4 ms·

Pardon my ignorance, but why do you need mcmc to analyze sensitivity? Isn't sensitivity something related to the derivative of the output with respect to the pa

Pardon my ignorance, but why do you need mcmc to analyze sensitivity? Isn't sensitivity something related to the derivative of the output with respect to the parameter?

shane_b 5y ago
I believe this is because not all derivatives are possible via calculation and mcmc gives close approximation. But I could misunderstand.
sanxiyn 5y ago
Yes, but we don't have a differentiable climate model. People are working on it, but it's not there yet. See https://clima.caltech.edu/ https://clima.caltech.edu/ and https://dj4earth.github.io/ https://dj4earth.github.io/.
contravariant 5y ago
There's no reason you couldn't take the expectation value of a derivative. Linearity of expectation means this will be equal the derivative of the expectation value. I may be missing the point though; I'm only versed in the mathematics.
- hexane360 5y ago
  Actually, because differentiation is a limiting process, this relies on the bounded convergence theorem or monotone convergence theorem. However, I imagine this would hold in most distributions under consideration.
  - contravariant 5y ago
    Indeed since expectation values make the constant functions integrable you'd have to do some pretty wild things before things goes wrong. In fact any bounded function (with bounded derivative) is going to be fine. I also wonder if differentiation doesn't have some nicer properties anyway, being a linear operation as well. Typically you can get quite far by showing things work for a sufficiently broad class of functions and just extending things from there. In particular you could use the Fourier transform, which trivially commutes with taking expectation values. You can always try something abhorrent like sin(e^(-1/x^2)) or something to make things go wrong though.
rich_sasha 5y ago
Perhaps this is to do with parameter calibration? If you vary parameters until you get best fit, then use them to forecast forward, you will hugely understate the variation, and very possibly also bias the estimate.
digikata 5y ago
When you have many parameters in multiple models with interdependent parameters in feedback cycles the Monte Carlo method lets you sample the space and explore faster. Especially in large scale simulations the execution speed limits exhaustively computing your way through the parameter space. And because there is likely no simple set gradients to follow, I’d suspect the Markov probabilities let you further navigate local maxima and minima more effectively. Something like a breadcrumb trail of past paths through the forest.

Pardon my ignorance, but why do you need mcmc to analyze sensitivity? Isn't sensitivity something related to the derivative of the output with respect to the parameter?

shane_b 5y ago

I believe this is because not all derivatives are possible via calculation and mcmc gives close approximation. But I could misunderstand.

sanxiyn 5y ago

Yes, but we don't have a differentiable climate model. People are working on it, but it's not there yet. See https://clima.caltech.edu/ https://clima.caltech.edu/ and https://dj4earth.github.io/ https://dj4earth.github.io/.

contravariant 5y ago

There's no reason you couldn't take the expectation value of a derivative. Linearity of expectation means this will be equal the derivative of the expectation value. I may be missing the point though; I'm only versed in the mathematics.

hexane360 5y ago
Actually, because differentiation is a limiting process, this relies on the bounded convergence theorem or monotone convergence theorem. However, I imagine this would hold in most distributions under consideration.
- contravariant 5y ago
  Indeed since expectation values make the constant functions integrable you'd have to do some pretty wild things before things goes wrong. In fact any bounded function (with bounded derivative) is going to be fine. I also wonder if differentiation doesn't have some nicer properties anyway, being a linear operation as well. Typically you can get quite far by showing things work for a sufficiently broad class of functions and just extending things from there. In particular you could use the Fourier transform, which trivially commutes with taking expectation values. You can always try something abhorrent like sin(e^(-1/x^2)) or something to make things go wrong though.

rich_sasha 5y ago

Perhaps this is to do with parameter calibration? If you vary parameters until you get best fit, then use them to forecast forward, you will hugely understate the variation, and very possibly also bias the estimate.

digikata 5y ago

When you have many parameters in multiple models with interdependent parameters in feedback cycles the Monte Carlo method lets you sample the space and explore faster. Especially in large scale simulations the execution speed limits exhaustively computing your way through the parameter space. And because there is likely no simple set gradients to follow, I’d suspect the Markov probabilities let you further navigate local maxima and minima more effectively. Something like a breadcrumb trail of past paths through the forest.