10. Extension: Zero and One Inflation

Score distributions in end-to-end self-driving carry a sharp spike at zero, the footprint of routes where the driving policy failed catastrophically, and a smaller spike at one for clean passes. Continuous flavors of IRT cannot put finite mass on a single point, so the fix is to peel each spike off into its own branch and let a vanilla IRT describe the interior. This section covers the GRM-style zero-and-one-inflated mixture, after Molenaar (2022).

10.1. Why a Plain Beta Cannot Absorb Boundary Spikes

NavHard scores live in \([0, 1]\) and are not smoothly distributed across that range. About one route attempt in ten ends with a hard zero (collision, off-road, scene incompletion), and a small share lands at exactly one.

Recall the Beta-IRT shape from the previous chapter: each cell is \(\mathrm{Beta}(a_{s,n}, b_{s,n})\) on \((0, 1)\) with

\[ a_{s,n} = \exp\!\bigl(\tfrac{1}{2}(\eta_{s,n} + o_n)\bigr), \qquad b_{s,n} = \exp\!\bigl(\tfrac{1}{2}(-\eta_{s,n} + o_n)\bigr), \qquad \eta_{s,n} = \alpha_n \theta_s - \beta_n, \]

where \(\theta_s\) is the subject’s ability, \(\beta_n\) is the route’s difficulty, \(\alpha_n\) is the route’s discrimination, and \(o_n\) is the route’s log-precision (see Beta-IRT in the previous chapter). With both shapes at least one, the density vanishes at the endpoints; push a shape below one and it blows up there as a continuous U-shape, never a point mass.

So a Beta cannot place finite mass on \(\{0\}\) or \(\{1\}\). The only continuous knob for chasing a boundary spike is to shrink the precision \(o_n\), which drives the per-cell Beta into a U-shape and ruins the interior fit. The fix is to peel each spike off into its own branch, leaving the continuous part free to shape the interior [3, 4].

10.2. Mixture Likelihood

We use the GRM-style inflation from Molenaar [1]. Two per-route thresholds, \(g_n^0 < g_n^1\), split the ability axis into three regions, and the inflation rates depend on the subject’s standing \(\alpha_n \theta_s\) on that axis through the same discrimination \(\alpha_n\) as the continuous branch:

\[ \Pr(X_{s,n} = 0 \mid \theta_s) = \sigma\!\bigl(g_n^0 - \alpha_n \theta_s\bigr), \qquad \Pr(X_{s,n} = 1 \mid \theta_s) = \sigma\!\bigl(\alpha_n \theta_s - g_n^1\bigr). \]

Weak subjects (\(\alpha_n \theta_s \ll g_n^0\)) pile up at zero; strong subjects (\(\alpha_n \theta_s \gg g_n^1\)) pile up at one; subjects in between split their mass into the interior Beta. We enforce \(g_n^1 > g_n^0\) through a softplus link, \(g_n^1 = g_n^0 + \mathrm{softplus}(g_n^{1,\mathrm{raw}})\), instead of a truncated prior, which keeps \(g_n^0\) and \(g_n^{1,\mathrm{raw}}\) unconstrained for the SVI guide.

The likelihood for a single cell is the three-branch mixture

\[\begin{split} \begin{aligned} \Pr(X_{s,n} = 0 \mid \theta_s) &= \sigma(g_n^0 - \alpha_n \theta_s), \\ \Pr(X_{s,n} = 1 \mid \theta_s) &= \sigma(\alpha_n \theta_s - g_n^1), \\ \Pr(0 < X_{s,n} < 1 \mid \theta_s) \cdot p_{s,n}(x) &= \bigl(1 - \sigma(g_n^0 - \alpha_n \theta_s) - \sigma(\alpha_n \theta_s - g_n^1)\bigr)\, \mathrm{Beta}(x;\, a_{s,n}, b_{s,n}), \end{aligned} \end{split}\]

with the three weights summing to one for every \(\theta_s\) because \(g_n^1 > g_n^0\). The continuous mean is the Molenaar Beta mean \(\sigma(\eta_{s,n})\), so weighting each branch by its probability gives the expected score:

\[ \mathbb{E}[X_{s,n}] = \sigma(\alpha_n \theta_s - g_n^1) + \bigl(1 - \sigma(g_n^0 - \alpha_n \theta_s) - \sigma(\alpha_n \theta_s - g_n^1)\bigr)\, \sigma(\eta_{s,n}). \]

The interior branch ranks subjects through \(\theta_s\) exactly as in plain Beta-IRT. The mixture only pulls the boundary mass off; the inflation rates also depend on \(\theta_s\), but through the same \(\alpha_n \theta_s\) that drives the interior, so they sharpen rather than override the ranking signal [2].

10.3. References

[1]

D. Molenaar, M. Cúri, and J. L. Bazán. Zero and one inflated item response theory models for bounded continuous data. Journal of Educational and Behavioral Statistics, 47(6):693–735, 2022.

[2]

C. Zopluoglu and J. R. Lockwood. A comparative study of item response theory models for mixed discrete-continuous responses. Journal of Intelligence, 12(3):26, 2024.

[3]

D. Lambert. Zero-inflated Poisson regression, with an application to defects in manufacturing. Technometrics, 34(1):1–14, 1992.

[4]

R. Ospina and S. L. P. Ferrari. Inflated beta distributions. Statistical Papers, 51(1):111–126, 2010.