Recent reports using conventional Susceptible, Exposed, Infected and Removed models suggest that the next wave of the COVID-19 pandemic in the UK could overwhelm health services, with fatalities exceeding the first wave. We used Bayesian model comparison to revisit these conclusions, allowing for heterogeneity of exposure, susceptibility and transmission. We used dynamic causal modelling to estimate the evidence for alternative models of daily cases and deaths from the USA, the UK, Brazil, Italy, France, Spain, Mexico, Belgium, Germany and Canada over the period 25 January 2020 to 15 June 2020. These data were used to estimate the proportions of people (i) not exposed to the virus, (ii) not susceptible to infection when exposed and (iii) not infectious when susceptible to infection. Bayesian model comparison furnished overwhelming evidence for heterogeneity of exposure, susceptibility and transmission. Furthermore, both lockdown and the build-up of population immunity contributed to viral transmission in all but one country. Small variations in heterogeneity were sufficient to explain large differences in mortality rates. The best model of UK data predicts a second surge of fatalities will be much less than the first peak. The size of the second wave depends sensitively on the loss of immunity and the efficacy of Find-Test-Trace-Isolate-Support programmes. In summary, accounting for heterogeneity of exposure, susceptibility and transmission suggests that the next wave of the SARS-CoV-2 pandemic will be much smaller than conventional models predict, with less economic and health disruption. This heterogeneity means that seroprevalence underestimates effective herd immunity and, crucially, the potential of public health programmes.
- mathematical modelling
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Hundreds of modelling papers have been published recently, offering predictions and projections of the current coronavirus outbreak; these range from peer-reviewed publications to rapid reports from learned societies.
Many, if not most, of these modelling initiatives commit to a particular kind of epidemiological model that precludes heterogeneity in viral exposure, susceptibility and transmission.
The ensuing projections can be fantastic in terms of fatalities and ensuing public health responses.
This study revisits the evidence for conventional epidemiological modelling assumptions using dynamic causal modelling and Bayesian model comparison.
It provides overwhelming evidence for heterogeneity, and the interaction between lockdown and herd immunity in suppressing viral transmission.
Heterogeneity of this sort means that low seroprevalence (<20%) is consistent with levels of population immunity that play a substantive role in attenuating viral transmission and, crucially, facilitating public health measures.
The UK has suffered one of the highest death rates in the world from SARS-CoV-2. Three recent analyses project an even larger second wave of infections—with the UK facing an overwhelmed health service and death rates far higher than the first wave, unless a series of national lockdowns are enforced.1 2 Okell et al (ibid) suggest that ‘the epidemic is still at a relatively early stage and that a large proportion of the population therefore remain susceptible’. The Academy of Medical Science report (Government Publications: research and analysis: COVID-19: preparing for a challenging winter 2020/21, 7 July 2020 (Paper prepared by the Academy of Medical Sciences) https://acmedsci.ac.uk/file-download/51353957) suggested a peak in hospital admissions and deaths in January/February 2021 with estimates of 119 900 (95% CI 24 500 to 251 000) hospital deaths between September 2020 and June 2021—double the number that occurred during the first wave in the spring of 2020. Davies et al 1 project a median unmitigated burden of 23 million (95% CI 13 to 30 million) clinical cases and 350 000 deaths (95% CI 170 000 to 480 000) due to COVID-19 in the UK by December 2021, with only national lockdowns capable of bringing the reproductive ratio near or below one. These kinds of projections have profound consequences for the national economy and the resulting health impacts of recession and unemployment.
This article challenges these projections and, in particular, the underlying assumptions that the risk of infection is homogeneous within the population. The role of pre-existing immunity, host genetics and overdispersion in nuancing viral transmission—and explaining the course of the pandemic in light of unlocking—calls for a more careful quantitative analysis.3–6 The role of heterogeneity in exposure, susceptibility and transmission is receiving more attention, especially in relation to the build-up of herd immunity.7 8 This article illustrates a formal approach to epidemiological modelling that may help resolve some prescient issues.
The pessimistic projections above consider two principal mechanisms that underlie the mitigation—and possible suppression—of the ongoing coronavirus epidemic: (i) a reduction in viral transmission due to lockdown and social distancing measures, and (ii) a build-up of population or herd immunity. Herd immunity can be read as the population immunity that is required to attenuate community transmission. For example, Okell et al (ibid) review three lines of argument and conclude that herd immunity is unlikely to explain differences in mortality rates across countries, thereby placing a strategic emphasis on lockdown to preclude a rebound of infections. This is in contrast with a herd immunity scenario, whereby immunity in the population will reduce transmission to pre-empt a second wave.5 8–10
We use their analyses as a vehicle to question the validity of projections based on conventional (Susceptible, Exposed, Infected and Removed (SEIR)) modelling assumptions. In particular, we deconstruct their arguments to show that the empirical observations they draw on are consistent with herd immunity. Furthermore, public health responses and herd immunity are not mutually exclusive explanations for mortality rates, they both contribute to the epidemiological process and contextualise each other in potentially important ways. In turn, this has implications for the timing of interventions such as lockdown and Find-Test-Trace-Isolate-Support (FTTIS). More generally, we question the commitment to conventional epidemiological models that have not been subject to proper model comparison.
Dynamic causal modelling
Dynamic causal modelling (DCM) is the application of variational Bayes to estimate the parameters of state-space models and, crucially, the evidence for alternative models of the same data.11 It was developed to model interactions among neuronal populations and has been used subsequently in radar, medical nosology and recently epidemiology.11–15 Variational Bayes is also known as approximate Bayesian inference and is computationally more efficient than Bayesian techniques based on sampling procedures (eg, approximate Bayesian computation), which predominate in epidemiological modelling.16–18 The particular dynamic causal model used here embeds an SEIR model of immune status into a model that includes all latent factors generating data; namely, location, infection symptom and testing status. Please see the foundational paper for structural details of the model used in this paper15 and the generic (variational Laplace) scheme used to estimate model parameters and evidence.
DCM differs from conventional epidemiological modelling in that it uses mean field approximations and standard variational procedures to model the evolution of probability densities.16 This contrasts with epidemiological modelling that generally uses stochastic realisations of epidemiological dynamics to approximate probability densities with sample densities.17 19–21 One advantage of variational procedures is that they are orders of magnitude more efficient, enabling end-to-end model inversion or fitting within minutes (on a laptop) as opposed to hours or days (on a supercomputer).17 More importantly, variational procedures provide an efficient way of assessing the quality of one model relative to another, in terms of model evidence (a.k.a., marginal likelihood).22 This enables one to compare different models using Bayesian model comparison (a.k.a. structure learning) and use the best model for nowcasting, forecasting or, indeed, test competing hypotheses about viral transmission.
More generally, Bayesian model comparison plays a central role in testing hypotheses given (often sparse or noisy) data. It eschews intuitive assumptions, about whether there are sufficient data to test this or that, by evaluating the evidence for competing hypotheses or models. If there is sufficient information in the data to disambiguate between two hypotheses, the difference in log-evidence will enable one to confidently assert one model is more likely than the other. Note that this automatically ensures that the model is identifiable, in relation to the model parameters or prior assumptions in question.
Dynamic causal models can be extended to generate any kind of epidemiological data at hand: for example, the number of positive antigen tests. This requires careful consideration of how positive tests are generated, by modelling latent variables such as the bias towards testing people with or without infection or, indeed, the time-dependent capacity for testing. In short, everything that matters—in terms of the latent (hidden) causes of the data—can be installed in the model, including lockdown, self-isolation and other processes that underwrite viral transmission. Model comparison can then be used to assess whether the effect of a latent cause is needed to explain the data—by withdrawing the effect and seeing if model evidence increases or decreases. Here, we leverage the efficiency of DCM to evaluate the evidence for a series of models that are distinguished by heterogeneity or variability in the way that populations respond to an epidemic. The dynamic causal model used for the analyses below is summarised in terms of its structure (figure 1) and parameters (table 1).
Parameters and priors
The prior expectations in table 1 should be read as the effective rates and time constants as they manifest in a real-world setting.23–28 The incubation period refers to the time constant corresponding to the rate at which one becomes symptomatic if infected—it does not refer to the period one is infected prior to developing symptoms. For example, early evidence indicates that by 14 days, approximately 95% of presymptomatic periods will be over.29 30 The priors for the non-susceptible and non-infectious proportion of the population are based on clinical and serological studies reported over the past few weeks.31 32 Please see the code base for a detailed explanation of the role of these parameters in transition probabilities among states. Although the (scale) parameters are implemented as probabilities or rates, they are estimated as log scale parameters, denoted by . Note that the priors are over log scale parameters and are therefore mildly informative. For example, a prior variance of 1/256 corresponds to a prior SD of 1/16. This means that the parameter in question can, a priori, vary by a factor of about 30%. Parameters with a variance of one can be regarded as essentially free parameters (that can vary over several orders of magnitude), for example, the effective population size, which is roughly the size of a large city. The default priors used for the current analyses are also listed in spm_COVID_priors.m (https://www.fil.ion.ucl.ac.uk/spm/covid-19/) and can be optimised using Bayesian model comparison (by comparing the evidence with models that have greater or lesser shrinkage priors).15
Notice that this model is more nuanced than most conventional epidemiological models. For example, immunity and testing are separate factors. This means that we have not simply added an observation model to an SEIR-like model; rather, testing now becomes a latent factor that can influence other factors (eg, the location factor via social distancing). Furthermore, there is a difference between the latent testing state and the reported number of new cases—that depends on sensitivity and specificity, via thresholds used for reporting.33 Separating the infection and symptom factors allows the model to accommodate asymptomatic infection34: to move from an asymptomatic to a symptomatic state depends on whether one is infected but moving from an infected to an infectious state does not depend on whether one is symptomatic. Furthermore, it allows for viral transmission prior to symptom onset.35 36
This particular dynamic causal model accommodates heterogeneity at three levels that can have a substantive effect on epidemiological trajectories. These effects are variously described in terms of overdispersion, super-spreading and amplification events.4 6 37 In the current model, heterogeneity was modelled in terms of three bipartitions (summarised in figure 2).
Heterogeneity in exposure
This was modelled in terms of an effective population size that is less than the total (census) population. The effective population comprises individuals who are in contact with other infected individuals. The remainder of the population are assumed to be geographically sequestered from a regional outbreak or are shielded from it. For example, if the population of the UK was 68 million, and the effective population was 39 million, then only 57% are considered to participate in the outbreak. Of this effective population, a certain proportion are susceptible to infection.
Heterogeneity in susceptibility
This was modelled in terms of a portion of the effective population that are not susceptible to infection. For example, they may have pre-existing immunity via cross-reactivity38–40 or particular host factors41 42 such as mucosal immunity.43 This non-susceptible proportion is assigned to the seronegative state at the start of the outbreak. Of the remaining susceptible people, a certain proportion can transmit the virus to others.
Heterogeneity in transmission
We modelled heterogeneity in transmission with a free parameter (with a prior of one half and a prior SD of 1/16). This parameter corresponds to the proportion of susceptible people who are unlikely to transmit the virus, that is, individuals who move directly from a state of being infected to a seronegative state (as opposed to moving to a seropositive state after a period of being infectious). We associated this transition with a mild infection44 that does not entail seroconversion, for example, recovery in terms of T-cell-mediated responses.39 41 In short, the seronegative state plays the role of a seropositive state of immunity for people who never become infectious, either because they are not susceptible to infection or have a mild infection (with or without symptoms, eg, children).
Modelling heterogeneity of susceptibility in terms of susceptible and non-susceptible individuals can be read as modelling the difference between old (susceptible) and young (non-susceptible) people.24 However, in contrast to models with age-stratification, the current model does not consider different contact rates between different strata (eg, contact matrices). Instead, we finesse this limitation with location-specific contact rates—parameterised by the number of people one is exposed to in different locations. Although full stratification is straightforward to implement (please see DEM_COVID_I.m for a Matlab demonstration that can be read as pseudocode), this simplified model of heterogeneity is sufficient to make definitive inferences about the joint contribution of lockdown and population immunity to viral transmission.
The prior means of non-susceptible and non-infectious people of 50% was chosen on the basis of secondary attack rates in well-defined cohorts.34 45 A prior variance of 1/256 corresponds to a range of between 42% and 58% (99% CI). Note that these priors are just constraints (that are also used as starting estimates). The posterior estimate can be markedly different from the prior if the data are sufficiently informative.
This kind of model is sufficiently expressive to reconcile the apparent disparity between morbidity/mortality rates and low seroprevalence observed empirically31 (https://www.gov.uk/government/publications/national-covid-19-surveillance-reports/sero-surveillance-of-covid-19). We will see below that Bayesian model comparison suggests there is very strong evidence46 for all three types of heterogeneity.
Given a suitable dynamic causal model one can use standard variational techniques to fit the empirical data and estimate model parameters. Having estimated the requisite model parameters, one can then reconstitute the most likely trajectories of latent states: namely, the probability of being in different locations, states of infection, symptom and testing states. An example is provided in figure 3 using daily cases and death-by-date data from the UK from 30 January 2020 to 1 September 2020.
Patients and public involvement
Patients or the public were not involved in the design, or conduct, or reporting, or dissemination plans of this research.
Having briefly established the form and nature of the quantitative modelling, we now apply it to daily reports of new cases and deaths from several countries. Our focus is not on the detailed structure of the model. Rather, we use the model to illustrate how Bayesian model comparison can be used to test some assumptions that underwrite conventional models. In this setting, we frame the results in the form of a commentary and restrict the analysis to data available at the time the above reports were published (ie, from 25 January 2020 to 15 June 2021).
Heterogeneity in exposure, susceptibility and transmission
In what follows, we use DCM to revisit some assumptions implicit in conventional epidemiological modelling. We follow the three lines of arguments rehearsed in Okell et al (ibid). The first can be summarised as: under herd immunity the cumulative mortality rate per million of the population should plateau at roughly the same level in different countries.
This is true if, and only if, the same proportion of the population can transmit the virus. In other words, a plateau to endemic equilibrium—based on the removal of susceptible people from the population—only requires people are susceptible to infection to be immune. If this proportion depends on the composition of a country’s population (ie, demography), mortality rates could differ from country to country. This can be illustrated by using models with heterogeneous population structures, of the sort summarised in figure 2. Figure 4 shows the data and ensuing predictions for 10 countries, using the format of Okell et al.2 These countries were chosen because, at time of writing, they had a well-defined first peak in conjunction with a high fatality rate, in relation to other countries.
Heterogeneity of transmission may be particularly important here.3 This is sometimes framed in terms of overdispersion or the notion of superspreading and amplification events.4 6 37 For example, if only 20% of the population were able to develop a sufficient viral load to infect others, then protective immunity in this subpopulation would be sufficient for an innocuous endemic equilibrium.47 Furthermore, if seroconversion occurs largely in the subpopulation spreading the virus,48 a sufficient herd immunity may only require a seroprevalence of around 10% of the effective population (right panel of figure 4). In short, one might challenge the assumption that COVID-19 is spread homogeneously across the population. Indeed, heterogeneity is becoming increasingly evident in high-risk settings, and in the variation in the period of infectivity across ages.
This begs the question: is there evidence for heterogeneity in the dispersion of SARS-CoV-2? And, if so, does this heterogeneity vary from country to country? Figure 5 answers this question using Bayesian model comparison. It shows—under the models in question—there is overwhelming evidence for heterogeneity of exposure, susceptibility and transmission. And that a substantial proportion of each country’s population does not contribute to viral transmission. These proportions vary from country to country, leading to the differential mortality rates as shown in figure 4.
In general, the effective population is roughly half of the census population, with some variation over countries. The non-susceptible and non-infectious proportions are roughly half of the effective and susceptible populations, respectively—varying between 45% and 65%. This variation underwrites the differences in fatality rates in figure 4. It could be argued that the estimates of the proportion of non-susceptible individuals is at odds with empirical data from contained outbreaks. For example, on the aircraft carrier Charles de Gaulle, about 70% of sailors were infected (https://en.wikipedia.org/wiki/COVID-19_pandemic_on_Charles_de_Gaulle). However, this argument overlooks the contribution of heterogeneity: there were no children on the Charles de Gaulle.
Okell et al (ibid) wrote: ‘If acquisition of herd immunity was responsible for the drop in incidence in all countries, then disease exposure, susceptibility, or severity would need to be extremely different between populations’. On the basis of the above quantitative modelling, this assumption transpires to be wrong: small variations in heterogeneity of exposure and susceptibility are sufficient to explain differences between countries. Our point here is that predicates or assumptions of this sort can be evaluated quantitatively in terms of model evidence.
Do countries that went into lockdown early experience fewer deaths in subsequent weeks?
This is the second argument made by Okell et al (ibid) for the unique role of lockdown in mitigating fatalities: however, exactly the same correlation—between cumulative deaths before and after lockdown—emerges under epidemiological models that entertain heterogeneity and herd immunity (see figure 4 (middle panel)). In short, had the authors tested the hypothesis that lockdown or herd immunity were necessary to explain the data, they would have found very strong evidence for both—and may have concluded that lockdown nuances the emergence of herd immunity (see figure 5).
Does a correlation between antibodies to SARS-CoV-2 (ie, seroprevalence) and COVID-19 mortality rates imply a similar infection fatality ratio over countries?
Conventional models generally assume this is the case.2 24 The problem with this assumption is that it precludes pre-existing immunity and loss of immunity (as in SEIR models).38 Population immunity could fall over a few months due to population flux and host factors, such as loss of neutralising antibodies.48 This is important because it means that seroprevalence could fall slowly after the first wave of infection (indeed, empirical seroprevalence is not increasing and may be decreasing in the UK (https://www.ons.gov.uk/peoplepopulationandcommunity/healthandsocialcare/conditionsanddiseases/bulletins/coronaviruscovid19infectionsurveypilot/18june2020%23antibody-data). In turn, this produces a non-linear relationship between the prevalence of antibodies and cumulative deaths at the time seroprevalence is assessed. This is illustrated by the curvilinear relationships in the right panel of figure 4 (under a loss of seroprevalence with a time constant of 3 months). If one associates infection fatality ratio (IFR) with the slope of fatality rates—as a function of seroprevalence—then the IFR changes over time. In short, the IFR changes as the epidemic progresses, as the proportion of susceptible and transmitting people falls, or those at highest risk succumb early in the pandemic. This begs the question: are quantities such as IFR fit for purpose when trying to model epidemiological dynamics?
What is the impact of different rates of loss of immunity?
So, what are the implications of heterogeneity for seroprevalence and a second wave? The Bayesian model comparisons in figure 5 speak to a mechanistic role for herd immunity in mitigating a rebound of infections. Note that this model predicts seroprevalences that are consistent with empirical community studies, without ever seeing these serological data (eg, in the UK if 11% of the effective population is seropositive and the effective population is 49% of the census population, we would expect 5.4% of people to have antibodies, which was the case at the time of analysis (https://www.gov.uk/government/publications/national-covid-19-surveillance-reports/sero-surveillance-of-covid-19).
Predictive validity of this sort generally increases with model evidence. This follows from the fact that the log-evidence is accuracy minus complexity. In other words, models with the greatest evidence afford an accurate account of the data at hand, in the simplest way possible. Unlike the Akaike and widely used Bayesian information criteria (AIC and BIC), the variational bounds on log-evidence used in DCM evaluate complexity explicitly.22 Models with the greatest evidence have the greatest predictive validity because they do not overfit the data. An example of the posterior predictions afforded by the current model is provided in figures 3 and 6 that indicate the timing and amplitude of a second wave in the UK. Figure 6 focuses on fatality rates under a couple of different scenarios; namely, under a rapid loss of antibody-mediated immunity and under an accelerated FTTIS programme.
These posterior predictions suggest that there may be a mild surge of fatalities over the autumn, peaking at about 100 per day. This second wave could be eliminated completely with an increase in the efficacy of contact tracing (FTTIS)—modelled as the probability of self-isolating, given one is infected but asymptomatic. It can be seen that even with a relatively low efficacy of 25%, elimination is possible by November, with convergence to zero fatality rates. Please see Friston et al 49 for a more comprehensive analysis. Note that in a month or two, death rates should disambiguate between these scenarios.
The CIs in figure 6 may appear rather tight. This reflects two issues. First, the well-known overconfidence problem with variational inference—discussed in Friston et al 15 and MacKay.50 Second, these posterior predictive densities are based on the entire timeseries, under a dynamic causal model that constrains the functional form of the trajectories. Put simply, this means that uncertainty about the future can be reduced substantially by data from the past.
Our reading of the epidemiological modelling literature suggests a systemic failure to formally evaluate the evidence for alternative models (eg, models with age stratification and heterogeneous contact structure). This may reflect the fact that agent-based, stochastic transmission models are notoriously difficult to evaluate in terms of their evidence.16 18 In contrast, the variational approaches used in DCM34–39 furnish a variational bound on model evidence that allows competing models to be assessed quickly and efficiently. The central role of model comparison is established in many disciplines and is currently attracting attention in epidemiology.51
Although model evaluation using the AIC (or widely used BIC) can be found in the epidemiological literature,52–54 this kind of comparison does not constitute proper model comparison. This is because the complexity part of model evidence is not estimated by the AIC (or widely used BIC). The model complexity corresponds to the df used to explain the data (technically, the KL divergence between the posterior and prior). The AIC and BIC approximate complexity with (functions of) the number of free parameters, irrespective of whether these parameters are used or not. This means that the AIC is not fit for purpose when comparing models in a clinical or epidemiological setting. Please see Penny22 for illustrations of the failure of the AIC (and BIC). In short, it appears that most of the predictions underwriting ‘scientific advice’ to governmental agencies are based on epidemiological models that have not been properly compared with alternative models. If there is no rebound in fatality rates in the next few months, the conclusions in Okell et al, Davies et al and the Academy of Medical Sciences report (ibid) will be put under some pressure. This pressure might license a more (model) evidence-based approach, using the requisite variational methods that predominate in statistical physics, machine learning and (dynamic) causal modelling.
The recurrent theme above is the danger of committing to one particular model or conception of the epidemiological process. In other fields—dealing with population dynamics—Bayesian model comparison is used to identify the best structure and parameterisation of models28–30 known as structure learning.31 32 Figure 5 offers an example of Bayesian model comparison in epidemiology, evincing very strong evidence for heterogeneity in responses to viral infection—and a synergistic role for social distancing and herd immunity.
Identifying the right epidemiological model has considerable public health and economic implications. While SARS-CoV-2 may not be eradicated, model selection suggests that any second wave will be much smaller than other models have projected, and the virus will become endemic rather than epidemic. The size of a second wave may depend sensitively on the efficacy of FTTIS programmes and the rate of loss of immunity. Recent evidence suggests T-cell immunity may be more important for long-term immunity with circulating SARS-CoV-2-specific CD8+ and CD4+ T cells identified in 70% and 100% of convalescent patients with COVID-19, respectively.39 Furthermore, 90% of people who seroconvert make detectible neutralising antibody responses that are stable for at least 3 months.55 If the above dynamic causal model is broadly correct, future national lockdowns may be unnecessary. As an endemic and potentially fatal virus, especially in elderly people and those with underlying conditions, attention to the details of FTTIS and shielding becomes all the more important. This emphasises the need for clear criteria for when and how to implement local lockdowns in ‘hotspot’ areas.
In summary, lockdown and social distancing have undoubtedly restricted the transmission of the virus. Model comparison suggests that these approaches remain an essential component of pandemic control, particularly at current levels of infections in the UK. However, extending the notion of ‘herd immunity’—to include seronegative individuals with lower susceptibility and/or lower risk of transmission—engenders an immune subpopulation that can change over time and country. The implicit immunity may reduce mortality and lower the risk of a second wave to a greater extent than predicted under many epidemiological models. On this view, herd immunity subsumes people who are not susceptible to infection or, if they are, are unlikely to be infectious or seroconvert; noting that SARS-CoV-2 can induce virus-specific T-cell responses without seroconversion. This reconciles the apparent disparity between reports of new cases, mortality rates and the low seroprevalence observed empirically. Crucially, Bayesian model comparison confirms that there is very strong evidence for the heterogeneity that underwrites this kind of herd immunity.8
Put simply, an effective herd immunity—that works hand-in-hand with appropriate public health and local lockdown measures—requires <20% seroprevalence. This seroprevalence has already reached in many countries and is sufficient to preclude a traumatic second wave, even under pessimistic assumptions about loss of humoral immunity endowed by antibodies.
Glossary of terms
Dynamic causal modelling: the application of variational Bayes to estimate the unknown parameters of state-space models and assess the evidence for alternative models of the same data.
Variational Bayes (a.k.a., approximate Bayesian inference): a generic Bayesian procedure for fitting and evaluating generative models of data by optimising a variational bound on model evidence. See https://en.wikipedia.org/wiki/Variational_Bayesian_methods.
Model evidence (a.k.a., marginal likelihood): the probability of observing some data under a particular model. It is called the marginal likelihood because its evaluation entails marginalising (ie, integrating) out dependencies on model parameters. Technically, model evidence is accuracy minus complexity, where accuracy is the expected log likelihood of some data and complexity is the divergence between posterior and prior densities over model parameters.
Variational bound (a.k.a., variational free energy): known as an evidence lower bound (ELBO) in machine learning because it is always less than the logarithm of model evidence. In brief, variational free energy converts an intractable marginalisation problem—faced by sampling procedures—into a tractable optimisation problem. This optimisation furnishes the posterior density over model parameters and ensures the ELBO approximates the log evidence for a model: https://en.wikipedia.org/wiki/Variational_Bayesian_methods. Crucially, the variational bound includes an explicit estimate of model complexity, in contrast to the AIC and BIC.22
Bayesian model comparison: a procedure to compare different models of the same data in terms of model evidence. The marginal likelihood ratio of two models is known as a Bayes factor: https://en.wikipedia.org/wiki/Bayes_factor.
Agent-based simulation models: an alternative to equation-based models, usually used to simulate scenarios that are richer than models based on population dynamics. Agent-based models simulate lots of individuals to create a sample distribution over outcomes. Evaluating the marginal likelihood from the ensuing sample distributions is extremely difficult, even with state-of-the-art estimators such as the harmonic mean: see https://radfordneal.wordpress.com/2008/08/17/the-harmonic-mean-of-the-likelihood-worst-monte-carlo-method-ever/.
Handling editor Seye Abimbola
Contributors All authors contributed equally to the writing and revision of this work. The figures were prepared by KF.
Funding The Wellcome Centre for Human Neuroimaging is supported by core funding from the Wellcome Trust (203147/Z/16/Z).
Competing interests None declared.
Patient and public involvement Patients and/or the public were not involved in the design, or conduct, or reporting, or dissemination plans of this research.
Patient consent for publication Not required.
Provenance and peer review Not commissioned; externally peer reviewed.
Data availability statement Data are available in a public, open access repository.
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