2.3. A conceptual model for replicate variations#
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Evaluating the risk Eq. 2.7 for each candidate model on a dataset \(\D_\test\) yields four scalars \(\hat{R}_A\) to \(\hat{R}_D\); since a lower risk should indicate a better model, a simple naive model selection rule would be
As with many selection criteria (e.g. AIC [34, 35], BIC [36], DIC [37, 38]), this rule is effectively binary: the third option, to reject neither model, has probability zero. It therefore always selects either \(\M_a\) or \(\M_b\), even when the evidence favouring one of the two is extremely weak. Another way to see this is illustrated at the top of Fig. 2.2: the lines representing the four values \(R_A\) through \(R_D\) have no error bars, so even minute differences suffice to rank the models.
The problem is that from a scientific standpoint, if the evidence is too weak, this ranking is likely irrelevant — or worse, misinformative. Hence our desire to assign uncertainty to each estimate of the risk. Ideally we would like to be able to compute tail probabilities like \(P(R_A < R_B)\), which would quantify the strength of the evidence in favour of either \(\M_A\) or \(\M_B\). Selecting a minimum evidence threshold \(\falsifythreshold\) would then allow us to convert Eq. 2.8 into a true ternary decision rule with non-zero probability of keeping both models:
As we explain in the Introduction, our goal is to select a model which can describe not just the observed data \(\D_\test\), but also new data generated in a replication experiment. In order for our criterion to be robust, the tail probabilities \(P(R_a < R_b)\) should account for uncertainty in the replication process.
Note that for a given candidate model \(\M_a\) and fixed \(\Mtrue\), we can always estimate the risk with high accuracy if we have enough samples, irrespective of the amount of noise intrinsic to \(\M_a\) or of misspecification between \(\M_a\) and \(\Mtrue\). Crucially however, we also do not assume the replication process to be stationary, so a replicate dataset \(\D_\test'\) may be drawn from a slightly different data-generating process \(\Mtrue'\). This can occur even when \(\Mtrue\) itself is stationary, reflecting the fact that it is often easier to control variability within a single experiments than across multiple ones. Ontologically therefore, the uncertainty on \(\hat{R}\) is a form of epistemic uncertainty arising from the variability across (experimental) replications.
To make this idea concrete, consider that the input \(I_{\mathrm{ext}}\) and noise \(ξ\) might not be stationary over the course of an experiment with multiple trials. We can represent this by making their parameters random variables, for example
and drawing new values of \((σ_o, σ_i, τ)\) for each trial (i.e. each replicate). Since it is a distribution over data-generating processes, we call \(Ω\) an epistemic distribution. For illustration purposes, here we have parametrised \(Ω\) in terms of two parameters of the biophysical model and one parameter of the observation model, thus capturing epistemic uncertainty within a single experiment. In general the parametrisation of \(Ω\) is a modelling choice, and may represent other forms of non-stationarity — for example due to variations between experimental setups in different laboratories.
Conceptually, we could estimate the tail probabilities \(P(R_a < R_b)\) by sampling \(J\) different data-generating processes \(\Mtrue^j\) from \(Ω\), for each then drawing a dataset \(\D_\test^j \sim \Mtrue^j\), and finally computing the empirical risks \(\hat{R}_a^j\) and \(\hat{R}_b^j\) on \(\D_\test^j\). The fraction of datasets for which \(\hat{R}_a^j < \hat{R}_b^j\) would then estimate the tail probability:
The issue of course is that we cannot know \(Ω\). First because we only observe data from a single \(\Mtrue\), but also because there may be different contexts to which we want to generalise: one researcher may be interested in modelling an individual LP neuron, while another might seek a more general model which can describe all neurons of this type. These two situations would require different epistemic distributions, with the latter one being in some sense broader.
However we need not commit to a single epistemic distribution: if we have two distributions, and we want to ensure that conclusions hold under both, we can instead use the condition
to attempt to reject \(\M_b\). In general, considering more epistemic distributions will make the model selection more robust, at the cost of discriminatory power. Epistemic distributions are not therefore prior distributions, since a Bayesian calculation is always tied to a specific choice of prior. (The opposite however holds: a prior can be viewed as a particular choice of epistemic distribution.)
We view epistemic distributions mostly as conceptual tools. For calculations, we will instead propose in the next sections a different type of distribution (\(\qproc\); technically a stochastic process), which is not on the data-generating process, but on the distribution of pointwise losses. Being lower-dimensional and more stereotyped than \(Ω\), we will be able to construct \(\qproc\) entirely non-parametrically, up to a scaling constant \(c\) (c.f. Fig. 1.1 and the section listing desiderata for \(\qproc\)). A later section we will then show, through numerical calibration and validation experiments, that \(\qproc\) also has nice universal properties, so that the only thing that really matters is the overall scale of the epistemic distributions. The constant \(c\) is matched to this scale by numerically simulating epistemic distributions as part of the calibration experiments.