# 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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### Is the Jacobian term needed if the prior is on the transformation parameter?

Suppose I have a strictly positive parameter $\sigma$ and I need to estimate it using the random walk Metropolis-Hasting algorithm. I know that I can do a parameter transform, i.e., $\beta=log(\sigma)$...
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### Jacobian Matrix of an Element wise operation on a Matrix

Is it right in saying that the Jacobian Matrix of a Matrix output of an elementwise operation to the same input is a diagonal matrix ? Context below. From ref 1 it is clear that when you have an ...
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### Derivation of ELBO in ADVI Paper, Jacobian of Elliptical Transformation

I've been following the ELBO derivations in the paper Automatic Differentiation Variational Inference and have a few questions. With the model $p(x,\theta)$, they first transform $\theta$ so that it ...
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### Change of variables by doing a transformation with a Jacobian versus finding an inverse

I have been solving one problem and there is something unclear to me in the solutions. Namely, let's consider a probability density $p_x(x)$ defined over a continuous variable $x$, and suppose that we ...
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### Integrating in log-space with a change of variable

I have a probability density function $P(f|\mu,\sigma) = \mathcal{N}(f|\mu,\sigma)$. I need to change the variable $f$ to $L = \log_{10}[f]$ so I can integrate it jointly with another PDF whose domain ...
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### Normalizing a custom Weight Shifted or Spiked Gaussian distribution

I have a custom weight shifted bivariate gaussian distribution that I wish to normalize. W is the weighted symmetric matrix that shifts the entire distribution and the λ below is the diagonal matrix ...
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### shape of the Jacobian tensor of the output of the layer w.r.t. the input 𝑿

Suppose we have a linear (i.e. fully-connected) layer, defined with in_features=1024 and out_features=2048. We apply this layer to an input tensor 𝑿 containing a batch of N=128 samples. What would ...
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### Maximum likelihood versus minimum distance estimator

Let's assume that I have data $X_1,\ldots X_n$ and a parametrized distribution $p(x,\theta)$. I can get an estimate of $\theta$ through maximum likelihood estimation $\theta^*_{ML}$. I can also get an ...
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### Vector Jacobian product in automatic differentiation

my questions is related to this post Higher Order of Vectorization in Backpropagation in Neural Network @shimao I don't really get the following claim (I know how the chain rule works and what is the ...
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### Jacobian for transformation of discrete random variables (intuition)

I am reading Blitzstein's introduction to probability. He states that, while a transformation of continuous r.v.s needs a Jacobian (or derivative), a transformation of discrete r.v.s does not. Is ...
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### Transformation of random normal variable $x/y^2$

If $x \sim N(\mu_x, \sigma_y)$ and $y \sim N(\mu_y, \sigma_y)$ what is the distribution of $x/y^2$ using Jacobian method?
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### 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 ...
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### 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?
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### 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 $Y_1=Y_1(X_1,X_2,X_3,X_4)$ ...
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