High-fidelity simulations of turbulent reacting flows enable scientific understanding of the physics and engineering design of practical systems. Whereas direct numerical simulation (DNS) is the most suitable numerical tool to understand the physics, under-resolved and large-eddy simulations offer a good compromise between accuracy and computational effort in the prediction of engineering flows. This compromise speeds up the computations but reduces the space-and-time accuracy of the prediction. The objective of this chapter is to (i) evaluate the predictability horizon of turbulent simulations with chaos theory, and (ii) enable the space-and-time accurate prediction of rare and transient events using a Bayesian statistical learning approach based on data assimilation. The methods are applied to DNS of Moderate or Intense Low-oxygen Dilution (MILD) combustion. The predictability provides an estimate of the time horizon within which the occurrence of ignition kernels and deflagrative modes, which are considered here as rare and transient events, can be accurately predicted. The accurate detection of ignition kernels and their evolution towards deflagrative structures are well-captured on a coarse (under-resolved) grid when data is assimilated from a costly refined DNS. Physically, such an accurate prediction is important to understand the stabilization mechanism of MILD combustion. These techniques enable the space-and-time-accurate prediction of rare and transient events in turbulent flows by combining under-resolved simulations and experimental data, for example, from engine sensors. This opens up new possibilities for on-the-fly calibration of reduced-order models for turbulent reacting flows.