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The strict minimum message length (SMML) principle links data compressionwith inductive inference. The corresponding estimators have many usefulproperties but they can be hard to calculate. We investigate SMML estimatorsfor linear regression models and we show that they have close connections tohyperbolic geometry. When equipped with the Fisher information metric, thelinear regression model with $p$ covariates and a sample size of $n$ becomes aRiemannian manifold, and we show that this is isometric to $(p+1)$-dimensionalhyperbolic space $\mathbb{H}^{p+1}$ equipped with a metric tensor which is $2n$times the usual metric tensor on $\mathbb{H}^{p+1}$. A natural identificationthen allows us to also view the set of sufficient statistics for the linearregression model as a hyperbolic space. We show that the partition of an SMMLestimator corresponds to a tessellation of this hyperbolic space.

SMML estimators for linear regression and tessellations of hyperbolic space