## 1 Introduction  
We provide new expressions relating the log likelihood ratio from signal processing, the minimum mean squared prediction error from time series analysis, and the entropy power from information theory. We then show how these new expressions for the log likelihood ratio invite new analyses and insights into problems in digital signal processing. To demonstrate their utility, we present applications to speech coding and show how the entropy power can explain results that previously escaped interpretation.  
The linear prediction model plays a major role in many digital signal processing applications, but none perhaps more important than linear prediction modeling of speech signals. [PERSON] introduced the log likelihood ratio as a distance measure for speech recognition [1], and it was quickly applied to evaluating the spectral match in other speech-processing problems [2; 3; 4; 5]. In time series analysis, the linear prediction model, called the autoregressive (AR) model, has also received considerable attention, with a host of results on fitting AR models, AR model prediction performance, and decision-making using time series analysis based on AR models. The particular results of interest here from time series analysis are the expression for the mean squared prediction error and the decomposition in terms of nondeterministic and deterministic processes [6; 7; 8].  
The quantity called entropy power was defined by [PERSON] [9], and has primarily been used in bounding channel capacity and rate distortion functions. Within the context of information theory and rate distortion theory, [PERSON] [10] and [PERSON] [11] derived an expression for the entropy power in terms of the power spectral density of a stationary random process. Interestingly, their expression for the entropy power is the same as the expression from time series analysis for the minimum mean squared prediction error. This fact was recognized by [PERSON] and [PERSON] in 1967, but the connection has never been formalized and exploited [12; 13].  
In this paper, we use the expression for the minimum mean squared prediction error to show that the log likelihood ratio equals the logarithm of the ratio of entropy powers, and then develop an expression for the log likelihood ratio in terms of the difference of differential entropies and the difference of two mutual informations. We specifically consider applications of linear prediction to speech coding, including codec performance analysis and speech codec design.  
In Section 2, we define and discuss expressions for the log likelihood ratio, and in Section 3, we define and explain the entropy power and its implications in information theory. Section 4 examines the mean squared prediction error from time series analysis and relates it to entropy power. The quantities of entropy power and mean squared prediction error are then used in Section 5 to develop the new expressions for the log likelihood ratio in terms of differential entropy and mutual information. Experimental results are presented in Section 6 for several speech coding applications, and in Section 7, a detailed discussion of the experimental results is provided, as the difference in differential entropies and mutual informations reveals new insights into codec performance evaluation and codec design. Section 8 provides the experimental lessons learned and suggestions for additional applications.