The stochastic dynamics of multistable perception poses an enduring challenge to our understanding of neural signal processing in the brain. We show that the emergence of perception switching and stability can be understood using principles of probabilistic Bayesian inference where the prior temporal expectations are matched to a scale-free power spectrum, characteristic of fluctuations in the natural environment. The optimal percept dynamics are inferred by an exact mapping of the statistical estimation problem to the motion of a dissipative quantum particle in a multi-well potential. In the bistable case the problem is further mapped to a long-ranged Ising model. Optimal inference in the presence of a 1/f noise prior leads to critical dynamics, exhibiting a dynamical phase transition from unstable perception to stable perception, as demonstrated in recent experiments. The effect of stimulus fluctuations and perception bias is also discussed.