Differential Entropy: Concepts and Modifications

• Differential entropy is defined for continuous distributions and can be negative due to its sensitivity to scaling and unit changes.
• Modified definitions, such as sign-adjusted and renormalized entropies, resolve issues of non-positivity and incompatibility with discrete approximations.
• These refined measures are crucial in information theory, enabling robust parameter estimation and consistent entropy analysis in practical applications.

Differential entropy is the natural extension of discrete Shannon entropy to continuous probability distributions and plays a fundamental role in information theory, statistics, and statistical physics. However, it presents significant conceptual and practical differences from its discrete counterpart, especially regarding its non-positivity, sensitivity to scaling, and lack of compatibility with discrete approximations. Modern research addresses these limitations through modified definitions, renormalization, and new estimation principles suited to both theoretical analysis and real-world applications.

1. Definition and Classical Properties

Given a probability density function (PDF) $p(x)$ on a continuous space, the standard differential Shannon entropy is defined as

$H_{SH}(p) = -\int p(x) \log p(x)\,dx.$

For Rényi entropy of order $\alpha > 0$, $\alpha \neq 1$, the expression is:

$$H_{R, \alpha}(p) = \frac{1}{1-\alpha} \log \left( \int p(x)^\alpha dx \right).$$

Unlike Shannon entropy for discrete distributions, which is always nonnegative and depends only on probability mass assignments, the differential entropy can take negative values and depends on the units and scaling of the random variable. For example, the entropy of a uniform distribution on $[0, a]$ is $\log a$, which can be negative for $a < 1$, and the entropy of a Gaussian scales logarithmically with its standard deviation $\sigma$, being negative for sufficiently small $\sigma$.

The correspondence between continuous and discrete entropy is also problematic. When a continuous distribution is discretized more finely, the discrete entropy diverges (often like $\log N$ as the number of bins $N \to \infty$), while the differential entropy remains bounded or negative, resulting in an incompatibility in their limiting behaviors (Mishura et al., 6 Aug 2025).

2. Issues with Classical Differential Entropy

The fundamental drawbacks of the classical definitions include:

• Non-positivity: $H_{SH}(p)$ can be negative for well-behaved, non-degenerate distributions.
• Discretization divergence: Discretized approximations to continuous distributions produce entropies diverging as $\log N$ (number of bins), destabilizing the continuum limit.
• Lack of invariance: Differential entropy is not invariant under change of scale or units, in stark contrast to discrete entropy.
• Poor discrete-continuous compatibility: The continuous and discrete expressions rarely agree even as the partition of the space is arbitrarily refined.

These issues are not unique to the Shannon case and similarly affect continuous Rényi entropy. In discrete settings, shifting or scaling the support does not change the entropy, but in the continuous case, entropy changes accordingly (Mishura et al., 6 Aug 2025, Petroni, 2014).

3. Modified and Renormalized Definitions

Alternative definitions aim to resolve non-positivity, incompatibility, and divergence, producing “modified” or “renormalized” continuous entropies. For Shannon entropy, proposed alternatives include:

• Sign-Adjusted Entropy: $_{SH}^{(1)}(p) = \int p(x)\log p(x)\,dx$ (simply $-H_{SH}$, interpreted as positive uncertainty).
• Positive-Truncated Entropy: $$_{SH}^{(2)}(p) = \int p(x) \big(-\log p(x)\big)_+ dx$$ ($(\cdot)_+$ is the positive part), ensuring non-negativity by zeroing out negative contributions.
• Log-Plus-One Entropy: $_{SH}^{(3)}(p) = \int p(x)\log(1/p(x) + 1)dx$, which smooths out the potential negativity even when $p(x)$ is large.
• Normalized Entropy: For a bounded density $p(x) \le M$, $_{SH}^{(4)}(p) = \int [p(x)/M]\log(M/p(x))dx$; this achieves zero only for the uniform distribution, strictly positive otherwise (Mishura et al., 6 Aug 2025).

Analogously, for Rényi entropy (order $\alpha$):

• Absolute Value-Adjusted: $$_{R,\alpha}^{(1)}(p) = \frac{1}{|1-\alpha|}\left|\log \int p(x)^\alpha dx\right|$$.
• Truncated Positive: $$_{R,\alpha}^{(2)}(p) = \frac{1}{|1-\alpha|}\big(\log \int p(x)^\alpha dx\big)_+$$.
• Log-Plus-One Modification: $$_{R,\alpha}^{(3)}(p) = \frac{1}{|1-\alpha|}\log\big(1 + \int p(x)^\alpha dx\big)$$.

Renormalized definitions also arise by introducing a reference scale (such as a dispersion $\kappa$, e.g. standard deviation or interquantile range), ensuring dimensionless arguments and invariance under rescaling:

$$\tilde{h} = -\int f(x) \log[\kappa f(x)] dx = h - \log \kappa,$$

as discussed in (Petroni, 2014). This adjustment cancels the divergence under discretization and renders the entropy invariant to linear scaling of $x$.

4. Compatibility with Discrete Approximations

The alternative (modified/renormalized) entropies are specifically chosen to be compatible with discrete approximations. For example, for the functional

$$_{SH}^{(1)}(p) = \lim_{N\to\infty} \sum_k \Delta F_k^N \log \left(\frac{\Delta F_k^N}{\Delta x_k^N}\right),$$

where $\Delta F_k^N = \int_{x_{k-1}^N}^{x_k^N} p(x) dx$ and $\Delta x_k^N = x_k^N - x_{k-1}^N$, the discretized version smoothly and finitely converges to the continuous entropy, as opposed to the divergent $\log N$ behavior in the standard discrete Shannon entropy. Correspondingly, for Rényi entropy,

$$\sim_{R,\alpha}^N = \frac{1}{1-\alpha}\log \sum_k (\Delta F_k^N)^\alpha (\Delta x_k^N)^{1-\alpha}$$

admits a well-defined limit, in contrast to the incompatibility of standard forms (Mishura et al., 6 Aug 2025).

Such compatibility ensures that these modified functionals smoothly interpolate between discrete and continuous settings. In many cases, the change in base measure or inclusion of scale-normalizing factors is what secures this convergence (Petroni, 2014).

5. Behavior on Standard Distributions

The behavior of both standard and modified entropies is instructive for common statistical distributions:

Distribution	Standard Shannon Entropy	Modified Entropies $(^{(i)})$
Gaussian ($N(0,\sigma^2)$)	$H_{SH} = \frac{1}{2}(1+\log2\pi) + \log\sigma$	$_{SH}^{(2)}$, $_{SH}^{(3)}$, $_{SH}^{(4)}$ are monotone increasing in $\sigma$, strictly positive and diverging as $\sigma\to\infty$; $_{SH}^{(1)}$ is nonmonotonic with a minimum at $\sigma\approx0.318$
Exponential ($\mu$)	$H_{SH} = 1 + \log\mu$	All modified entropies are strictly positive; monotone in $\mu$, with cut-off behavior in some forms

For the Gaussian, standard differential entropy can be negative for small variance, but all $_{SH}^{(i)}$ for $i=2,3,4$ are strictly positive and monotonic in $\sigma$. Similar positivity and monotonicity are observed for the exponential. The modified Rényi entropies are also engineered to remain positive, often featuring a "switching point" where the value is precisely zero, beyond which they display strictly monotonic behavior (Mishura et al., 6 Aug 2025).

6. Implications and Applications in Information Theory

Modified and renormalized formulations of differential entropy are particularly valuable in several respects:

• They enable practical entropy and information measurement in contexts where traditional expressions yield negative or divergent results.
• Their invariance under scaling and compatibility with discrete estimators facilitate robust parameter estimation, statistical analysis, and physical modeling.
• The strict positivity aligns with physical intuition, e.g., entropy as an extensive, non-negative measure of uncertainty.
• Their parameter monotonicity supports their use in estimation and signal processing, where an entropy measure reflecting broader distributional spread is crucial for detection, compression, and reconstruction tasks.

The universal adoption of these refinements is justified given their ability to bridge theoretical and applied domains, offering entropy measures that are well-behaved in both continuous and discrete regimes (Mishura et al., 6 Aug 2025, Petroni, 2014).

7. Summary Table: Standard vs Modified Differential Entropy

Property/Issue	Standard Differential Entropy	Modified/Renormalized Forms
May be negative	Yes	No (engineered for positivity)
Sensitive to scaling	Yes	No (with dispersion normalization)
Discrete compatibility	No (diverges under refinement)	Yes (smooth partition limit)
Monotonicity	No (may lose monotonicity in spread)	Yes (most forms monotonic)

These developments are crucial in applications where merging discrete and continuous frameworks is required, and where positivity and invariance are mathematically or physically necessary—themes that continue to drive research on the foundations of entropy for both classical and generalized probability distributions.