Caron 2012 - Identifiability, Hyperparameters and Further Extensions

Summary

BT/PL likelihoods are invariant to rescaling $\lambda$, so only ${\pi }_i={\lambda }_i/{\sum }_k{\lambda }_k$ is identified; with a $\mathcal{G}(a,b)$ prior $\pi \sim \text{Dirichlet}(a,\dots ,a)$. The paper adds a rescaling step to improve Gibbs mixing, shows how to learn $a$, and sketches EM/Gibbs for degree-sequence graphs, feature (Tversky/Restle) choice and a positive-weight classifier.

Overview

This is the “plumbing” section: what is identified, how the prior scale interacts with the sampler, and which other models share the exponential-race trick.

Main Content

5.1 Identifiability

Dirichlet-Gamma decomposition

Let $\Lambda ={\sum }_i{\lambda }_i$ and ${\pi }_i={\lambda }_i/\Lambda$. The likelihood depends only on $\pi$, so $p(\pi ,\Lambda \mid D)=p(\pi \mid D)p(\Lambda )$. Under ${\lambda }_i\stackrel{iid}{\sim }\mathcal{G}(a,b)$: $\pi \sim \text{Dir}(a,\dots ,a)$, $\Lambda \sim \mathcal{G}(Ka,b)$ independently, and the MAP of $\Lambda$ is $(aK-1)/b$.

Practical consequences.

1. Rescaling step. After each Gibbs sweep, normalise ${\lambda }^{(t)}$ and rescale by a fresh draw ${\Lambda }^{(t)}\sim \mathcal{G}(Ka,b)$: ${\lambda }_i^{\ast (t)}={\lambda }_i^{(t)}{\Lambda }^{(t)}/{\sum }_j{\lambda }_j^{(t)}$. This removes a slowly mixing random-walk along the non-identified scale direction. It is useless if only $\pi$ is reported.
2. Fixing the scale. Setting $b=Ka-1$ makes the MAP satisfy ${\sum }_k{\lambda }_k=1$ (used in all experiments). The inverse-scale $b$ is not likelihood-identified so no prior is placed on it.
3. Alternative EM on $\pi$. With $\pi \sim \text{Dir}(a,\dots ,a)$ add latent negative-binomial counts $M_{ij}\sim \text{NB}(n_{ij},{\pi }_i+{\pi }_j)$ and category labels $C_{ijk}$ with $P(C_{ijk}=l)\propto {\pi }_l$ for $l\ne i,j$, giving ${\pi }_k^{(t)}\propto a-1+w_k+{\pi }_k^{(t-1)}{\sum }_{i\ne k}{\sum }_{j\ne i,k}{n}_{ij}/({\pi }_i^{(t-1)}+{\pi }_j^{(t-1)})$, renormalised to sum 1. It is identified but converges slowly when some ${\pi }_k$ are tiny (the sum becomes large) and the matching Gibbs sampler mixes slowly too.

The vault-wide analogue in log-skill (logit) parameterisations: use sum-to-zero or an anchor element; see E67’s “sum-to-zero vs anchoring identification”.

5.2 Hyperparameter estimation

Posterior of the shape $a$

Given $\lambda$: $p(a\mid \lambda )\propto p(a)\underset{l_1(a)}{\underbrace{(b^K\prod _i{\lambda }_i{)}^a}}\underset{l_2(a)}{\underbrace{\Gamma (a{)}^{-K}}}$.

Sample with the slice-type auxiliary variable scheme of Damien et al. (1999): $U_1\sim \mathcal{U}(0,l_1(a))$, $U_2\sim \mathcal{U}(0,l_2(a))$, then $a\mid u\propto p(a)1\{l_1(a)>u_1,l_2(a)>u_2\}$; or a random-walk Metropolis step on $\log a$: propose $a^{\star }=e^{\sigma z}a$, accept with probability $\min \{1,\frac{p(a^{\star })}{p(a)}(\frac{\Gamma (a)}{\Gamma (a^{\star })}{)}^K(b^K\prod {\lambda }_i{)}^{a^{\star }-a}\}$. Large $a$ pulls all skills together (strong shrinkage); small $a$ lets skills spread.

5.3 Further extensions

Degree-sequence random graph

Edge $(i,j)$ present with probability ${\lambda }_i{\lambda }_j/(1+{\lambda }_i{\lambda }_j)$ (Holland-Leinhardt; Chatterjee et al.). Latent $Z_{ij}\sim \text{Exp}({\lambda }_i+1/{\lambda }_j)$; the ${\lambda }_i$ conditional is a generalized inverse Gaussian, and the EM step solves a quadratic.

Feature-based choice (Restle 1961; Tversky 1972)

Items carry binary features $f_i\in \{0,1{\}}^K$; ${\pi }_{ij}=\frac{{\sum }_k{\lambda }_k{f}_{ik}(1-f_{jk})}{{\sum }_k{\lambda }_k{f}_{ik}(1-f_{jk})+{\sum }_k{\lambda }_k{f}_{jk}(1-f_{ik})}$, i.e. only features not shared matter. EM/Gibbs follow the group-comparison construction.

Positive-weight classifier

$P(Y_i=k)=\frac{\exp (X_i{)}^{\top }{\lambda }_k}{{\sum }_l\exp (X_i{)}^{\top }{\lambda }_l}$ with $X_i\in {\mathbb{R}}^p$ and ${\lambda }_k\in {\mathbb{R}}_{+}^p$, an alternative to multinomial logit; latent $Z_i\sim \text{Exp}({\sum }_l\exp (X_i{)}^{\top }{\lambda }_l)$ gives EM and Gibbs.

Examples

Why the rescaling step matters

With a Gamma prior, one Gibbs sweep multiplies every ${\lambda }_i$ by a similar factor driven by $Z$; across sweeps the overall scale $\Lambda$ random-walks. Resampling $\Lambda \sim \mathcal{G}(Ka,b)$ after normalising makes successive $\pi$ draws less correlated. If you only store $\pi$ you can skip it.

Connections

• Dirichlet shrinkage on $\pi$ is the analogue of a hierarchical normal prior on log-skills (Hierarchical Models).
• Tversky choice with features is a pairwise analogue of attribute-based demand in Discrete Choice Models.

See Also

Caron 2012 - Experiments NASCAR and Chess · Caron 2012 - Bayesian Inference for Generalized Bradley-Terry Models - Overview