By Bruce Edmonds
“Just as physical tools and machines extend our physical abilities, models extend our mental abilities, enabling us to understand and control systems beyond our direct intellectual reach” (Calder & al. 2018)
There is a modelling norm that one should be able to completely understand one’s own model. Whilst acknowledging there is a trade-off between a model’s representational adequacy and its simplicity of formulation, this tradition assumes there will be a “sweet spot” where the model is just tractable but also good enough to be usefully informative about the target of modelling – in the words attributed to Einstein, “Everything should be made as simple as possible, but no simpler”1. But what do we do about all the phenomena where to get an adequate model2 one has to settle for a complex one (where by “complex” I mean a model that we do not completely understand)? Despite the tradition in Physics to the contrary, it would be an incredibly strong assumption that there are no such phenomena, i.e. that an adequate simple model is always possible (Edmonds 2013).
There are three options in these difficult cases.
- Do not model the phenomena at all until we can find an adequate model we can fully understand. Given the complexity of much around us this would mean to not model these for the foreseeable future and maybe never.
- Accept inadequate simpler models and simply hope that these are somehow approximately right3. This option would allow us to get answers but with no idea whether they were at all reliable. There are many cases of overly simplistic models leading policy astray (Adoha & Edmonds 2017; Thompson 2022), so this is dangerous if such models influence decisions with real consequences.
- Use models that are good for our purpose but that we only partially understand. This is the option examined in this paper.
When the purpose is empirical the last option is equivalent to preferring empirical grounding over model simplicity (Edmonds & Moss 2005).
Partially Understood Models
In practice this argument has already been won – we do not completely understand many computer simulations that we use and rely on. For example, due to the chaotic nature of the dynamics of the weather, forecasting models are run multiple times with slightly randomised inputs and the “ensemble” of forecasts inspected to get an idea of the range of different outcomes that could result (some of which might be qualitatively different from the others)4. Working out the outcomes in each case requires the computational tracking of a huge numbers of entities in a way that is far beyond what the human mind can do5. In fact, the whole of “Complexity Science” can be seen as different ways to get some understanding of systems for which there is no analytic solution6.
Of course, this raises the question of what is meant by “understand” a model, for this is not something that is formally defined. This could involve many things, including the following.
- That the micro-level – the individual calculations or actions done by the model each time step – is understood. This is equivalent to understanding each line of the computer code.
- That some of the macro-level outcomes that result from the computation of the whole model is understood in terms of partial theories or “rules of thumb”.
- That all the relevant macro-level outcomes can be determined to a high degree of accuracy without simulating the model (e.g. by a mathematical model).
Clearly, level (1) is necessary for most modelling purposes in order to know the model is behaving as intended. The specification of this micro-level is usually how such models are made, so if this differs from what was intended then this would be a bug. Thus this level would be expected of most models7. However, this does not necessarily mean that this is at the finest level of detail possible – for example, we usually do not bother about how random number generators work, but simply rely on its operation, but in this case we have very good level (3) of understanding for these sub-routines.
At the other extreme, a level (3) understanding is quite rare outside the realm of physics. In a sense, having this level of understanding makes the model redundant, so would probably not be the case for most working models (those used regularly)8. As discussed above, there will be many kinds of phenomena for which this level of understanding is not feasible.
Clearly, what many modelers find useful is a combination of levels (1) & (2) – that is, the detailed, micro-level steps that the model takes are well understood and the outcomes understood well enough for the intended task. For example, when using a model to establish a complex explanation9 (of some observed pattern in data using certain mechanisms or structures) then one might understand the implementation of the candidate mechanisms and verify that the outcomes fit the target pattern for a range of parameters, but not completely understand the detail of the causation involved. There might well be some understanding, for example how robust this is to minor variations in the initial conditions or the working of the mechanisms involved (e.g. by adding some noise to the processes). A complete understanding might not be accessible but this does not stop an explanation being established (although a better understanding is an obvious goal for future research or avenue for critiques of the explanation).
Of course, any lack of a complete, formal understanding leaves some room for error. The argument here is not deriding the desirability of formal understanding, but is against prioritising that over model adequacy. Also the lack of a formal, level (3), understanding of a model does not mean we cannot take more pragmatic routes to checking it. For example: performing a series of well-designed simulation experiments that intend to potentially refute the stated conclusions, systematically comparing to other models, doing a thorough sensitivity analysis and independently reproducing models can help ensure their reliability. These can be compared with engineering methods – one may not have a proof that a certain bridge design is solid over all possible dynamics, but practical measures and partial modelling can ensure that any risk is so low as to be negligible. If we had to wait until bridge designs were proven beyond doubt, we would simply have to do without them.
Layering Models to Leverage some Understanding
As a modeller, if I do not understand something my instinct is to model it. This instinct does not change if what I do not understand is, itself, a model. The result is a model of the original model – a meta-model. This is, in fact, common practice. I may select certain statistics summarising the outcomes and put these on a graph; I might analyse the networks that have emerged during model runs; I may use maths to approximate or capture some aspect of the dynamics; I might cluster and visualise the outcomes using Machine Learning techniques; I might make a simpler version of the original and compare them. All of these might give me insights into the behaviour of the original model. Many of these are so normal we do not think of this as meta-modelling. Indeed, empirically-based models are already, in a sense, meta-models, since the data that they represent are themselves a kind of descriptive model of reality (gained via measurement processes).
This meta-modelling strategy can be iterated to produce meta-meta-models etc. resulting in “layers” of models, with each layer modelling some aspect of the one “below” until one reaches the data and then what the data measures. Each layer should be able to be compared and checked with the layer “below”, and analysed by the layer “above”.
An extended example of such layering was built during the SCID (Social Complexity of Immigration and Diversity) project10 and illustrated in Figure 1. In this a complicated simulation (Model 1) was built to incorporate some available data and what was known concerning the social and behavioural processes that lead people to bother to vote (or not). This simulation was used as a counter-example to show how assumptions about the chaining effect of interventions might be misplaced (Fieldhouse et al. 2016). A much simpler simulation was then built by theoretical physicists (Model 2), so that it produced the same selected outcomes over time and aa range of parameter values. This allowed us to show that some of the features in the original (such as dynamic networks) were essential to get the observed dynamics in it (Lafuerza et al. 2016a). This simpler model was in turn modelled by an even simpler model (Model 3) that was amenable to an analytic model (Model 4) that allowed us to obtain some results concerning the origin of a region of bistability in the dynamics (Lafuerza et al. 2016b).
Figure 1. The Layering of models that were developed in part of the SCID project
Although there are dangers in such layering – each layer could introduce a new weakness – there are also methodological advantages, including the following. (A) Each model in the chain (except model 4) is compared and checked against both the layer below and that above. Such multiple model comparisons are excellent for revealing hidden assumptions and unanticipated effects. (B) Whilst previously what might have happened was a “heroic” leap of abstraction from evidence and understanding straight to Model 3 or 4, here abstraction happens over a series of more modest steps, each of which is more amenable to checking and analysis. When you stage abstraction the introduced assumptions are more obvious and easier to analyse.
One can imagine such “layering” developing in many directions to leverage useful (but indirect) understanding, for example the following.
- Using an AI algorithm to learn patterns in some data (e.g. medical data for disease diagnosis) but then modelling its working to obtain some human-accessible understanding of how it is doing it.
- Using a machine learning model to automatically identify the different “phase spaces” in model results where qualitatively different model behaviour is exhibited, so one can then try to simplify the model within each phase.
- Automatically identifying the processes and structures that are common to a given set of models to facilitate the construction of a more general, ‘umbrella’ model that approximates all the outcomes that would have resulted from the set, but within a narrower range of conditions.
As the quote at the top implies, we are used to settling for partial control of what machines do because it allows us to extend our physical abilities in useful ways. Each time we make their control more indirect, we need to check that this is safe and adequate for purpose. In the cars we drive there are ever more layers of electronic control between us and the physical reality it drives through which we adjust to – we are currently adjusting to more self-drive abilities. Of course, the testing and monitoring of these systems is very important but that will not stop the introduction of layers that will make them safer and more pleasant to drive.
The same is true of our modelling, which we will need to apply in ever more layers in order to leverage useful understanding which would not be accessible otherwise. Yes, we will need to use practical methods to test their fitness for purpose and reliability, and this might include the complete verification of some components (where this is feasible), but we cannot constrain ourselves to only models we completely understand.
If the above seems obvious, then why am I bothering to write this? I think for a few reasons. Firstly, to answer the presumption that understanding one’s model must have priority over all other considerations (such as empirical adequacy) so that sometimes we must accept and use partially understood models. Secondly, to point out that such layering has benefits as well as difficulties – especially if it can stage abstraction into more verifiable steps and thus avoid huge leaps to simple but empirically-isolated models. Thirdly, because such layering will become increasingly common and necessary.
In order to extend our mental reach further, we will need to develop increasingly complicated and layered modelling. To do this we will need to accept that our understanding is leveraged via partially understood models, but also to develop the practical methods to ensure their adequacy for purpose.
 These are a compressed version of his actual words during a 1933 lecture, which were: “It can scarcely be denied that the supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience.” (Robinson 2018)
 Adequate for whatever our purpose for it is (Edmonds & al. 2019).
The weasel words I once heard from a Mathematician excusing an analytic model he knew to be simplistic were: that, although he knew it was wrong, it was useful for “capturing core dynamics” (though how he knew that they were not completely wrong eludes me).
 For an introduction to this approach read the European Centre for Medium-Range Weather Forecasts’ fact sheet on “Ensemble weather forecasting” at: https://www.ecmwf.int/en/about/media-centre/focus/2017/fact-sheet-ensemble-weather-forecasting
 In principle, a person could do all the calculations involved in a forecast but only with the aid of exterior tools such as pencil and paper to keep track of it all so it is arguable whether the person doing the individual calculations has an “understanding” of the complete picture. Lewis Fry Richardson, who pioneered the idea of numerical forecasting of weather in the 1920s, did a 1-day forecast by hand to illustrate his method (Lynch 2008), but this does not change the argument.
 An analytic solution is when one can obtain a closed-form equation that characterises all the outcomes by manipulating the mathematical symbols in a proof. If one has to numerically calculate outcomes for different initial conditions and parameters this is a computational solution.
 For purely predictive models, whose purpose is only to anticipate an unknown value to a useful level of accuracy, this is not strictly necessary. For example, how some AI/Machine learning models work may not clear at the micro-level, but as long as it works (successfully predicts) this does not matter – even if its predictive ability is due to a bug.
 Models may still be useful in this case, for example to check the assumptions made in the matching mathematical or other understanding.
 For more on this use see (Edmonds et al. 2019).
 For more about this project see http://cfpm.org/scid
Bruce Edmonds is supported as part of the ESRC-funded, UK part of the “ToRealSim” project, 2019-2023, grant number ES/S015159/1 and was supported as part of the EPSRC-funded “SCID” project 2010-2016, grant number EP/H02171X/1.
Calder, M., Craig, C., Culley, D., de Cani, R., Donnelly, C.A., Douglas, R., Edmonds, B., Gascoigne, J., Gilbert, N. Hargrove, C., Hinds, D., Lane, D.C., Mitchell, D., Pavey, G., Robertson, D., Rosewell, B., Sherwin, S., Walport, M. and Wilson, A. (2018) Computational modelling for decision-making: where, why, what, who and how. Royal Society Open Science, DOI:10.1098/rsos.172096.
Edmonds, B. (2013) Complexity and Context-dependency. Foundations of Science, 18(4):745-755. DOI:10.1007/s10699-012-9303-x
Edmonds, B. and Moss, S. (2005) From KISS to KIDS – an ‘anti-simplistic’ modelling approach. In P. Davidsson et al. (Eds.): Multi Agent Based Simulation 2004. Springer, Lecture Notes in Artificial Intelligence, 3415:130–144. DOI:10.1007/978-3-540-32243-6_11
Edmonds, B., le Page, C., Bithell, M., Chattoe-Brown, E., Grimm, V., Meyer, R., Montañola-Sales, C., Ormerod, P., Root H. & Squazzoni. F. (2019) Different Modelling Purposes. Journal of Artificial Societies and Social Simulation, 22(3):6. DOI:10.18564/jasss.3993
Fieldhouse, E., Lessard-Phillips, L. & Edmonds, B. (2016) Cascade or echo chamber? A complex agent-based simulation of voter turnout. Party Politics. 22(2):241-256. DOI:10.1177/1354068815605671
Lafuerza, LF, Dyson, L, Edmonds, B & McKane, AJ (2016a) Simplification and analysis of a model of social interaction in voting, European Physical Journal B, 89:159. DOI:10.1140/epjb/e2016-70062-2
Lafuerza L.F., Dyson L., Edmonds B., & McKane A.J. (2016b) Staged Models for Interdisciplinary Research. PLoS ONE, 11(6): e0157261. DOI:10.1371/journal.pone.0157261
Lynch, P. (2008). The origins of computer weather prediction and climate modeling. Journal of Computational Physics, 227(7), 3431-3444. DOI:10.1016/j.jcp.2007.02.034
Robinson, A. (2018) Did Einstein really say that? Nature, 557, 30. DOI:10.1038/d41586-018-05004-4
Thompson, E. (2022) Escape from Model Land. Basic Books. ISBN-13: 9781529364873
Edmonds, B. (2023) The inevitable “layering” of models to extend the reach of our understanding. Review of Artificial Societies and Social Simulation, 9 Feb 2023. https://rofasss.org/2023/02/09/layering
© The authors under the Creative Commons’ Attribution-NoDerivs (CC BY-ND) Licence (v4.0)