Clarify interpretation of noise distributions #656
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| - Probability density function (PDF) | ||
| * - Gaussian distribution | ||
| * - | Gaussian distribution | ||
| | (i.e., :math:`m` is normally distributed as :math:`m \sim \mathcal{N}(y, \sigma)`) |
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| | (i.e., :math:`m` is normally distributed as :math:`m \sim \mathcal{N}(y, \sigma)`) | |
| | (i.e., :math:`m` is normally distributed as :math:`m \sim \mathcal{N}(y, \sigma^2)`) |
| \pi(m|y,\sigma) = \frac{1}{\sqrt{2\pi}\sigma}\exp\left(-\frac{(m-y)^2}{2\sigma^2}\right) | ||
| * - | Log-normal distribution | ||
| | (i.e., :math:`\log(m)` is normally distributed) | ||
| | (i.e., :math:`\log(m)` is normally distributed as :math:`\log(m) \sim \mathcal{N}(\log(y), \sigma)`) |
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| | (i.e., :math:`\log(m)` is normally distributed as :math:`\log(m) \sim \mathcal{N}(\log(y), \sigma)`) | |
| | (i.e., :math:`\log(m)` is normally distributed as :math:`\log(m) \sim \mathcal{N}(\log(y), \sigma^2)`) |
| Note that, for all continuous distributions, the simulated value is modeled | ||
| as the median of the noise distribution; i.e., measurements are assumed to | ||
| be equally likely to lie above or below the model output. |
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Possibly not true for the prior distributions; I didn't check. It's implied here that this note doesn't apply to prior distributions, but here's a suggestion just to clarify that
| Note that, for all continuous distributions, the simulated value is modeled | |
| as the median of the noise distribution; i.e., measurements are assumed to | |
| be equally likely to lie above or below the model output. | |
| Note that, for all PEtab noise distributions, the simulated value is modeled | |
| as the median of the noise distribution; i.e., measurements are assumed to | |
| be equally likely to lie above or below the model output. |
| * - Laplace distribution | ||
| - ``laplace`` | ||
| - | ``laplace`` | ||
| | (i.e., :math:`m` is Laplace distributed as :math:`m \sim \mathcal{L}(y, \sigma)`) |
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Move to first column. GitHub doesn't let me make the suggestion but you can see the issue here: https://petab--656.org.readthedocs.build/en/656/v2/documentation_data_format.html#noise-distributions
| \pi(m|y,\sigma) = \frac{1}{2\sigma}\exp\left(-\frac{|m-y|}{\sigma}\right) | ||
| * - | Log-Laplace distribution | ||
| | (i.e., :math:`\log(m)` is Laplace distributed) | ||
| | (i.e., :math:`\log(m)` is Laplace distributed as :math:`\log(m) \sim \mathcal{L}(\log(y), \sigma)`) |
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I would be fine with shortening all of these to make the column narrower, and to explicitly write Laplace since I guess it doesn't have a commonly used symbol unlike normal
e.g.:
| | (i.e., :math:`\log(m)` is Laplace distributed as :math:`\log(m) \sim \mathcal{L}(\log(y), \sigma)`) | |
| | (i.e., :math:`\log(m) \sim \mathrm{Laplace}(\log(y), \sigma)`) |
| Denote by :math:`m` the measured value, | ||
| :math:`y:=\text{observableFormula}` the simulated value | ||
| (the location parameter of the noise distribution), | ||
| (the median of the noise distribution), |
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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.
| :math:`y:=\text{observableFormula}` the simulated value | ||
| (the location parameter of the noise distribution), | ||
| (the median of the noise distribution), | ||
| and :math:`\sigma` the scale parameter of the noise distribution |
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Same problem here, right? For the log-normal distribution, sigma wouldn't be the scale but its logarithm?
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When implementing support for
LogLaplacein 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 $\log(m) \sim \mathcal{N}(\log(y), \sigma)$ , which implies $m \sim \mathcal{LN}(\log(y), \sigma)$ . For this
LogNormaldistribution in PEtab we haveLogNormal, the median isy(expof first argument). A similar interpretation holds forLogLaplace. Overall, this PR aims to clarify this.