Time-Frequency Analysis as Probabilistic Inference

This paper proposes a new view of time-frequency analysis framed in terms of probabilistic inference. Natural signals are assumed to be formed by the superposition of distinct time-frequency components, with the analytic goal being to infer these components by application of Bayes' rule. The framework serves to unify various existing models for natural time-series; it relates to both the Wiener and Kalman filters, and with suitable assumptions yields inferential interpretations of the short-time Fourier transform, spectrogram, filter bank, and wavelet representations. Value is gained by placing time-frequency analysis on the same probabilistic basis as is often employed in applications such as denoising, source separation, or recognition. Uncertainty in the time-frequency representation can be propagated correctly to application-specific stages, improving the handing of noise and missing data. Probabilistic learning allows modules to be co-adapted; thus, the time-frequency representation can be adapted to both the demands of the application and the time-varying statistics of the signal at hand. Similarly, the application module can be adapted to fine properties of the signal propagated by the initial time-frequency processing. We demonstrate these benefits by combining probabilistic time-frequency representations with non-negative matrix factorization, finding benefits in audio denoising and inpainting tasks, albeit with higher computational cost than incurred by the standard approach.