We present a new statistical technique for the estimation of the high frequency components (4-8 kHz) of speech signals from narrow-band (0-4 kHz) signals. The magnitude spectra of broadband speech are modelled as the outcome of a Polya Urn process, that represents the spectra as the histogram of the outcome of several draws from a mixture multinomial distribution over frequency indices. The multinomial distributions that compose this process are learnt from a corpus of broadband (0-8 kHz) speech. To estimate high-frequency components of narrow-band speech, its spectra are also modelled as the outcome of draws from a mixture-multinomial process that is composed of the learnt multinomials, where the counts of the indices of higher frequencies have been obscured. The obscured high-frequency components are then estimated as the expected number of draws of their indices from the mixture-multinomial. Experiments conducted on bandlimited signals derived from the WSJ corpus show that the proposed procedure is able to accurately estimate the high frequency components of these signals.