I have a transformation that goes from lower dimension to higher dimension, and I was wondering how to add the Jacobian adjustment. I have the following model with parameters \mu and \sigma. p \in \mathbb{R}^k with p ~ Dirichlet(\alpha) with alpha being priors that are provided. For each i, p_i …

As mentioned in @spinkney ’s link there is no such thing as a Jacobian for a low to high dimensional mapping. You have to first work out a bijective (one-tone) transformation with a well-defined Jacobain and then marginalize out the excess dimension in a separate step. See https://arxiv.org/abs/101…

Bumping this post in hope anyone can help! I’ve been stuck on this issue for some time with little progress.

Could you tell more about your model? Jacobian adjustments are usually not recommended–instead you should build a generative model. So put your prior not on p_i but on \mu and \sigma. Think about this: the fact that you know t_i implies a very strong relationship between different p_i–a relation…

Maybe this answers your question. I remember thinking it was interesting and filed it away: [image] Nonlinear transformation from R² to R Modeling You can’t transform probability density functions between two real spaces of different dimensions – the Jacobian update rule i…

I’m interested in solving the type of problem thats similar to this: Suppose there’s a bowl with a random number of jelly beans (non-integer number allowed) that’s normally distributed with mean mu and stdev sigma. The number of jelly beans can be put into K brackets: [-\infty,t_1], [t_1,t_2] , [t…

To me, it is not clear what is given data and what is model parameter.

The given data are the \alpha's and the t_i's . So we have priors on the CDFs, which are functions of the parameters \mu and \sigma. We try to use those priors along with the result to get parameter estimates of \mu and \sigma.

Not without adding the log determinant. In the 2-D case (ie when p \in \mathbb{R^2}) the deteriminant is easy to calculate and then with a weak prior on \mu the model converges.

As the answer of spinkney, the Jacobian is only defined for a map between Euclidean spaces of a same dimension. f(y|\theta) d\theta = f(y|\theta(\theta')) |\det( \frac{\partial \theta}{\partial \theta'} )| d \theta' . I guess, you assume a map (\mu,\sigma) \to (p_1,p_2) is a transformation of loca…

You’re right that the map I said goes to dimension 1. I misspoke; what I meant to say is that when p \in \mathbb{R^3} then the Jacobian is two dimensional and then the model converges well. My current approach is to investigate the behaviour of the Dirichlet under different assumptions. If you look…

Jacobian Adjustment with low to high dimension mapping

Modeling

spinkney July 12, 2020, 9:06pm 4

Maybe this answers your question. I remember thinking it was interesting and filed it away:

Topic		Replies	Views
Do I need Jacobian adjustment here? Modeling jacobian-adjustment	19	3540	April 27, 2023
Do I need Jacobian adjustment? Modeling specification	5	955	November 9, 2019
Jacobian adjustment Modeling	4	1554	December 17, 2018
Jacobian adjustment needed? Modeling fitting-issues , specification	4	972	February 14, 2022
Understanding Jacobian adjustments (Stan user guide) & multiple transformations of one parameter Modeling specification , jacobian-adjustment	3	612	January 29, 2021

Jacobian Adjustment with low to high dimension mapping

Related topics