If MFCC vectors are speaker independent, then how can we reconstruct unique speech signals from them? Reconstructing a speech signal from a collection of MFCC vectors seems to work pretty well, but I've heard that one advantage of MFCCs is speaker-independence i.e. they are more-or-less the same across different speakers for a given phoneme. How then can a speech signal, with all its speaker-dependent idiosyncrasies (accent, etc.) be reconstructed from a supposedly speaker-independent MFCC vector? Are MFCCs not in fact speaker-independent then? If not, what does determine speaker-independence vs. speaker dependence?

Thanks.
First of all there is some serious "cheating" in the MFCC reconstruction experiment you linked to: not only the MFCCs are used, but also the voiced/unvoiced bit and the pitch.

MFCC are not speaker independent. In fact, they are used for speaker identification/verification tasks!

The speaker "idiosyncracies" are both in their prosody (preserved by this reconstruction experiment because the pitch is provided as a side-information to the reconstruction process) and in the articulation/timbre (preserved by the MFCC).

Two ingredients are needed to get MFCCs to work for speaker-independent recognition:


Vocal tract length normalization. A linear transformation (matrix multiplication of the MFCC vector) can map relatively well the MFCC sequences of two speakers speaking the same sentence. So even if MFCC are not speaker-independent one can optimize for a transformation matrix that "flattens out" the speaker-specific details.
Acoustic modelling. Using a large number of gaussians (or any classifier with large capacity) for a specific acoustic unit allows it to capture all the variations.