Model-assisted interval designs such as the Keyboard design are transparent and easy to implement in phase I oncology trials. However, interim decisions based solely on data from the current dose may overlook informative signals from neighbouring doses, leading to unnecessary escalation or de-escalation. We propose the shared Keyboard design, a Bayesian model-assisted design that replaces the independent beta--binomial updating scheme at each dose with a posterior induced by a Beta kernel process using kernel-weighted pseudo-counts. The design preserves the decision structure of the Keyboard design while enabling controlled borrowing across nearby doses. To prioritise overdose control, we propose an asymmetric kernel that assigns greater weight to toxicities observed at higher doses during escalation. We further extend the proposed design to accommodate adaptive dose insertion when the initial dose grid is inadequate and time-to-event outcomes when late-onset toxicities are present. Extensive simulation studies demonstrate substantial improvements in both accuracy and safety for identifying the maximum tolerated dose. In settings involving dose insertion, the proposed design identifies inserted target doses more effectively than adaptive dose modification while maintaining a comparable modification rate.

In this part, we state the orthogonal decomposition Property, motivate its importance with a pedagogical example, and finally prove Proposition 1, which enables the decomposition property in the context of HSIC attribution method. A.1 Orthogonal Decomposition Property Let x = {x1,..., xn}2Xn be a set of n univariate random input variables. For any subset A = {l1,...,l |A|} { 1,...,n}, we denote xA =( xl1,..., xl|A|) the vector of input variables with indices in A. Let y the random output variable defined by y = f(x), F the RKHS defined by the kernel kA: X|A|! R and G the RKHS defined by the kernel l: Y! R. In [11], the author shows that for any choice of kernel l, if we respect some constraints on the kernel kA, we can construct indices HSIC (xA,y) that satisfy the following decomposition property. The constraints on the kernel kA are detailed in the main document and in the last section of this appendix.

In this document we provide supplementary materials that we are not able to fit into the main manuscriptduetothepagelimit. The dimensions of the hidden features of the three-layer GCN are set toF, F/2, and F respectively. The dataset is separated into ten parts. We generate ten validation accuracy curves when setting each of parts as the validation one. The ten curves are then averaged.

Weevaluate ourmodel onbothsynthetic andin-the-wild Internet imagery and demonstrate its advantages in terms of generalization and accuracy.

Their primary objective is to efficiently support the insertion of new elements with assigned priorities and the extraction of the highest priorityelement.

Understanding how these systems arrive at their decisions is a necessary first step before these biases can be corrected.

Earlier schemes could only handle stochastic substitutions anddeletions, andthus weareaiming foramore natural and appealing robustness guarantee that holds with respect to editdistance.