\(\renewcommand{\var}{\operatorname{Var}} \renewcommand{\corr}{\operatorname{Corr}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\bb}[1]{\mathbb{#1}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\mm}[1]{\mathrm{#1}} \renewcommand{\dist}[1]{\mathcal{#1}} \renewcommand{\rv}[1]{\mathsf{#1}} \renewcommand{\vrv}[1]{\vv{\rv{#1}}} \renewcommand{\disteq}{\stackrel{d}{=}} \renewcommand{\gvn}{\mid} \renewcommand{\Ex}{\mathbb{E}} \renewcommand{\Pr}{\mathbb{P}}\)

Approximating probability distributions by a Gaussian with the same mode. Thanks to limit theorems this is not always a terrible idea, especially since Neural networks seem pretty keen to converge to Gaussians in various senses.

Specifically, we (possibly locally) approximate some posterior density of interest using a Gaussian \[ p(\theta \mid \mathcal{D}) \approx \mathcal{N}\left(\theta_{\mathrm{MAP}}, \Sigma\right). \] The Laplace trick in general uses a local curvature estimate to choose the covariance.

I am particularly keen to see it work for probabilistic neural nets, where it is a classic methods (Mackay 1992). There are many variants of the technique for different assumptions. We often see this applied to estimating the posterior predictive density, which is usually compact enough to be tractable, but also we can apply it to input uncertainty and even parameters under certain simplifications (Foong et al. 2019; Immer, Korzepa, and Bauer 2021).

## Basic

In classic Laplace approximation, we assume that the parameters of our model have a Gaussian distribution (independently why not) both *a prior* and *a posteriori*.
Specifically we then attempt to approximate these using local curvature of the likelihood function at some maximum a posteriori estimate.

⚠️ The next bit needs a do-over. I completely lost track of the prior and the hyperparameters.

Writing that in symbols, the basic idea is that we hold \(x \in \mathbb{R}^{n}\) fixed and use the Jacobian matrix \(J(x):=\left.\nabla_{\theta} f(x ; \theta)\right|_{\theta_{\mathrm{MAP}}} \in \mathbb{R}^{d \times n}\), to the network as \[ f(x ; \theta) \approx f\left(x ; \theta_{\mathrm{MAP}}\right)+J(x)^{\top}\left(\theta-\theta_{\mathrm{MAP}}\right) \] where the variance is now justified using a Taylor expansion and some smoothness assumptions on \(f\). Under this approximation, since \(\theta\) is by assumption Gaussian, \(\theta \sim \mathcal{N}\left(\theta_{\mathrm{MAP}}, \Sigma\right)\), it follows that the marginal distribution over the network output \(f(x)\) is also Gaussian, given by \[ f(x) \mid x, \mathcal{D} \sim \mathcal{N}\left(f\left(x ; \theta_{\mathrm{MAP}}\right), J(x)^{\top} \Sigma J(x)\right). \] For more on this, see e.g. (Bishop 2006, 5.167, 5.188). It can be essentially a gratis Laplace approximation in the sense that if I have fit the networks I can already calculate those Jacobians so I am probably 1 line of code away from getting some kind of uncertainty estimate. However, I have no particular guarantees that it is well calibrated — is it meaningfully estimating the “true” uncertainty in the model subject to all the uncertainties with this simplified model structure?

## Last Layer Laplace

a.k.a. Neural Linear models.

AFAICT this case is the simplest. We are concerned with the density over the predictive, so we start with a neural network. Then we treat the neural network as a feature generator in all the layers up to the last one, and treat the last layer probabilistically, as an adaptive-basis regression or classification problem, to get a decent learnable predictive uncertainty. I think this was implicit in Mackay (1992), but it was named in Snoek et al. (2015), critiqued and extended in Lorsung (2021).

For a simple practical example, see the Probflow tutorial.

Under a last-layer Laplace approximation, we write the joint model as \(\vrv{y}= \vrv{r}^{\top}\Phi(\vrv{u})\) so the joint distribution is \[\begin{align*} \left.\left[\begin{array}{c} \vrv{y} \\ \vrv{r} \end{array}\right]\right|\vrv{u} &\sim\dist{N}\left( \left[\begin{array}{c} \vv{m}_{\vrv{y}}\\ \vv{m}_{\vrv{r}} \end{array}\right], \left[\begin{array}{cc} \mm{K}_{\vrv{y}\vrv{y}} & \mm{K}_{\vrv{y}\vrv{r}}^{\top} \\ \mm{K}_{\vrv{y}\vrv{r}} & \mm{K}_{\vrv{r}\vrv{r}} \end{array}\right] \right) \end{align*}\] with \[\begin{align*} \vv{m}_{\vrv{y}} &=\vv{m}_{\vrv{r}}^{\top}\Phi(\vrv{u}) \\ \mm{K}_{\vrv{y}\vrv{r}} &=\Phi(\vrv{u}) \mm{K}_{\vrv{r}\vrv{r}}\\ \mm{K}_{\vrv{y}\vrv{y}} &= \Phi(\vrv{u})\mm{K}_{\vrv{r}\vrv{r}} \Phi^{\top} (\vrv{u})+ \sigma^2\mm{I}. \end{align*}\] Here \(\vrv{r}\sim \dist{N}\left(\vv{m}_{\vrv{r}}, \mm{K}_{\vrv{r}\vrv{r}}\right)\) is the random weighting, and \(\Phi(\vrv{u})\) is called the feature map.

## Learnable Laplace approximations

**NB** ⚠️⛔️☣️: This sections is *vague half-arsed notes* of *dubious accuracy*.

Agustinus Kristiadi and team have created various methods for low-overhead neural uncertainty quantification via Laplace approximation that have greater flexibility for adaptively choosing the type and manner of approximation. See, e.g. Painless Uncertainty for Deep Learning and their papers (Kristiadi, Hein, and Hennig 2020, 2021).

Kristiadi, Hein, and Hennig (2021) generalises to learnable uncertainty to, for example, allow the distribution to reflect uncertainty about datapoints drawn far from the training distribution.
They define an augmented *Learnable Uncertainty Laplace Approximation* (LULA) network \(\tilde{f}\) with more parameters \(\tilde{\theta}=\theta_{\mathrm{MAP}}, \hat{\theta}.\)

Let \(f: \mathbb{R}^{n} \times \mathbb{R}^{d} \rightarrow \mathbb{R}^{k}\) be an \(L\)-layer neural network with a MAP-trained parameters \(\theta_{\text {MAP }}\) and let \(\widetilde{f}: \mathbb{R}^{n} \times \mathbb{R}^{\widetilde{d}} \rightarrow \mathbb{R}^{k}\) along with \(\widetilde{\theta}_{\text {MAP }}\) be obtained by adding LULA units. Let \(q(\widetilde{\theta}):=\mathcal{N}\left(\tilde{\theta}_{\mathrm{MAP}}, \widetilde{\Sigma}\right)\) be the Laplace-approximated posterior and \(p\left(y \mid x, \mathcal{D} ; \widetilde{\theta}_{\mathrm{MAP}}\right)\) be the (approximate) predictive distribution under the LA. Furthermore, let us denote the dataset sampled i.i.d. from the data distribution as \(\mathcal{D}_{\text {in }}\) and that from some outlier distribution as \(\mathcal{D}_{\text {out }}\), and let \(H\) be the entropy functional. We construct the following loss function to induce high uncertainty on outliers while maintaining high confidence over the data (inliers): \[ \begin{array}{rl} \mathcal{L}_{\text {LULA }}\left(\widetilde{\theta}_{\text {MAP }}\right)&:=\frac{1}{\left|\mathcal{D}_{\text {in }}\right|} \sum_{x_{\text {in }} \in \mathcal{D}_{\text {in }}} H\left[p\left(y \mid x_{\text {in }}, \mathcal{D} ; \widetilde{\theta}_{\text {MAP }}\right)\right] \\ &-\frac{1}{\left|\mathcal{D}_{\text {out }}\right|} \sum_{x_{\text {out }} \in \mathcal{D}_{\text {out }}} H\left[p\left(y \mid x_{\text {out }}, \mathcal{D} ; \widetilde{\theta}_{\text {MAP }}\right)\right] \end{array} \] and minimize it w.r.t. the free parameters \(\widehat{\theta}\).

I am assuming that by the *entropy functional* they mean the entropy of the normal distribution,
\[
H(\mathcal{N}(\mu, \sigma)) = {\frac {1}{2}}\ln \left((2\pi \mathrm {e} )^{k}\det \left({\boldsymbol {\Sigma }}\right)\right)
\]
but this looks expensive due to that determinant calculation in a (large) \(d\times d\) matrix.
Or possibly they mean some general entropy with respect to some density \(p\),
\[H(p)=\mathbb{E}_{p}\left[-\log p( x)\right]\]
which I suppose one could estimate as
\[H(p)=\frac{1}{N}\sum_{i=1}^N \left[-\log p(x_i)\right]\] without taking that normal Laplace approximation at this step, if we could find the density, and assuming the \(x_i\) were drawn from it.

The result is a slightly weird hybrid fitting procedure that requires two loss functions and which feels a little *ad hoc*, but maybe it works?

## By stochastic weight averaging

A Quasi-Bayesian extension of a gradient descent trick called Stochastic Weight Averaging (Izmailov et al. 2018, 2020; Maddox et al. 2019; Wilson and Izmailov 2020). AFAICT it precludes using a prior? Any may not actually be a Laplace approximation per se?

## Variational

This is a different objective, no longer centred at a MAP estimate. See Variational Gaussian Approximation.

## For model selection

**NB** ⚠️⛔️☣️: This sections is *vague half-arsed notes* of *dubious accuracy*.

We can estimate marginal likelihood with respect to hyperparameters by Laplace approximation, apparently. See Immer et al. (2021).

## In function spaces

Where the Laplace approximations are Gaussian Processes over some index space. TBC. See Piterbarg and Fatalov (1995); Wacker (2017); Alexanderian et al. (2016); Alexanderian (2021) and possibly Solin and Särkkä (2020) and possibly the INLA stuff.

## INLA

Integrated nested Laplace approximation leverages the GP-as-SDE idea to generalise Matérn-type covariances to interesting domains and non-stationarity.

## Second order gradient matrices

## Other covariance factorisations

I think the *Kronecker-factored Approximate Curvature* (K-FAC) is a famous one.
There are others?
(Flaxman et al. 2015; Martens and Grosse 2015; Ritter, Botev, and Barber 2018)

## Tools

A toolkit combining all the NN Laplace tricks is introduced in Agustinus Kristiadi’s Modern Arts of Laplace Approximations is an excellent start.

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