Questions tagged [jacobian]

For statistical questions involving the Jacobian matrix (or determinant) of first partial derivatives. For purely mathematical questions about the Jacobian it is better to ask at math SE https://math.stackexchange.com/.

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Maximum Likelihood - Normal Errors - When is the Jacobian needed?

I am considering the following non-linear model $$h(z) - \lambda_0 - \lambda_1 z - \lambda_2x = v$$ where $v \sim \mathcal N(0,\sigma^2)$ unobserved error and where $\lambda_j$ are unknown ...
151 views

Express the density of a function of two random variables using the Gradient and the joint density

I would like to know if it is possible to express the density $f_Z(z)$ of a function $Z = g(X,Y)$ of two continuous "nice" random variables $X$ and $Y$ only using the joint density $f_{XY}(x,y)$ and ...
2k views

Calculating the Jacobian of a neural network

I'm trying to calculate confidence intervals for a neural network (rather than prediction intervals). I'm following this paper, which treats them in the same framework as any parametric (parameter-...
133 views

Suppose I have some continuous random variable $X$. Further, suppose I am interested about a transformed random variable $Y = g(X)$ where $g$ is some increasing function. If I know the CDF of $X$, I ...
22 views

How do you solve for inverses when doing the Jacobian transformation method?

Hopefully a very basic question. This is from a textbook exercise I'm doing to prepare for a class. Suppose I have two independent random variables, X and Y, and I am trying to find the joint ...
38 views

Showing a useful result for Wisharts and Multivariate Beta random matrices

Let $\mathbf{A} \sim \text{Wishart}_m\left(k_a,\mathbf{V} \right)$ and $\mathbf{B} \sim \text{Wishart}_m\left(k_b,\mathbf{V} \right)$ be two full rank Wishart random matrices. Define  \mathbf{S} = \...
21 views

Change of variables: 4-dimensional PDF to 2-dimensional PDF

I have a 4-dimensional joint-PDF between variables $X_1,X_2,X_3,X_4$ which are all Gaussian. I want to transform this into a 2-dimensional joint-PDF between new variables $Y_1=Y_1(X_1,X_2,X_3,X_4)$ ...
246 views

How to understand Jacobian Matrix from the geometric perspective?

I found a good lecture about Jacobian Matrix which was part of a statistics course. However, it was published 20 years ago and lack of explanation. As a beginner of statistics, I'm not able to find ...
65 views

Wrong vector calculus in lecture note 5 of cs224n, Stanford

I am studying NLP via cs224n from Stanford. I am reading this lecture note now. When you refer to the 5th page, they want to derive the gradient with respect to W for RNN, to show the mathematical ...
601 views

Meaning of Jacobian of the transformation for pdf of function of random vectors

I am studying multivariate statistics and I don't understand the meaning of Jacobian of the transformation for pdf of function of random vectors. If I have a random vector, let's say bivariate, (X,Y)...
49 views

What are some interesting parameterizations of $4 \times 4$ correlation matrices, and also perhaps their associated jacobians?

I am studying (mainly using Mathematica) some constrained integration problems in which the six-dimensional convex set of $4 \times 4$ correlation matrices plays a central role. In light of this, I ...
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Finding the log Jacobian of a transformation from lower to high dimensions, specifically in the case of normal cdf to Dirichlet?

I have random variables $\mu$ and $\sigma$ that I have transformed and I am interested in finding their joint distribution given the following information I have. Particularly, I need help finding the ...
Let $X$ and $Y$ be two independent random variables with the density functions: $f(x) = 3 x^2$, for $0<x<1$, $0$ elsewhere $g(y) = 4y^3$, for $0 <y<1$, $0$ elsewhere Give $\mathbb E(x/... 0answers 24 views Calculating a Weighted Standard Error of the Fit for Nonlinear Regression I have a data set of$N$points to which I have fit an equation of$n$parameters$\theta_{1..n}$such that$y_i \sim f(x_i; \theta_{1..n})$. These data$(x_{1..N},y_{1..N})$have been provided with ... 0answers 12 views Autograd theory question I have attached an image of the mathematical description of calculating the gradient for the cost function from Pytorch. 1.) Is$\vec{y}$the output of the network? 2.) What is$v$in terms of a ... 0answers 105 views Is Covariance Matrix analogous to Jacobian Matrix? In probability theory covariance matrix denote how each variable relates to other in a pairwise manner. So 1 would mean they are identical and 0 would mean they are independent and are not related. Is ... 1answer 33 views Jacobian for function including cubic spline I am trying to fit a measured spectrum with a linear combination of end-member spectra which are approximated by cubic spline functions ($f_1$and$f_2$). I also need to incorporate terms that account ... 0answers 203 views Transformation of random variables and Jacobian When transforming 2+ continuous random variables, you use a Jacobian matrix and compute the determinant. Do you also compute the Jacobian for discrete random variables? 0answers 55 views Projection of multivariate distribution to lower dimensional subspace Say that$X \in \mathbb{R}^n$is a vector of$n$r.v.'s with pdf$p(x_1,\ldots,x_n)$. Let's consider now the linear map$Y = A X$where$Y \in \mathbb{R}^m$with$m < n$. I am seeking$p(y_1,\ldots,...
Suppose $X$ is distributed with a unimodal pdf $f(x)$ and let $Y = g(X)$ for some strictly monotone function $g$. Hence $g$ is invertible. Is there an analytically tractable relationship between the ...