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I am reading Elements of Information Theory (2nd Ed.) by Cover. I have something that I couldn't figure out.

On page 248, theorem 8.3.1

If the density f(x) of the random variable X is Riemann integrable, then $H(X^Δ)+log(⁡Δ)→h(f)=h(X)$, as $Δ→0$.
Thus, the entropy of an n-bit quantization of a continuous random variable X is approximately h(X) + n.

The continuous random variable X is divided into bins of length $\Delta$.
$X^Δ=x_i\quad if\quad i\Delta\le X<(i+1)\Delta$. Where $\Delta f(x_i)=\int_{i\Delta}^{(i+1)\Delta}f(x)dx$.
The book gives a proof of the theorem, but what I didn't understand is how the last conclusion is reached. Clearly, if the range of X is [0, 1), the length of the bin $\Delta$ is related to n. $-log_2(\Delta)=n$ if the range is [0, 1). But this condition is not required in the theorem. Of course, if the range is finite, when $\Delta$ approaches 0, $-log_2(\Delta) \rightarrow n$. Also, the equation only holds when $\Delta$ is close to 0, which is also not stated in the conclusion. ("Thus, the entropy of an n-bit quantization of a continuous random variable X is approximately h(X) + n.")

Did I miss something, or is there some other proofs of that statement?

Thanks a lot.

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In the book, by a "$n-$bit quantization" they mean how many binary (fractional) digits of precision we have.

When we speak in decimal base, we say we have "5 decimal digits" of precision (or that 5 decimal fractional digits are meaningful) when we quantize the real line in intervals of length $1/10000=1/10^5$. (Granted, the word precision is also used with different meanings; often it means the amount of significant digits). Similarly, to have 5 bits of precision (or a $5-$bit quantization) we must quantize the real range in intervals of length $\Delta=1/2^5$. Hence, indeed, $-\log_2 \Delta = n$.

See also the second item in example 8.3.1.

I'm not sure if this answers your doubt.

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  • $\begingroup$ Thank you. I had thought it was the total number of digits or bits required to describe the value, something like significant digits. $\endgroup$ – LLS Apr 12 at 1:20

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