Jacobian term shouldn't be included when transforming loglikelihoods

[frankiepe]

When working in a transformed parameter space to perform maximum likelihood estimation, we don't need to include the Jacobian term provided we are not also transforming the data.

This follows from the "invariance property of the ML estimator": https://en.wikipedia.org/wiki/Maximum_likelihood_estimation

[MichaelClerx]

I think this means we need to update the Transformation class to have a transform_logpdf and a transform_loglikelihood.
To do that, we'll need to start distinguishing between unnormalised log pdfs (which take a random variable as input) and loglikelihoods (which take a non-random variable as input, plus data sampled from some distribution (but fixed)). Might have some far-reaching consequences software wise...

See #457

The use case for this is performing MLP using the Transform class and an optimiser: at the moment that uses the transform_logpdf method even if you pass it a likelihood

[MichaelClerx]

Thoughts @ben18785 , @martinjrobins , @chonlei ?

[martinjrobins]

is this just for MLP? How about if the likelihood is part of a log posterior, do you transform its output then?

[MichaelClerx]

No but you'd have [changed prior] * [unchanged likelihood] = [changed pdf]. So the likelihood itself would be treated the same as in MLE, but the posterior would still change due to the prior changing.

[MichaelClerx]

See https://theoryandpractice.org/stats-ds-book/distributions/invariance-of-likelihood-to-reparameterizaton.html

[MichaelClerx]

And https://en.wikipedia.org/wiki/Maximum_a_posteriori_estimation#Limitations

[martinjrobins]

No but you'd have [changed prior] * [unchanged likelihood] = [changed pdf]. So the likelihood itself would be treated the same as in MLE, but the posterior would still change due to the prior changing.

so the likelihood is multiplied by a factor (lets say A where A = dp(sigma)/dsigma is the jacobian of the pdf) due to the change of variables, but MLE is unaffected by this since A is constant wrt the parameters sigma. The prior has an A that might be dependent on the parameters, so this needs to be transformed. More broadly, any logpdf that has a jacobian that is constant wrt the parameters sigma does not need to be transformed. So perhaps a pints logpdf can have a function that returns a bool indicating if its jacobian is constant wrt the parameters? Then the Transformation class can just check this?