[dj3l4l]

The Bel is a unitless quantity. Yes, by convention, in certain fields, it applies to the logarithm of the ratio of powers. But in other fields (for example, quantifying a change in the degree of evidence for a hypothesis, as in Bayesian probability theory) it is applied to a ratio of different quantities (in the Bayesian case, a ratio of probabilities). There is no reason why dB can't be used for any unit, and its meaning is incomplete until the denominator of the ratio within the logarithm is known.

This is the gripe that is being conveyed in this article. Mathematically, the Bel is unitless. It is only by additional context that one can understand the value of the denominator in the logarithm.

[kazinator]

Mathematically, the Bel is unitless because it is a function applied to a ratio of measurements. Whatever units those measurements have disappear due to the cancelation.

If that measurement is linked to another one in a nonlinear way, that nonlinearity doesn't disappear: the fact that if the unitless fraction in which certain units canceled out is 2, then the corresponding unitless fraction obtained via different units is 4.

Just because the units disappeared in a fraction doesn't mean they are not relevant.

[paipa]

The denominator isn't the issue. The context-dependent base of the logarithm is, which makes 1 Bel = 10x for some things and 1 Bel = 3.16x for others.

I've never heard of decibels used in probability theory. Did they adopt it with the same baked-in bastardizations? Please tell me +10dB(stdev) = +10dB(variance) isn't a thing.

[dj3l4l]

The problem stated in the article is that the unitless quantity of 1 Bel is effectively applied only to power ratios. It is of course true that one can transfer a scaling of powers into the base of the logarithm when we are trying to figure out what effective scaling of voltages corresponds to 1 Bel of power scaling, but it is ultimately more meaningful to state that a Bel is "only a scaling of power", which is a statement about the units of the two variables in the logarithm (but ultimately, once we know the denominator that belongs to the definition, the numerator is also known, so we only need to know the reference denominator).

In Bayesian probability theory, there is a quantity known as the "evidence". It is defined as e(D|H) = 10 * log_10 (O(D|H)), where O(D|H) is the odds of some data, D, given the hypothesis, H.

The odds are the ratio of the probability of the data given that the hypothesis H is true, over the probability of the data given that the hypothesis is false, or: O(D|H) = P(D|H)/P(D|NOT(H)).

Taking the logarithm of the odds allows us to add up terms instead of multiplying the probability ratios when we are dividing D into subsets; so we can construct systems that reason through additive increases or decreases in evidence, as new data "arrives" in some sequence.

The advantage of representing the evidence in dB is that we often deal with changes to odds that are difficult to represent in decimal, such as the difference between 1000:1 (probability of 0.999, or an evidence of 30dB) and 10000:1 (probability of 0.9999, or evidence of 40dB).

This use of evidence has been around at least since the 60s. For example, you can find it in Chapter 4 of "Probability Theory - The Logic of Science" by E.T. Jaynes.

[paipa]

The puzzling thing is why they chose to adopt a term used to describe "only a scaling of power" to talk about log10 values. If 1 Bel simply meant 10x, I'd get it, but Bel has baggage and also means sqrt(10)x.

Is odds a power-like or amplitude-like quantity? If you can't tell, dB isn't the most fortunate choice. It's not like mathematicians need fake units to talk about unitless ratios and their logarithms.

[dj3l4l]

The useage of dB in the case of probability is exactly as the 1 Bel -> 10x meaning, but that is the technical meaning, without further context of the unit in the logarithm. There is no need to conceptualise this further as involving power or amplitude (which does not apply in probability).

I think that the unit having been popularised in the telecommunications industry just meant that every other instance of a log_10 ratio in physics lead to a realisation that it was a Bel. For Bayesian odds, this was probably because even the development of Bayesian probability was largely advanced by physicists (E.T. Jaynes being a famous example), who also were trained, and often worked, in signal processing of some kind or another. But I doubt they would have thought about this "power-ratios-only" adherence that is more the conception of telecommunications engineers, as opposed to physicists.

[kazinator]

(Funny you should mention that because while writing my grandparent comment above I vaguely ruminated about some kind of example involving standard deviation and variance, due to those being linked by squaring.)

Even if there isn't a +10dB(stddev), logarithmic graphs are a thing, in many disciplines. You just refer to the axes as "log <whatever>". Any time you are dealing with data which has a wide dynamic range, especially with scale-invariant patterns.

Back in the realm of electronics and signal processing, we commonly apply logarithm to the frequency domain, for Bode plots and whatnot. I've not heard of a word being assigned to the log f axis; it's just log f.

[BlueTemplar]

One parallel comment mentions 'octaves' being used for that.