7. Supplementary Discussion

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7.1. Other forms of uncertainty

As we say in the main text, there are two main sources of epistemic uncertainty on an estimate of the risk: limited number of samples and variability in the replication process. This work focusses on the estimating replication uncertainty, and numerically studying its effect as a function of sample size. We have eschewed a formal treatment of the effect of the latter, since this is a well-studied problem and good estimation procedures already exist.

For example, a bootstrap procedure can be used to estimate the uncertainty on a statistic (here the risk \(R\)) from a single dataset \(\D\), by recomputing the statistic on multiple surrogate datasets obtained by resampling \(\D\) with replacement (Fig. 7.1b). Alternatively, if we have access to good candidate models, we can use those models as simulators to generate multiple synthetic datasets (Fig. 7.1c). The distribution of risks over those datasets is then a direct estimate of its uncertainty due to finite samples.

In the limit of infinite data, both of these methods produce a risk “distribution” which collapses onto a precise value, independent of any discrepancy between model and true data-generating process. In contrast, that discrepancy defines the spread of risk distributions in Fig. 2.2, which do not collapse when \(L \to \infty\).

Aleatoric uncertainty also does not vanish in the large \(L\) limit, but manifests differently. It sets a lower bound on the spread of pointwise losses a model can achieve (Fig. 7.1a). In terms of our formalism therefore, the aleatoric uncertainty determines the shape of the PPFs \(\qs\) and \(\qt\), epistemic uncertainty (due to finite samples) translates to uncertainty on those shapes, and epistemic uncertainty (across replicates) affects the metric variance of the HB process around \(\qs\).

Fig. 7.1 Aleatoric and finite-size uncertainty a) Loss distribution of individual data points – \(\{Q_a(t_k, V^{\LP{\!\!}}(t_k; \Mtrue))\}\) – for the dataset and models shown in Fig. 2.1 and \(a \in \{A,B,C,D\}\). For a model which predicts the data well, this is mostly determined by the aleatoric uncertainty. b) Bootstrap estimate of finite-size uncertainty on the risk, obtained using case resampling [73]: for each model, the set of losses was resampled 1200 times with replacement. Dataset and colours are the same as in (a). c) Synthetic estimate of finite-size uncertainty on the risk, obtained by evaluating Eq. 2.6 on 400 different simulations of the candidate model (differing by the random seed). Here the same model is used for simulation and loss evaluation. Dataset sizes \(L\) determine the integration time, adjusted so all datasets contain the same number of spikes. All subpanels use the same vertical scale. b–c) The variance of the \(R\)-distributions, i.e. the uncertainty on \(R\), goes to zero as \(L\) is increased. a–c) Colours indicate the model used for the loss. Probability densities were obtained by a kernel density estimate (KDE). [source]