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We develop a novel approach in exploring the joint dependence of halo bias onmultiple halo properties using Gaussian process regression. Using a$\Lambda$CDM $N$-body simulation, we carry out a comprehensive study of thejoint bias dependence on halo structure, formation history and environment. Weshow that the bias is a multivariate function of halo properties that fallsinto three regimes. For massive haloes, halo mass explains the majority of biasvariation. For early-forming haloes, bias depends sensitively on the recentmass accretion history. For low-mass and late-forming haloes, bias depends moreon the structure of a halo such as its shape and spin. Our framework enables usto convincingly prove that $V_\mathrm{max}/V_\mathrm{vir}$ is a lossy proxy offormation time for bias modelling, whereas the mass, spin, shape and formationtime variables are non-redundant with respect to each other. Combining mass andformation time largely accounts for the mass accretion history dependence ofbias. Combining all the internal halo properties fully accounts for the densityprofile dependence inside haloes, and predicts the clustering variation ofindividual haloes to a $20\%$ level at $\sim 10\mathrm{Mpc}h^{-1}$. When anenvironmental density is measured outside $1\mathrm{Mpc}h^{-1}$ from the halocentre, it outperforms and largely accounts for the bias dependence on theinternal halo structure, explaining the bias variation above a level of $30\%$.

The multidimensional dependence of halo bias in the eye of a machine: a tale of halo structure, assembly and environment