Modern, low emission combustion systems with improved fuel-air mixing are more prone to combustion instabilities and therefore use advanced control methods to balance minimum NOx emissions and and the presence of thermoacoustic combustion instabilities. The exact operating conditions at which the system becomes encounters an instability is uncertain because of sources of stochasticity, such as turbulent combustion, and the influence of hidden variables, such as un-measured wall temperatures or differences in machine geometry within manufacturing tolerances. Practical systems tend to be more elaborate than laboratory systems and tend to have less instrumentation, meaning that they suffer more from uncertainty induced by hidden variables. In many commercial systems, the only direct measurement of the combustor comes from a dynamic pressure sensor. In this study we train a Bayesain Neural Network (BNN) to predict the probability of onset of thermoacoustic instability at various times in the future, using only dynamic pressure measurements and the current operating condition. We show that, on a practical system, the error in the onset time predicted by the BNNs is 45% lower than the error when using the operating condition alone and more informative than the warning provided by commonly used precursor detection methods. This is demonstrated on two systems: (i) a premixed hydrogen/methane annular combustor, where the hidden variables are wall temperatures that depend on the rate of change of operating condition, and (ii) full scale prototype combustion system, where the hidden variables arise from differences between the systems.