11/25/2023 0 Comments Unit of entropy![]() ![]() But the conclusion follows, that both Shannon and differential entropy is unitless. That rises a problem, while $p(x)$ certainly is unitless, since probability is an absolute number, the density $f(x)$ measures probability pr unit of $x$, so if unit of $x$ is $\text$. Again, from general principles (see lognormal distribution, standard-deviation and (physical) units for discussion and references) the arguments of transcendental functions like $\log$ must be unitless. This leaves us with the unit of measurement of $\log p(x), \log f(x)$ respectively. Entropy change S T q r e v Unit of entropy is J K 1 m o l 1 The units of entropy are J K 1 m o l 1, which basically means joules of energy per unit heat (in Kelvin) per mol. Now, from general principles the unit of measurement of the expectation (mean, average) of a variable (random or not) is the same as the unit of measurement of the variable itself. Where $f$ is the probability density function of a continuous random variable. If the if the logarithm is to the base e e, the unit of entropy is the nat. As with entropy, the base of the logarithm defines the units of mutual information. I ( X, Y) log p X, Y ( X, Y) p X ( X) p Y ( Y). ![]() H_d(X) = -\int f(x) \log f(x) \ dx = -\E_X \log f(X) The mutual information of X X and Y Y is the random variable I(X, Y) I ( X, Y) defined by. Where $p$ is the probability mass function of a discrete random variable. Some details: We treat Shannon (discrete) and differential (continuous) entropy separately. A measurement of length in meters can be written in decimal or binary, that does not really change the unit of measurement of length used. ) in the physical sense, it corresponds more to writing the measurement in decimal or binary number systems. But these are not really measurements units (like meter, kg. First, sometimes units are given as "nats" or "bits", for the cases of use of natural logs/binary logs, respectively. We also say that H(X) is approximately equal to how much information we learn on average from one instance of the random variable X. Still, there are some details to elaborate. 2 Entropy Denition The entropy of a discrete random variable X with pmf pX(x) is H(X) X x p(x)logp(x) E log(p(x)) (1) The entropy measures the expected uncertainty in X. Gave the answer in comments, entropy is unitless. Information theory is a subfield of mathematics concerned with transmitting data across a noisy channel. ![]()
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