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A sufficient condition would be that the tails of $f$ decay as $O(x^{-1-\epsilon})$, which is most distributions you might encounter over $\mathbb{R}$.

That being said, the differential entropy of a continuous pdf isn't really a meaningful physical, or information theoretical, quantity. The equation isn't homogeneous: changing the units in which you measure $x$ changes the entropy (dimensional analysis can lead to a surprising amount of mathematical insights, even outside of physics).

What does make sense is the KL-divergence of one distribution with respect to another. For a pdf with a compact support, you're implicitly looking at the divergence with respect to the uniform distribution. However, there is no "uniform" distribution over the real line and thus the concept isn't meaningful.

A sufficient condition would be that the tails of $f$ decay as $O(x^{-1-\epsilon})$, which is most distributions you might encounter over $\mathbb{R}$.

That being said, the differential entropy of a continuous pdf isn't really a meaningful physical, or information theoretical, quantity. The equation isn't homogeneous: changing the units in which you measure $x$ changes the differential entropy (dimensional analysis can lead to a surprising amount of mathematical insights, even outside of physics).

What does make sense is the KL-divergence of one distribution with respect to another. For a pdf with a compact support, you're implicitly looking at the divergence with respect to the uniform distribution. However, there is no "uniform" distribution over the real line and thus the concept isn't meaningful.

A sufficient condition would be that the tails of $f$ decay as $O(x^{-1-\epsilon})$, which is most distributions you might encounter over $\mathbb{R}$.

That being said, the entropy of a continuous pdf isn't a meaningful physical, or information theoretical, quantity. The equation isn't homogeneous: changing the units in which you measure $x$ changes the entropy (dimensional analysis can lead to a surprising amount of mathematical insights, even outside of physics).

What does make sense is the KL-divergence of one distribution with respect to another. For a pdf with a compact support, you're implicitly looking at the divergence with respect to the uniform distribution. However, there is no "uniform" distribution over the real line and thus the concept isn't meaningful.

A sufficient condition would be that the tails of $f$ decay as $O(x^{-1-\epsilon})$, which is most distributions you might encounter over $\mathbb{R}$.

That being said, the differential entropy of a continuous pdf isn't really a meaningful physical, or information theoretical, quantity. The equation isn't homogeneous: changing the units in which you measure $x$ changes the entropy (dimensional analysis can lead to a surprising amount of mathematical insights, even outside of physics).

What does make sense is the KL-divergence of one distribution with respect to another. For a pdf with a compact support, you're implicitly looking at the divergence with respect to the uniform distribution. However, there is no "uniform" distribution over the real line and thus the concept isn't meaningful.

A sufficient condition would be that the tails of $f$ decay as $O(x^{-1-\epsilon})$, which is most distributions you might encounter over $\mathbb{R}$.

That being said, the entropy of a continuous pdf isn't a meaningful physical, or information theoretical, quantity. The equation isn't homogeneous, changing the units in which you measure $x$ changes the entropy (a surprising amount of mathematical insights can be had by dimensional analysis, even outside of physics).

What does make sense is the KL-divergence of one distribution with respect to another. For a pdf with a compact support, you're implicitly looking at the divergence with respect to the uniform distribution. However, there is no "uniform" distribution over the real line and thus the concept isn't meaningful.

A sufficient condition would be that the tails of $f$ decay as $O(x^{-1-\epsilon})$, which is most distributions you might encounter over $\mathbb{R}$.

That being said, the entropy of a continuous pdf isn't a meaningful physical, or information theoretical, quantity. The equation isn't homogeneous: changing the units in which you measure $x$ changes the entropy (dimensional analysis can lead to a surprising amount of mathematical insights, even outside of physics).

What does make sense is the KL-divergence of one distribution with respect to another. For a pdf with a compact support, you're implicitly looking at the divergence with respect to the uniform distribution. However, there is no "uniform" distribution over the real line and thus the concept isn't meaningful.