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/.

Filter by
Sorted by
Tagged with
1
vote
0answers
17 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 ...
2
votes
0answers
40 views

Are Jacobian adjustments necessary when the target parameter is a difference between two parameters in Stan?

I want to model the index called Delta P (e.g., p.144 of this paper), which is basically a difference between two proportions (i.e., $\frac{n_1}{N_1}$ - $\frac{n_2}{N_2}$), as a function of a ...
0
votes
0answers
9 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 ...
23
votes
0answers
2k views

What is the difference between the Jacobian, Hessian and the Gradient? [migrated]

I know there is a lot of topic regarding this on the internet, and trust me, I've googled it. But things are getting more and more confused for me. From my understanding, The gradient is the slope of ...
1
vote
1answer
25 views

How to estimate the PDF of the logarithm of a uniformly distributed random variable?

This is a question I have to solve and need help with. I know it's usual to give pointers and hints so the OP can follow from there. Thus, I'll appreciate all input that shows me the way to go. Let $...
0
votes
0answers
10 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 ...
0
votes
0answers
64 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 ...
2
votes
1answer
56 views

Jacobian of Inverse Gaussian Transformation in Schwarz & Samanta (1991)

In the sample size $n=2$ case when transforming $\{x_1, x_2\}$ to $\{\bar{x}, s\}$ (where $X_1, X_2 \overset{iid}{\sim} IG(\mu, \lambda)$, $\bar{X}=\frac{\sum_i^2 X_i}{n}$, and $S=\sum_i^2 (\frac{1}{...
2
votes
0answers
21 views

Jacobian matrix X going from x to the y coordinate system [closed]

In Christoper Bishop's book Pattern Recognition and Machine learning, they use Jacobian to convert covariance matrix x to y. However, according to definitions on Wikipedia, the definition of Jacobian ...
3
votes
2answers
54 views

Given X and Y are independent ~N(0,1), what is the distribution of $ Z=X^2 + Y^2 $

Our joint pdf is $f(x,y) = \frac{1}{\sqrt{2π}} e^\frac{x^2+y^2}{2}$ Now we let $ U = X^2 + Y^2 $ and $ V = Y$, we can then get our Jacobian as $ J = \frac{1}{\sqrt{u-v^2}} $ Since this ...
0
votes
1answer
32 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 ...
3
votes
2answers
213 views

How to calculate the Jacobian of the transformation ( for covariance matrix)

I'm reading this Paper about a separation strategy for modeling covariance matrices with focus on Bayesian analysis. Direct decomposition of covariance matrix is as follows: $\Sigma = \text{diag}(S)\,...
8
votes
2answers
599 views

How can I obtain a Cauchy distribution from two standard normal distributions?

I am interested in Let $X\sim N(0,1), Y \sim N(0,1)$ independently. Show $\frac{X}{X+Y}$ is a Cauchy random variable. My work: $f_{X,Y}(x,y)=\frac{1}{2\pi} e^{\frac{-1}{2}(x^2+y^2)}, -\infty&...
4
votes
0answers
81 views

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 ...
1
vote
0answers
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} = \...
3
votes
1answer
229 views

Reason for absolute value of Jacobian determinant in change-of-variable formula?

When we have a random variable $x$ with a probability density $p(x)$, and a function $y = f(x)$ that is differentiable and can be solved for $x = g(y)$, the change of variable formula leads us to a ...
0
votes
0answers
159 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?
1
vote
0answers
20 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 $Y_1=Y_1(X_1,X_2,X_3,X_4)$ ...
0
votes
0answers
47 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,...
1
vote
0answers
225 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 ...
1
vote
0answers
15 views

Is it possible to find the joint distribution of a random vector if only the distribution of scalar many-to-one transformation is known? [duplicate]

Theoretical Exercise: I'd like to derive the joint distribution $p_{\boldsymbol{X}}$ of a random vector $\boldsymbol{X} \in \mathbb{R}^K$ if only the distribution of a scalar many-to-one ...
3
votes
1answer
265 views

Higher Order of Vectorization in Backpropagation in Neural Network

I am learning a machine learning class online from Stanford, namely CS 229. There is one section about deep learning and back-propagation in deep learning. The network looks like: The forward ...
0
votes
0answers
27 views

Highest posterior interval and monotone changes of variables

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 ...
1
vote
3answers
165 views

Computing directly the pdf of $Y=X^2$ for the pdf $f_X(x) = \frac{2}{9}(x+1)$

In the book Statistical inference by Casella and Berger, given that the pdf of X is $f_X(x) = \frac{2}{9}(x+1)$ for $-1 \le x \le 2$, we want to find the pdf of $Y=X^2$. But it says we are not able to ...
0
votes
1answer
153 views

For a posterior $p(\theta |y)$, if I specify a one-to-one transformation $\phi = g(\theta)$, how can I apply the transformation? [duplicate]

Suppose I have a posterior distribution, $p(\theta \mid y)$, where $y$ was my data and $\theta$ is a random variable with some prior distribution. If I specify a one-to-one transformation $\phi = g(\...
1
vote
0answers
64 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 ...
3
votes
1answer
351 views

Shouldn't it be Jacobian Descent?

Whenever we have a multiclass prediction the classifier generates a vector output. Per the definition of a Jacobian we are actually taking Jacobian steps towards a local minimum - so should it ...
1
vote
1answer
232 views

Prediction Interval for Regression Models with Weights

Courtesy Prediction Interval for Neural Net With Hessian :: nnet in R I would like to understand how to manually derive a point-wise prediction confidence interval applicable to neural models. ...
3
votes
1answer
340 views

Weighted Least Squares weights not changing Jacobian matrix

I currently have 4 data points and the following Jacobian matrix $A$ and cost vector $b$ $ A = \begin{bmatrix} -0.7867 & 0.0464& -0.6155 & 1.0000 \\ -0.3751 & 0.4299 &...
4
votes
2answers
5k views

What is the jacobian for a neural network

I know that a Jacobian is a matrix holding all of the first-order derivatives for a vector-valued-function. What is the Jacobian of a neural network, though? What are the inputs and what are the ...
2
votes
0answers
142 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 ...
18
votes
2answers
474 views

Suppose $Y_1, \dots, Y_n \overset{\text{iid}}{\sim} \text{Exp}(1)$. Show $\sum_{i=1}^{n}(Y_i - Y_{(1)}) \sim \text{Gamma}(n-1, 1)$

What is the easiest way to see that the following statement is true? Suppose $Y_1, \dots, Y_n \overset{\text{iid}}{\sim} \text{Exp}(1)$. Show $\sum_{i=1}^{n}(Y_i - Y_{(1)}) \sim \text{Gamma}(n-1, ...
1
vote
0answers
581 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)...
1
vote
0answers
48 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 ...
1
vote
0answers
124 views

Normalization constant for many to one mapping (Laplace distribution)

Suppose $\alpha=U\beta$ where $U$ is $N\times K$ with $N > K$. What is the probability density function (PDF) of $\beta$, $p(\beta)$, given that we know that it is proportional to $q$, the PDF of $\...
2
votes
0answers
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-...
6
votes
2answers
1k views

Sample random variables conditional on their sum

Let $(X_1, \dots, X_n)$ be an iid sample of random variables with a known continuous distribution. I would like to simulate such a sample, conditional on the value of its sum, that is: $$ X_1, \dots, ...
2
votes
1answer
159 views

IS $\int_{-\infty}^\infty e^{-\beta\cdot g(x)}g(x)^{\alpha-1}\text{d}x={\Gamma(\alpha)\over \beta^\alpha}\ \ ?$ [closed]

Is the following statement true: Let $g(x)$ be some non negative continuous function of $x$.We know that$$\int_{0}^\infty e^{-\beta x}x^{\alpha-1}dx={\Gamma(\alpha)\over \beta^\alpha}$$ Does ...
18
votes
1answer
9k views

Derivation of change of variables of a probability density function?

In the book pattern recognition and machine learning (formula 1.27), it gives $$p_y(y)=p_x(x) \left | \frac{d x}{d y} \right |=p_x(g(y)) | g'(y) |$$ where $x=g(y)$, $p_x(x)$ is the pdf that ...
1
vote
1answer
1k views

Jacobian and covariance matrix

Anyone know the Bishop's book in 2.53 they use Jacobian to convert covariance matrix x to y. $$J_{ij}=\dfrac{\partial x_i}{\partial y_i}=U_{ji} \qquad{(2.53)}$$ $$\int_{\bf x} f({\bf x})d{\bf x} = \...
6
votes
1answer
8k views

Relation between Covariance matrix and Jacobian in Nonlinear Least Squares

I saw that CovB = inv(J'*J)*MSE in a MATLAB documentation here at http://www.mathworks.com/help/stats/nlinfit.html However, I cant find any sources for the ...
1
vote
0answers
229 views

Delta method with mix of continuous and discrete variables

This is my first question on Cross Validated so please bear with me if my question is lagging in any dimension. My question regards how to evaluate a Jacobian matrix when one variable is binary. I ...
3
votes
1answer
377 views

Change of Variable technique for two variables?

If, $\theta_1 = \ln \frac p{1-p}$ $\theta_2 = \ln \frac q{1-q}$ $\theta_2|\theta_1 \sim N(\theta_1, \sigma^2)$ which means $f(\theta_1,\theta_2) \propto e^{\frac{-(\theta_1-\theta_2)^2}{2\...
2
votes
0answers
133 views

About deriving PDFs from CDFs

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 ...
1
vote
1answer
104 views

Sampling from an (almost) multivariate normal over matrices

Consider $n$ points in the euclidean plane, $p_i = (x_i,y_i)_{1\leq i \leq n}$. Now consider a $2 \times 2$ matrix $M = \left(\begin{array}{cc}a & b\\c& d\end{array}\right)$ a vector $r = \...
9
votes
1answer
356 views

If $X_1,X_2$ are independent beta then show $\sqrt{X_1X_2}$ is also beta

Here is a problem that came in a semester exam in our university few years back which I am struggling to solve. If $X_1,X_2$ are independent $\beta$ random variables with densities $\beta(n_1,n_2)$ ...
4
votes
1answer
2k views

conditional probability, change of variable and Jacobian

I have a question, and I am guessing that the question arises due to my lack of good understanding in the change of variable technique. I would like to evaluate $f_X(x)$. When $f_Y(y)$ exists, I can ...
4
votes
2answers
2k views

Transformation of variables (Metropolis Hastings)

Say I have a bunch of data from a Poisson distribution and I want to find out my posterior i.e. I'm data fitting: $p(\lambda | X) \sim p(X|\lambda)p(\lambda)$ where $p(X|\lambda) = \frac{\exp(-\...
1
vote
0answers
715 views

Jacobian matrix in neural network

How do you calculate the Jacobian matrix using the results (weights and biases) of a neural network after training? I am working in MATLAB, if anyone has any code suggestions, that would be helpful as ...
3
votes
1answer
719 views

Sufficient statistics for a continuous distribution

Below is a discussion of sufficient statistics for a continuous distribution, taken from the third edition of Lehmann's Testing Statistical Hypotheses. I understand the discussion until the underlined ...