Theory of Gibbs posterior concentration

Notes on some research in progress.

Olivier Binette

Consider the statistical learning framework where we have data \(X\sim Q\) for some unknown distribution \(Q\), a model \(\Theta\) and a loss function \(\ell_\theta(X)\) measuring a cost associated with fitting the data \(X\) using a particular \(\theta\in\Theta\). Our goal is to use the data to learn about parameters which minimize the risk \(R(\theta) = \mathbb{E}[\ell_\theta(X)]\). Here are two standard examples.

Density estimation. Suppose we observe independent random variables \(X_1, X_2, \dots, X_n\). Here the model \(\Theta\) parametrizes a set \(\mathcal{M} = \{p_\theta : \theta \in \Theta \}\) of probability density functions (with respect to some dominating measure on the sample space), and our loss for \(X = (X_1, \dots, X_n)\) is defined as \[ \ell_\theta(X) = - \sum_{i=1}^n \log p_\theta(X_i). \] If, for instance, the variables \(X_i\) are independent with common distribution with density function \(p_{\theta_0}\) for some \(\theta_0 \in \mathbb{\Theta}\), then it follows from the positivity of the Kullback-Leibler divergence that \(\theta_0 \in \arg\min _ \theta \mathbb{E}[\ell _ \theta(X)]\). That is, under identifiability conditions, our learning target is the true data-generating distribution.

If the model is misspecified, roughly meaning that there is no \(\theta_0\in \Theta\) such that \(p_{\theta_0}\) is a density of \(X_i\), then our framework sets up the learning problem to be about the parameter \(\theta_0\) which is such that \(p_{\theta_0}\) mininizes the Kullback-Leibler divergence between \(p_{\theta_0}\) and the true marginal distribution of the \(X_i\)’s.

Regression. Here our observations take the form \((Y_i, X_i)\), the model \(\Theta\) parameterizes regression functions \(f_\theta\) and we can consider a sum of squared errors loss \[ \ell_\theta(X) = \sum_{i=1}^n(Y_i - f_\theta(X_i))^2. \]

Gibbs posterior distributions

Gibbs Learning approaches this problem from a pseudo Bayesian point of view. While typically a Bayesian approach would require the specification of a full data-generating model, here we replace the likelihood function by the pseudo-likelihood function \(\theta \mapsto e^{-\ell_\theta(X)}\). Given a prior \(\pi\) on \(\Theta\), the Gibbs posterior distribution is then given by \[ \pi(\theta \mid X) \propto e^{-\ell_\theta(X)} \pi(\theta) \] and satisfies \[ \pi(\cdot \mid X) \in \text{argmin}_{\hat \pi} \left\{ \mathbb{E}_{\theta \sim \hat \pi}[\ell_\theta(X)] + D_{\text{KL}}(\hat \pi \mid \pi) \right\} \] whenever these expressions are well defined.

In the context of integrable pseudo-likelihoods, the above can be re-interpreted as a regular posterior distributions built from density functions \(f _ \theta(x) \propto e^{-\ell _ \theta(x)}\) and with a prior \(\tilde \pi\) satisfying \[ \frac{d\tilde \pi}{d\pi}(\theta) \propto \int e^{-\ell_\theta(x)}\,dx =: c(\theta). \] However, the reason we cannot apply standard asymptotic theory to the analysis of Gibbs posterior is that the quantity \(c(\theta)\) will typically be sample-size dependent. That is, if \(X=X^n=(X_1, X_2, \dots, X_n)\) for i.i.d. random variables \(X_i\) and if the loss \(\ell_\theta\) separates as the sum \[ \ell_\theta(X) = \sum_{i=1}^nl_{\theta}(X_i), \] then \(c(\theta) = \left(\int e^{-l_\theta(x_1)} \, dx_1\right)^n\). This data-dependent prior, tilting \(\pi\) by the function \(c(\theta)^n\), is what allows Gibbs learning to target general risk-minimizing parameters rather than likelihood Kullback-Leibler minimizers.

Some of my ongoing research, presented as a poster at the O’Bayes conference in Warwick last summer, focused on understand the theoretical behaviour of Gibbs posterior distributions. I studied the posterior convergence and finite sample concentration properties of Gibbs posterior distributions under the large sample regime with additive losses \(\ell_\theta^{(n)}(X_1, \dots, X_n) = \sum_{i=1}^n\ell_\theta(X_i)\). I’ve attached the poster (joint work with Yu Luo) below and you can find the additional references here.

Note that this is very preliminary work. We’re still in the process of exploring interesting directions (and I have very limited time this semester with the beginning of my PhD at Duke).