Quantifying the informativity of emission lines to infer physical conditions in giant molecular clouds - I. Application to model predictions

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Volume 691 (November 2024)

A&A, 691 (2024) A109

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Table 2

Differential entropy for a few common distributions.

Distribution on Θ		Differential entropy h (Θ)
	𝒩(μ, ∑) μ ∈ ℝ^D, ∑ ∈ ℝ^D^×D	$\frac{1}{2} \log_{2} [(2 π e)^{D} \| Σ \|]$ $\[\frac{1}{2} ~\log _{2}\left[(2 \pi e)^{D}\|\Sigma\|\right]\]$

General Θ ∈ ℝ^D D ≥ 1	Unif (C) C ⊂ ℝ^D	log₂ Vol C

	log 𝒩(μ, ∑) μ ∈ ℝ^D, ∑ ∈ ℝ^D^×D	$\frac{1}{2} \log_{2} [(2 π e)^{D} \| Σ \| e^{2 \sum_{d = 1}^{D} μ_{d}}]$ $\[\frac{1}{2} ~\log _{2}\left[(2 \pi e)^{D}\|\Sigma\| e^{2 \sum_{d=1}^{D} \mu_{d}}\right]\]$

	𝒩(μ, σ²) μ ∈ ℝ, σ > 0	$\frac{1}{2} \log_{2} [2 π e σ^{2}]$ $\[\frac{1}{2} ~\log _{2}\left[2 \pi e ~\sigma^{2}\right]\]$

Univariate Θ ∈ ℝ D = 1	Unif (a, b) a < b	log₂ [b − a]

	log 𝒩(μ, σ²) μ ∈ ℝ, σ > 0	$\frac{1}{2} \log_{2} [2 π e σ^{2} e^{2 μ}]$ $\[\frac{1}{2} ~\log _{2}\left[2 \pi e ~\sigma^{2} ~e^{2 \mu}\right]\]$

Notes. As is introduced in Sect. 2.1, 𝒩 denotes a Gaussian distribution, Unif a uniform distribution and log 𝒩 a lognormal distribution. Vol(C) is the volume of a set C, |∑| is the determinant of a covariance matrix ∑.

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