## 5 Connection to the Log Likelihood Ratio  
To connect the entropy rate power to the log likelihood ratio, we focus on autoregressive processes or the linear prediction model, which is widely used in speech processing and speech coding. We recognize \(BVB^{T}\) as the \(MMSPE_{B}\) when predicting a processed or coded signal with the linear prediction coefficients optimized for that signal, and we also see that \(AVA^{T}\) is the \(MSPE\), not the minimum, when predicting the processed signal with the coefficients optimized for the original unprocessed signal. The interpretation of \(BVB^{T}\) as an entropy rate power is direct, since we know that, for a random variable \(X\) with zero mean and variance \(\sigma^{2}\), the Gaussian distribution lower bounds its differential entropy, so \(h(X)\leq\frac{1}{2}\log(2\pi e\sigma^{2})\), and thus it follows by definition that \(Q\leq\sigma^{2}\). Therefore, \(Q_{B}=BVB^{T}\), since \(\sigma^{2}=MMSPE_{B}\) here. However, \(AVA^{T}\) is not the minimum \(MSPE\) for the process being predicted, because the prediction coefficients were calculated on the unprocessed signal but used to predict the processed signal, so it does not automatically achieve any lower bound. What we do is _define_\(AVA^{T}\) as the \(MMSPE_{A}\) of a new process, so that for this new process, we can obtain a corresponding entropy rate power as \(Q_{A}=AVA^{T}\). This is equivalent to associating the suboptimal prediction error with a whitened nondeterministic component [6; 8]. In effect, the resulting \(MMSPE_{A}\) is the maximum power in the nondeterministic component that can be associated with \(AVA^{T}\). Using the expression for entropy power \(Q\) from Equation (4), substituting for both \(Q_{A}\) and \(Q_{B}\), and taking the logarithm of the two expressions, we have that  
\[d=\log\frac{Q_{A}}{Q_{B}}=2[h(X|P_{A})-h(X|P_{B})], \tag{10}\]  
where \(h(X|P_{A})\) is the differential entropy of the signal generated by passing the processed signal through a linear prediction filter using the linear prediction coefficients computed from the unprocessed signal, and \(h(X|P_{B})\) is the differential entropy of the signal generated by passing the processed signal through a linear filter using the linear coefficients calculated from the processed signal. [PERSON] and [PERSON] [3] have used such linear filter analogies for the log likelihood ratio in terms of a test signal and a reference signal before for spectral matching studies, and other log likelihood ratios can be investigated based on switching the definitions of the test and reference. It is now evident that the log likelihood ratio has an interpretation beyond the usual viewpoint of just involving a ratio of prediction errors or just as a measure of a spectral match. We now see that the log likelihood ratio is interpretable as the difference between two differential entropies, and although we do not know the form or the value of each differential entropy, we do know their difference. We can provide an even more intuitive form of the log likelihood ratio: since we have \(Q_{B}=BVB^{T}\) as the \(MMSPE\) when predicting the processed signal with the linear prediction coefficients optimized for that signal, and we have that \(Q_{A}=AVA^{T}\) is the \(MMSPE\) when predicting the processed signal with the coefficients optimized for the original unprocessed signal, we can add and subtract \(h(X)\), the differential entropy of the processed signal, to Equation (10) to obtain  
\[d=2[I(X;P_{B})-I(X;P_{A})], \tag{11}\]  
where, as before, we have used the notation \(P_{B}\) for the predictor, obtaining \(MMSPE_{B}=BVB^{T}\) and \(P_{A}\) for the predictor, and generating \(MMSPE_{A}=AVA^{T}\).