundaries, and the boundary reached determines the response while the time taken to reach it determines the reaction time. From that single mechanism a model such as Ratcliff’s diffusion account predicts, from the same parameters, both which choice is made and the full distribution of how long it takes — including the characteristic finding that errors can be systematically faster or slower than correct responses (Ratcliff, 1978). The third demonstration runs this accumulation process: adjusting the drift rate (the quality of the evidence) and the boundary (how much evidence is required) reshapes both the speed and the accuracy of the simulated decisions at once, exposing the speed–accuracy trade-off that any adequate model of choice must reproduce.
Evidence accumulation and the speed–accuracy trade-off
On each trial, noisy evidence drifts from the centre toward the upper (correct) or lower (error) boundary; the boundary reached is the choice and the number of steps is the reaction time. Raise the drift to sharpen the evidence; raise the boundary to demand more of it before deciding.
A discrete random-walk approximation to the diffusion model, with fixed non-decision time 250 ms. The seed is fixed, so the traces and summary are identical on every load; the parameters, not chance, drive the change.
Worked Example
Consider choosing between two models of the same reaction-time dataset. Model A has 2 free parameters and, at its best fit, yields a maximised log-likelihood of −520. Model B has 5 free parameters and fits a little better, with a maximised log-likelihood of −517. Model B fits the data more closely — it always will, having more freedom — so fit alone would pick it. The Akaike information criterion asks whether the improvement is worth the extra flexibility.
The criterion is AIC = 2k − 2 ln L, where k is the number of parameters and ln L the maximised log-likelihood; the lower value is preferred. For Model A, AIC = 2(2) − 2(−520) = 4 + 1040 = 1044. For Model B, AIC = 2(5) − 2(−517) = 10 + 1034 = 1044. The two are equal: Model B’s three extra parameters bought exactly the 3-unit gain in log-likelihood needed to offset their cost, and no more. On this criterion there is nothing to choose between them, and the principle of parsimony then favours the simpler Model A — the same three points of fit that looked like an advantage disappear once the flexibility that produced them is charged for. Had Model B’s log-likelihood been only −519, its AIC would have been 1048, decisively worse despite the better raw fit.
Discussion
The reason to build models is not that they are true — every model is known to be false in detail — but that they are precise, and precision is what makes a theory answerable to evidence. A model states a position clearly enough to be wrong, exposes the assumptions a verbal theory can keep hidden, and forces rival explanations onto common ground where their predictions can be compared quantitatively rather than rhetorically (Guest & Martin, 2021). Much of the value is realised before any data are collected, in the act of construction itself: writing a theory as a mechanism reveals whether it is even coherent, and often shows that an informally compelling account cannot in fact do what it was supposed to (van Rooij & Baggio, 2021).
The recurring danger is mistaking fit for confirmation. Because a sufficiently flexible model can fit almost anything, a good fit is informative only in proportion to the risk the model ran of fitting badly, and a model’s worth is measured by its predictions on data it has not seen, weighed against its complexity, rather than by how tightly it hugs the data used to build it (Roberts & Pashler, 2000; Pitt & Myung, 2002). Held together by these disciplines — parsimony, falsification, out-of-sample prediction — modelling is less a way of proving theories right than a way of finding out, efficiently, which ones are wrong.
Current Directions
Contemporary modelling has consolidated around a set of methodological standards meant to make published models reproducible and their comparisons fair. The move is away from demonstrating that a model can fit a phenomenon and toward showing that it survives stringent tests: that its parameters are recoverable from simulated data, that it can be distinguished from its competitors, and that it predicts held-out observations (Wilson & Collins, 2019). Falsification, rather than accumulation of good fits, is increasingly treated as the point of the exercise (Palminteri et al., 2017).
A parallel development is the convergence of psychological modelling with neuroscience and machine learning under the banner of cognitive computational neuroscience, which seeks models that are simultaneously accountable to behaviour and to neural data, and that specify all three of Marr’s levels at once (Kriegeskorte & Douglas, 2018). At the same time, a methodological reappraisal has questioned whether automatic model-selection statistics can substitute for scientific judgement, arguing that choosing between models is a matter of theory and context that no single number can settle (Navarro, 2019). Running through both is a renewed insistence that formal modelling and theory building are the same activity, not separate stages (Guest & Martin, 2021; van Rooij & Baggio, 2021).
Common Misconceptions
- “A model that fits the data well is a good model.”
- Fit alone is weak evidence. A flexible model fits noise as readily as signal, and one that fits the noise predicts new data worse than a simpler model. A close fit counts only in proportion to the risk the model ran of fitting badly (Roberts & Pashler, 2000; Pitt & Myung, 2002).
- “A model has to be realistic to be useful.”
- A model is a deliberate simplification, and its power often comes from what it leaves out. A model detailed enough to reproduce every feature of a system is as hard to understand as the system itself and usually generalises worse (van Rooij & Baggio, 2021).
- “Computational and neural models compete to explain the same thing.”
- They usually address different levels. A computational-level account of what is computed is compatible with an algorithmic account of how and a neural account of where; they are complementary rather than rival (Marr, 1982; Kriegeskorte & Douglas, 2018).
- “Modelling is a technical add-on once the theory is settled.”
- Building the model is often where the theorising happens. Casting a verbal theory as a mechanism regularly reveals hidden assumptions or outright incoherence, so the model shapes the theory rather than merely dressing it up (Guest & Martin, 2021).
Glossary
- AIC (Akaike information criterion).
- A model-selection statistic that adds a penalty of two per free parameter to twice the negative maximised log-likelihood, so that added complexity must be repaid by improved fit; the lower value is preferred.
- Bayesian model.
- A model that casts cognition as inference under uncertainty, combining prior knowledge with noisy evidence to derive what an ideal reasoner should conclude, against which human behaviour is compared.
- Cognitive architecture.
- A model that specifies a fixed set of memory and control mechanisms held to underlie cognition generally, so that a model of any one task is assembled from the same standing components.
- Computational level.
- In Marr’s scheme, the level of analysis specifying what problem a system solves and why, independent of the algorithm or hardware that solves it.
- Connectionist model.
- A model in which computation is carried by networks of simple units linked by weighted connections, so that behaviour emerges from parallel interaction rather than from explicit rules.
- Falsification.
- The testing of a model by attempts to make it fail; a model earns credence by surviving such attempts and by out-predicting rivals, not by accumulating good fits.
- Free parameter.
- A quantity in a model whose value is not fixed in advance but estimated from data by fitting; more free parameters mean greater flexibility and a greater risk of overfitting.
- Goodness of fit.
- A measure of how closely a model’s predictions match observed data, such as the sum of squared error or the likelihood; a necessary but insufficient basis for choosing a model.
- Idealisation.
- The deliberate omission or simplification of features judged inessential, by which a model gains tractability; the choice of what to leave out is central to modelling.
- Likelihood.
- The probability of the observed data under a model at given parameter values; maximising it is a standard way of fitting a model.
- Model comparison.
- The principled choice among rival models, weighing each model’s fit against its complexity so as to favour the one expected to predict new data best rather than the one that fits existing data best.
- Overfitting.
- The fitting of noise as well as signal by an over-flexible model, which improves its fit to the sample at hand while worsening its prediction of fresh data.
- Parsimony.
- The preference, other things equal, for the simpler of two models; formalised in model-selection criteria that penalise added parameters.
- Sequential-sampling model.
- A process model of choice in which noisy evidence accumulates over time toward a decision boundary, jointly predicting which response is made and how long it takes.
- Symbolic model.
- A model that treats cognition as computation over discrete symbols manipulated by rules, as in production-system accounts of problem solving.
Key Researchers
John R. Anderson
(living). Cognitive psychologist at Carnegie Mellon University; creator of the ACT-R cognitive architecture, a unified theory that assembles task models from a fixed set of memory and control mechanisms (Anderson et al., 2004). Faculty · Wikipedia
David Marr
(1945–1980). Computational neuroscientist at MIT; his three levels of analysis — computational, algorithmic, and implementational — gave the field a lasting framework for saying what a model is and is not claiming (Marr, 1982). Wikipedia · Biography
James L. McClelland
(living). Cognitive scientist at Stanford University; co-developer of parallel distributed processing and the interactive-activation model, which showed how context effects arise from a network without explicit rules (McClelland & Rumelhart, 1981). ORCID · Faculty · Google Scholar · Wikipedia
Roger Ratcliff
(living). Mathematical psychologist at the Ohio State University; originator of the diffusion decision model, a sequential-sampling account that jointly predicts the accuracy and the timing of two-choice decisions (Ratcliff, 1978). ORCID · Faculty · Google Scholar
Herbert A. Simon
(1916–2001). Cognitive scientist at Carnegie Mellon University; with Allen Newell, established the information-processing view of cognition, modelling human problem solving as search through a space of states (Newell & Simon, 1972). Wikipedia · Nobel biography
Joshua B. Tenenbaum
(living). Computational cognitive scientist at MIT; a leader of the Bayesian program in cognitive science, modelling learning and inference as probabilistic reasoning over structured knowledge (Tenenbaum et al., 2011). Faculty · Google Scholar
Frequently Asked Questions
What is a psychological model?
A psychological model is a simplified, explicit representation of a mental or behavioural process, built so that its consequences can be worked out and compared with data. It captures the features thought to matter and sets the rest aside, standing in for the real system in a form that can be reasoned about precisely.
Why express a theory as a formal model rather than in words?
Because a formal model fixes its predictions before the data arrive and cannot be quietly bent to fit afterward. A verbal theory’s loose terms let it absorb almost any result, whereas an equation or a program entails specific predictions that can be shown to be wrong, which is what makes the theory testable.
What are the main kinds of psychological model?
Broad theoretical classes include statistical and associative-learning models, symbolic information-processing architectures, connectionist networks, and probabilistic or Bayesian models. These cut across narrower, subject-based categories such as the health-behaviour models that the MeSH vocabulary lists under the heading.
What are Marr’s three levels of analysis?
They are three distinct levels at which any information-processing system can be described: the computational level (what problem is solved and why), the algorithmic level (what representations and procedures solve it), and the implementational level (how it is physically realised). A claim at one level does not fix the others.
What does it mean to fit a model?
Fitting means finding the values of a model’s free parameters that make its predictions come closest to the observed data, usually by minimising an error measure or maximising the likelihood. The best-fitting parameters are the ones under which the model best reproduces what was actually observed.
Why is a good fit not enough to trust a model?
Because a model with enough flexibility can fit almost any dataset, including its noise, and a model that fits the noise predicts new data poorly. A good fit is informative only to the degree that the model was genuinely at risk of fitting badly, so fit must be weighed against complexity.
What is overfitting?
Overfitting is what happens when an over-flexible model captures the random noise in a sample as though it were structure. Its fit to that sample looks excellent, but it generalises poorly, predicting fresh observations worse than a simpler model that ignored the noise.
How are competing models compared?
By weighing each model’s fit against its complexity, using criteria such as the Akaike or Bayesian information criteria that penalise extra parameters, and by testing which model better predicts data it has not seen. The goal is the model expected to predict best, not the one that hugs the existing data most tightly.
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