Testing Dependent Censoring via Mutual-Information Uncertainty and Deep Latent Imputation

Learning a survival prediction model can be viewed as regression with the added complication of censoring. Each subject $x_i$ has a true event time $E_i$ and a censoring time $C_i$, yet we only observe $$\begin{gathered} T_i= \\ min(E_i,C_i) \end{gathered}$$ and $$\begin{gathered} delt{a}_i= \\ mathbf1(E_i\le C_i) \end{gathered}$$. Many standard methods assume conditional independence $$\begin{gathered} E \\ perpC \\ mid \\ mathbfX \end{gathered}$$, which may not hold in practice. Assessing this dependence is challenging, as for each subject we observe either $E_i$ or $C_i$, but never both. To address this, for each $t>0$ we define indicator variables ${\tilde{E}}_i(t)=1(E_i>t)$ and ${\tilde{C}}_i(t)=1(C_i>t)$, taking values in $0,1,?$, where "?" denotes censoring-induced missingness. Using $$\begin{gathered} {{\tilde{E}}_i(t),{\tilde{C}}_i(t) \\ }_t \end{gathered}$$ across multiple time points, we develop a model-assisted diagnostic for testing $E\perp C\mid \mathbf{X}$. Our statistic is the Conditional Mutual Information (CMI) uncertainty range, ΔIₜ = Iᵗₘₐₓ − Iᵗₘᵢₙ, defined as the width of the CMI values attainable over all feasible completions of the missing "?" entries. Here, Iᵗₘₐₓ and Iᵗₘᵢₙ are the corresponding maximum and minimum feasible CMI values. Under independence, ΔIₜ follows a characteristic null distribution, whereas dependent censoring induces structured deviations. To compute these bounds efficiently, we formulate the problem as a decomposable integer program and solve it exactly via a polynomial-time algorithm. To calibrate the statistic, we combine this uncertainty measure with a deep variational frailty model that imputes latent event and censoring marginals, followed by stratified permutation to generate a model-assisted null distribution under conditional independence. Synthetic experiments show calibrated behavior under independent censoring and increasing power for several dependent-censoring mechanisms, while real-data experiments reveal heterogeneous evidence of possible dependent censoring.