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# Questions tagged [inverse-problem]

In science an inverse problem is the problem of calculating from a set of observations the causes that produced the observations. Examples are tomography and seismic reconstruction, and many others. Use this tag for statistical methods used with inverse problems.

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### Fitting a Gaussian function to Poisson noisy data

Let $A$, $\mu$, $\sigma$ be some positive, a priori unknown parameters. Define a Gaussian function $f$ as $$f(x) = A \mathrm{exp}\left(-\frac{1}{2} \left( \frac{x-\mu}{\sigma}\right)^2\right).$$ One ...
• 141
1 vote
0 answers
10 views

### Issues with gradient of standard deviation in GPR using skopt.learning.gaussian_process

I'm currently working on a Gaussian Process Regression (GPR) model using the implementation provided in skopt.learning.GaussianProcessRegressor which is a wrapper for the sklearn implementation. This ...
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0 votes
0 answers
29 views

### Can estimated value plus one standard deviation be less than true value in a synthetic experiment?

I perform a synthetic calculation and then an inversion, that is, I have actual value in the synthetic calculation and I estimate it in the inverse problem stage. The inverse problem gives me the ...
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0 votes
1 answer
105 views

### Solve equation for a given set of parameters

I'm modelling a process for which the probability of the event (stop) not happening before time $t$ is $e^{-\lambda\cdot t }|\lambda>0$. When the event happens, the process stops running for a ...
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1 vote
2 answers
312 views

• 116
1 vote
1 answer
35 views

### Estimate the Image Using Multi Many Realizations of Its Convolution with a Known Filters Using Wiener Filter

Suppose we have a corrupted image $Y = H*X + \epsilon$ that is formed by taking an image $X$, convolving it with a point-spread function $H$, and adding gaussian noise $\epsilon$. Then we know that ...
2 votes
1 answer
92 views

### How to invert/infer a parameter in nonlinear conditional expectation function

I wouldn't be surprised if this question has already been asked, as it sounds like a standard bookwork result. However, I'm not sure I know the language to describe it, and when I type in the the ...
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0 votes
0 answers
111 views

### Inverse Neural Networks

Suppose there is a series of transformation applied to the random variable $z_0$ such that $$z_M = f_{\theta_{M}} \circ f_{\theta_{M-1}} \circ \ldots \circ f_{\theta_{1}}(z_0) =: f_{\theta}(z_0).$$ ...
0 votes
1 answer
111 views

### Validation of inverse problem solution based on Bayesian method

Recently, I read a paper about the inverse problem and parameter estimation. The main approach of the paper is based on the Bayesian method. The answer in this method is a posterior probability ...
1 vote
0 answers
120 views

### Is it possible to recover original normal distributions from observations of sums of normally distributed variables?

I have been trying to solve the following problem for several weeks. I would greatly appreciate it if you help me solve the following problem. Problem description Assume that there are seven iid ...
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2 votes
0 answers
87 views

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2 votes
1 answer
2k views

### L-curve method for regularization parameter selection

I work on PDE inverse problems and I'm interested in how these can be viewed as problems of statistical inference. I'm looking for some model parameters $m$ which minimize the misfit with some data $d$...
5 votes
0 answers
225 views

### Making sense of standard deviation after sampling using Cholesky

I have an inverse problem with over 65,000 degrees of freedom. I am using Bayesian formulation to solve this problem. After using the optimization algorithm to obtain MAP solution, I want to calculate ...
• 151
1 vote
1 answer
1k views

### Least square regression with L1 regularization and non-negativity constraint

There are two functions associated by the model $a(x) = \int_{k_1}^{k_2} b(k)\exp(-kx)dk$ where $a(x)$ is the experimental data I have, and $b(k)$ is the information I want to get. Or I can write ...
• 131
2 votes
0 answers
144 views

### Confusion related to inverse problems in statistics [closed]

I am getting started with inverse problems in statistics. However, I didn't something related to it. I was reading this paper http://math.uni-heidelberg.de/studinfo/reiss/CavalierInvProb.pdf. It ...
• 6,807
3 votes
1 answer
423 views

### Regression with an unknown dependent variable

I want to know if there is any literature about the following regression problem: $$Y=X\beta +\epsilon$$ where $Y$ is unknown. But, i know $X$ and the OLS estimator of $\beta$  \hat{\beta}=(X^\top ...
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