Detecting bifurcations in noisy and/or high-dimensional physical systems is an important problem in nonlinear dynamics. Near bifurcations, the dynamics of even a high dimensional system is typically dominated by its behavior on a low dimensional manifold. Since the system is sensitive to perturbations near bifurcations, they can be detected by looking at the apparent deterministic structure generated by the interaction between the noise and low-dimensional dynamics. We use minimal hidden Markov models built from the noisy time series to quantify this deterministic structure at the period-doubling bifurcations in the two-well forced Duffing oscillator perturbed by noise. The apparent randomness in the system is characterized using the entropy rate of the discrete stochastic process generated by partitioning time series data. We show that as the bifurcation parameter is varied, sharp changes in the statistical complexity and the entropy rate can be used to locate incipient bifurcations.