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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36 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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40 views

Inverse Neural Networks

Suppose there is a series of transformation applied to the random variable $z_0$ such that $$ z_M = f_{\theta_{M}} \circ f_{\theta_{M-1}} \circ \ldots \circ f_{\theta_{1}}(z_0) =: f_{\theta}(z_0). $$ ...
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1answer
68 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 ...
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10 views

MDP optimal policy inverse problem

Given a map $\pi: S \to A$, is there an MDP with state ans action spaces $S,A$ such that it has $\pi$ as an optimal policy if we suppose the MDP is over an infinite time horizon and the optimality ...
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11 views

Variance estimate of solution of inverse problem by nonlinear least squares with multi-dimensional model function?

The inverse problem I solve has the following basic outline: a number $n_g$ of Gaussian kernels ($\textbf{x}$) of some width $\sigma$ and each with an amplitude $s_j$ is propagated in n-d space to $\...
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81 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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52 views

Inverse problem with normal distributions and max

Consider $n$ independent normally distributed random variables $$X_i\sim N(\mu_i,\sigma_i^2)$$ and denote $Y = \max\limits_{1\leq i\leq n}\{X_i\}$. We can define the probabilities, for each $1\leq i\...
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21 views

How to best summarize multivariate and unimodal but asymmetric posterior pdf generated with an MCMC?

To do parameter inference in an inverse problem, I use a Metroplois Hastings MCMC to sample the posterior pdf of the model parameters $\theta$ of the forward model $f(\theta)$. Where the forward model ...
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2answers
94 views

Distribution of population size $n$ given binomial sampled quantity $k$ and selection probability $\pi$

Given a drawn (without replacement) sample size $k$ from a binomial distribution with known probability parameter $\pi$, is there a function which gives distribution of likely population size $n$ from ...
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1answer
50 views

Existence of least squares and maximum likelihood estimators?

In statistical parameter estimation where there is a deterministic and stochastic component to the observation-generating model, do least squares and maximum likelihood estimators always exist? ...
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0answers
27 views

What is the inference behind the momentum variable and the Kinetic energy for a weakly non-linear inverse problems in the HMC method?

We generate an auxiliary momentum variable in the HMC method to provide gradient for the propagation of trajectory (m, p) (model or position, momentum) in the phase space. If we look into Newton's ...
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1answer
64 views

Probability of Crossing a Threshold as a Function of Time (Forward Model)

I know the state of some object at a given time. Let's assume that the state is temperature. At each time, I also know the mean and variance of that temperature. I would like to obtain a probability ...
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14 views

Can we get the input from a multilayer perceptron based on the output of one of its hidden layers?

I was reading a relatively new paper that proposed to split a nerual networks layers into groups and sending each group to different nodes to train them in a distributed manner. In order to not send ...
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111 views

Neural Network Inversion and its consequences

I am currently looking at Federated Learning. Here is a good example from google. The idea is that training happens on multiple devices. This means on one hand that training data never leaves a user (...
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69 views

Population Monte Carlo Algorithm using L2 Distance Measure/ Likelihood Distribution

I am currently struggling with some concepts of the Population Monte Carlo Framework. Initially, I came across this set of algorithms as I am currently trying to infer parameters from a 7D ...
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56 views

How to properly solve for the inverse problem of OLS? [duplicate]

In textbook ordinary least squares we want to find a vector of coefficients $b_{k+1\times n}$ such that the sum of the squared deviations of what's observed ($y_{n\times 1}$) from what's assumed to be ...
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1answer
90 views

What is the error of my regression? [closed]

I'm conducting a polynomial of a third degree upon a diode measurement where Amplification was measured against Voltage. It's a very exponential behavior. However, I used the ...
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578 views

Is the covariance matrix a diagonal matrix with variances on the diagonals?

I am a geophysicist learning about geophysical inverse problems. In many papers, the authors discuss the "covariance matrix" as it applies to the inverse problem. In most geophysical applications, ...
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1answer
93 views

Inferring a Markov chain from its invariant measure

Given a probability measure $p$ on $\{1,\dots,n\}$ assumed to be the invariant measure of some irreducible ergodic Markov chain with unknown transition matrix $P$, i.e., $p = pP$, what (if any) ...
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34 views

Approximation of fractional function that has real-power numerator

I have the function $f(x)=\frac{(1+x)^k}{1+ax}$, where $x>0, 0<a<k<1$. The function has only one maximum at $x_0=\frac{a-k}{a(k-1)}$, increases on the left of $x_0$ and decreases on the ...
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348 views

How to select the regularization parameter between two losses?

In deep learning, the total loss commonly consists of a task-specific loss and a weight regularized loss: loss = loss_specific + lambda * reg_loss In my case (...
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111 views

Machine learning/Deep learning to solve inverse tomographic problem

The typical simplifiled representation of a tomographic system is $y = Ax$, where $y$ is the collected data (sinogram in CT), $A$ is the projection matrix, and $x$ is the unknown image. The ...
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1answer
38 views

Marginal Posterior Likelihood-Solving inverse Problem

For a university project, we were required to code our own Parallel Tempering Algorithm and use it to solve an Inverse Problem with 4 Parameters. Unfortunately, I'm not sure if I'm too stupid or have ...
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346 views

Solving an inverse problem with machine learning

I am running up against a very tough inverse problem that I suspect might be solvable using machine learning. Here is the problem. Overview I am studying an object $X$ which, internally, is ...
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228 views

Connection between MCMC and Optimization for Inverse/Parameter-Estmation Problems

I've been considering two approaches to solving inverse/parameter-estimation problems, and I'm curious to the connection and/or difference between the two approaches. Set up: Say we have a forward ...
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146 views

Accuracy of exponential decay parameter estimates comparing different models

I am wondering what approach is more accurate for estimating parameters between two different models of exponentially decaying signal data. The signal decays very rapidly and only a few samples can ...
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311 views

How to compare posterior distributions for different observed data? KL-divergence?

So I'm solving an inverse problem with the Bayesian approach $p(u | y) \propto p(y| u )p(u)$. Assuming I have two datasets $y_1$ and $y_2$, what can be said about the difference in the posteriors $p(...
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1answer
1k 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 $d$...
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1answer
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 ...
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132 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 ...
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1answer
312 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 ...