Clarify interpretation of noise distributions

[sebapersson]

When implementing support for LogLaplace in PEtab.jl, I realized that the interpretation of the noise distributions in the spec is not entirely clear. In particular, for the supported distributions, the model output is not assumed to be the mean or location of the data distribution, but rather its median.

For example, let ($m$) be the measured value, $y := \text{observableFormula}$ the simulated value, and $\sigma$ the noise. For the LogNormal distribution in PEtab we have $\log(m) \sim \mathcal{N}(\log(y), \sigma)$, which implies $m \sim \mathcal{LN}(\log(y), \sigma)$. For this LogNormal, the median is y (exp of first argument). A similar interpretation holds for LogLaplace. Overall, this PR aims to clarify this.

[dilpath]

Thanks!

[dweindl]

Thanks, you are right, that wasn't worded correctly.

Fine to merge with Dilan's suggestions, but with respect to #654, I am wondering if we really want to refer here to the median, or just generally to something like "first parameter in canonical notation", or just nothing at all.

[sebapersson]

"first parameter in canonical notation"

This is not correct either, as with log-normal we have log(m) not m entering the formula, so I would like to keep median (I think it helps to think about what the noise distribution means). As for #654, I think the way to handle it would simply be to split the noise distribution into two sections, discrete and continuous.

[dweindl]

Same problem here, right? For the log-normal distribution, sigma wouldn't be the scale but its logarithm?

[sebapersson]

Actually, for normal distributions it is the standard distribution (no log), while for Laplace it is the scale. I have made this clearer.

[dilpath]

Thanks, now just for consistency and to make the table narrower, I'll shorten the other descriptions too.

Removes the horizontal scrollbar on the table for me: https://petab--656.org.readthedocs.build/en/656/v2/documentation_data_format.html#noise-distributions