Equal-width bin entropy ($H_e(Q_n)-\ln n⟶h_e(X)$)

Partition $(0,1)$ into $n$ bins of width $\delta =1/n$, and let the discrete random variable $Q_n$ record the bin containing $X$. This is a quantization of $X$. Its probability mass function isand its discrete information entropy is $H_e(Q_n)=-{\sum }_j{p}_j\ln p_j$. Let ${\bar{f}}_j=p_j/\delta$ be the average probability density within bin $j$. Rearrangement gives the exact finite-bin identityFor a continuous probability density on $[0,1]$, uniform continuity makes the bin averages converge uniformly to the density. Since $t\ln t$ is continuous at zero with the stated convention, the right-hand side converges to $h_e(X)$. Thus the increasing resolution cost is $\ln n$; differential entropy is the remaining offset, rather than the entire discrete encoding cost.

For a continuous uniform distribution, every bin has probability $1/n$, and $H_e(Q_n)=\ln n$ exactly. For the continuous uniform distribution's squared transformation above, the exact bin probabilities areThe diagram computes these masses directly, including the bin next to the integrable singularity. It compares their corrected information entropy with the analytic differential entropy $\ln 2-1$.