2.4. Model discrepancy as a baseline for non-stationary replications#
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To keep the notation in the following sections more general, we use the generic \(x\) and \(y\) as independent and dependent variables. To recover expressions for our neuron example, substitute \(x \to t\), \(y \to \tilde{V}^{\LP{\!\!}}\), \(\X \to \Tspace\) and \(\Y \to \Vspace\). Where possible we also use \(A\) and \(B\) as a generic placeholder for a model label.
Our goal is to define a selection criterion which is robust against variations between experimental replications, but which can be computed using knowledge only of the candidate models and the observed empirical data. To do this, we make the following assumption:
EMD assumption (version 1)
Candidate models represent that part of the experiment which we understand and control across replications.
More precisely, in the next section we define the empirical model discrepancy function \(δ^{\EMD}_{A}: (0, 1) \to \RR_{\geq 0}\) such that if model \(\M_A\) exactly reproduces the observations, then \(δ^{\EMD}_{A}\) is identically zero. This function therefore measures the discrepancy between model predictions and actual observations. Since we expect misspecified models to have positive discrepancy (\(\int_0^1 δ^{\EMD}_{A}(Φ) dΦ > 0\)), in the following we treat the discrepancy \(δ^{\EMD}_A\) as a measure of misspecification.
Under our EMD assumption, misspecification in a model corresponds to experimental conditions we don’t fully control, and which could therefore vary across replications of the experiment. Concretely this means that we can reformulate the EMD assumption as
EMD assumption (version 2)
The variability of \(R_A\) across replications is predicted by the model discrepancy \(δ^{\EMD}_A\).
We also assume that the data-generating process \(\Mtrue\) within one replication is strictly stationary with finite correlations, i.e. that all observations are identically distributed but may be correlated. For simplicity in fact we treat the samples as i.i.d., since if necessary this could be done by thinning. See references 7 or 39 for discussions on constructing estimators from correlated time series.
Below we further assume a particular linear relationship between \(δ^{\EMD}_A\) and the replication uncertainty: the function \(δ^{\EMD}_A\) is scaled by the aforementioned sensitivity factor \(c\) to determine the variance of the stochastic process \(\qproc_A\) (which we recall induces the \(R_A\)-distribution for the risk of model \(\M_A\)). The parameter \(c \in \RR_+\) therefore represents a conversion from model discrepancy to epistemic uncertainty. A practitioner can use this parameter to adjust the sensitivity of the criterion to misspecification: a larger value of \(c\) will emulate more important experimental variations. Since the tail probabilities (Eq. 2.11) we want to compute will depend on \(c\), we will write them
The EMD rejection rule
For a chosen rejection threshold \(\falsifythreshold \in (0.5, 1]\), reject model \(\M_A\) if there exists a model \(\M_B\) such that \(\Bemd{AB;c} < \falsifythreshold\) and \(\hat{R}_A > \hat{R}_B\).
The second condition (\(\hat{R}_A > \hat{R}_B\)) ensures the rejection rule remains consistent, even if the \(R\)-distributions become skewed. (See comment below Eq. 2.3.)
As an illustration, Table 2.1 gives the value of \(\Bemd{}\) for each candidate model pair in our example from Fig. 2.1 and Fig. 2.2. As expected, models that were visually assessed to be similar also have \(\Bemd{}\) values close to \(\tfrac{1}{2}\). In practice one would not necessarily need to compute the entire table, since the \(\Bemd{}\) satisfy dice transitivity [40, 41]. In particular this implies that for any threshold \(\falsifythreshold > \varphi^{-2}\) (where \(\varphi\) is the golden ratio), we have
Since any reasonable choice of threshold will have \(\falsifythreshold > 0.5 > \varphi^{-2}\), we can say that whenever the comparisons \(\Bemd{AB;c}\) and \(\Bemd{BC;c}\) are both greater than \(\sqrt{\falsifythreshold}\), we can treat them as transitive. We give a more general form of this result in the Supplementary Methods.
A |
B |
C |
D |
|
|---|---|---|---|---|
A |
0.500 |
0.483 |
0.846 |
0.821 |
B |
0.517 |
0.500 |
0.972 |
0.940 |
C |
0.154 |
0.028 |
0.500 |
0.463 |
D |
0.179 |
0.060 |
0.537 |
0.500 |