2.5. \(δ^{\mathrm{EMD}}\): Expressing misspecification as a discrepancy between CDFs#
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We can treat the loss function for a given model \(\M_A\) as a random variable \(Q(x,y;\M_A)\) where the \((x,y)\) are sampled from \(\Mtrue\). A key realisation for our approach is that the CDF (cumulative distribution function) of the loss suffices to compute the risk. Indeed, we have for the CDF of the loss
where \(H\) is the Heaviside function with \(H(q) = 1\) if \(q \geq 0\) and \(0\) otherwise.
Since \(\Mtrue\) is unknown, a crucial feature of Eq. 2.15 is that \(\Phis_A(q)\) can be estimated without needing to evaluate \(p\bigl(x,y \mid \Mtrue \bigr)\); indeed, all that is required is to count the number of observed data points in \(\D_\test\) which according to \(\M_A\) have a loss less than \(q\). Moreover, since \(Q\) returns a scalar, the data points \((x_i, y_i)\) can have any number of dimensions: the number of data points required to get a good estimate of \(\Phis_A(q)\) does not depend on the dimensionality of the data, for the same reason that estimating marginals of a distribution requires much fewer samples than estimating the full distribution. Finally, because the loss is evaluated using \(\M_A\), but the expectation is taken with respect to a distribution determined by \(\Mtrue\), we call \(\Phis_A\) the mixed CDF.
We can invert \(\Phis_A\) to obtain the mixed PPF[1] (percent point function, also known as quantile function, or percentile function):
which is also a 1-d function, irrespective of the dimensionality of \(\X\) or \(\Y\). We can then rewrite the risk as a one dimensional integral in \(\qs_A\):
To obtain Eq. 2.17, we simply used Fubini’s theorem to reorder the integral of Eq. 2.15 and marginalized over all slices of a given loss \(\qs_A(Φ)\). The integral form (first line of Eq. 2.17) is equivalent to averaging an infinite number of samples, and therefore to the (true) risk, whereas the average over observed samples (second line of Eq. 2.17) is exactly the empirical risk defined in Eq. 2.3.
In practice, to evaluate Eq. 2.17, we use the observed samples to compute the sequence of per-sample losses \(\{Q(x_i,y_i;\M_A)\}_{x_i,y_i \in \D_\test}\). This provides us with a sequence of losses, which we use as ordinate values. We then sort this sequence so that we have \(\{Q_i\}_{i=1}^L\) with \(Q_i \leq Q_i+1\), and assign to each the abscissa \(Φ_i = i/L+1\), such that losses are motonically increasing and uniformly distributed on the [0, 1] interval. This yields the empirical PPF of the loss — the “empirical” qualifier referring to this construction via samples, as opposed to an analytic calculation. Interpolating the points then yields a continuous function which can be used in further calculations. All examples in this paper linearly interpolate the PPF from \(2^{10}=1024\) points.
In Fig. 2.3 we show four examples of empirical PPFs, along with their associated empirical CDFs. We see that the statistics of the additive observational noise affects the shape of the PPF: for noise with exponential tails, as we get from Gaussian or Poisson distributions, we have strong concentration around the minimum value of the loss followed by a sharp increase at \(Φ=1\). For heavier-tailed distributions like Cauchy, loss values are less concentrated and the PPF assigns non-negligible probability mass to a wider range of values. The dimensionality of the data also matters. High-dimensional Gaussians are known to place most of their probability mass in a thin shell centered on the mode, and we see this in the fourth column of Fig. 2.3: the sharp increase at \(Φ=0\) indicates that very low probability is assigned to the minimum loss.
Since by construction, the abscissae \(Φ\) of an empirical PPF are spaced at intervals of \(1/L\), the Riemann sum for the integral in Eq. 2.17 reduces to the sample average. More importantly, we can interpret the risk as a functional in \(\qs_A(Φ)\), which will allow us below to define a generic stochastic process that accounts for epistemic uncertainty.
Fig. 2.3 Loss PPF for different models: each column corresponds to a different model. The PPF (bottom row) is the inverse of the CDF (top row). For calculations we interpolate \(2^{10}=1024\) points (cyan line) to obtain a smooth function; for illustration purposes here only 30 points are shown. The data for the first two columns were generated with the neuron model described at the top of our Results, where the additive noise follows either a Gaussian or Cauchy distribution. The black body radiation data for the third column were generated from a Poisson distribution using Eq. 2.34 with \(s = 2^{14}\) and \(λ\) in the range \(6\ \mathrm{µm}\text{ to }20\ \mathrm{µm}\). Here the true noise is binomial, but the loss assumes a Gaussian. The fourth column shows an example where the data are high-dimensional; the same 30 dimensional, unit variance, isotropic Gaussian is used for both generating the data and evaluating the loss. In all panels the loss function used is the log likelihood under the model. [source]#
Up to this point with Eq. 2.17 we have simply rewritten the usual definition of the risk. Recall now that in the previous section, we proposed to equate replication uncertainty with misspecification; specifically we are interested in how differences between the candidate model \(\M_A\) and the true data-generating process \(\Mtrue\) affect the loss PPF, since this determines the risk. Therefore we also compute the PPF of \(Q(x,y; \M_A)\) under its own model (recall from Eq. 1.1 that \(\M_A\) must be a probabilistic model):
from which we obtain the PPF:
The only difference between \(\qt_A\) and \(\qs_A\) is the use of \(p(x,y \mid \M_A)\) instead of \(p(x,y \mid \Mtrue )\) in the integral. In practice this integral would also be evaluated by sampling, using \(\M_A\) to generate a dataset \(\D_{\synth,A}\) with \(L_{\synth,A}\) samples. Because in this case the candidate model is used for both generating samples and defining the loss, we call \(\qt_A\) (\(\Phit_A\)) the synthetic PPF (CDF).
The idea is that the closer \(\M_A\) is to \(\Mtrue\), the closer also the synthetic PPF should be to the mixed PPF — indeed, equality of the PPFs (\(\qt_a = \qs_A\)) is a necessary condition for equality of the models (\(\M_A = \Mtrue\)). Therefore we can quantify the uncertainty due to misspecification, at least insofar as it affects the risk, as the absolute difference between \(\qt_A\) and \(\qs_A\):
We refer to \(δ^{\EMD}_A\) as the empirical model discrepancy (EMD) function because it measures the discrepancy between two empirical PPFs.
It is worth noting that even a highly stochastic data-generating process \(\Mtrue\), with a lot of aleatoric uncertainty, still has a well defined PPF — which would be matched by an equally stochastic candidate model \(\M_A\). Therefore the discrepancies measured by \(δ^{\EMD}_A\) are a representation of the epistemic uncertainty. These discrepancies can arise either from a mismatch between \(\M_A\) and \(\Mtrue\), or simply having too few samples to estimate the mixed PPF \(\qs_A\) exactly; either mechanism contributes to the uncertainty on the expected risk of replicate datasets. We illustrate some differences between aleatoric and epistemic uncertainty in Fig. 7.1, and include further comments on how different types of uncertainties relate to risk in the Supplementary Discussion.