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Recent attempts to address the Higgs mass naturalness problem have led to the development of approaches based on a dynamical Higgs mass selection during the cosmological evolution of the universe. Instead of providing a screening mechanism for the quantum corrections to the Higgs mass, the idea here is to allow the latter to take a large range of values in different patches of the Universe or at different times of its cosmological evolution and supplement this with some sort of dynamical selection mechanism which singles out the vacuum with the observed small Higgs mass. In this talk I will present a scenario capable of combining the dynamical Higgs mass selection with the anthropic selection of the cosmological constant. The proposed selection mechanism is based on vacuum statistics during inflation, which is defined in general by the interplay of tunnelling rates, Hubble jumps, classical rolling, and volume expansion, that eventually maximize the probability to observe the vacuum with the needed hierarchically small Higgs mass. Computation of probabilities during inflation is famously problematic because of the measure problem, and I will show that the two most polar probability measures are capable of producing the desired statistical distribution of vacua. Although having much less spectacular experimental signatures compared to the traditional scenarios for the Higgs mass naturalness, the dynamical solutions to the Higgs mass problem are generically expected to be testable in the current or near future experiments, which I will briefly discuss.