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.

Filter by
Sorted by
Tagged with
2
votes
0answers
36 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\...
0
votes
0answers
18 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 ...
1
vote
2answers
85 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 ...
0
votes
1answer
26 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? ...
1
vote
0answers
19 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 ...
1
vote
1answer
63 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 ...
0
votes
0answers
32 views

Finding weight value for smooth constrained least squares that comes closest with a priori solution?

I need to solve a system of $n.k$ equations of form $t_{ij} = a_i + b_j + x_{ij}/v$, with $i = 1, ..., n$ and $j = 1, ..., k$. A least squares approach can be used, as in $d = Gm$. My data is $t_{ij}$...
0
votes
0answers
19 views

Some Questions on Data Assimilation Methods

I am currently learning about data assimilation methods from this document: https://www.ecmwf.int/sites/default/files/elibrary/2002/16928-data-assimilation-concepts-and-methods.pdf I had a few ...
2
votes
0answers
13 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 ...
2
votes
0answers
81 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 (...
1
vote
0answers
64 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 ...
1
vote
0answers
47 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 ...
1
vote
1answer
81 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 ...
1
vote
0answers
426 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, ...
4
votes
1answer
76 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) ...
0
votes
0answers
33 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 ...
1
vote
0answers
346 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 (...
2
votes
0answers
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 ...
0
votes
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 ...
1
vote
0answers
343 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 ...
2
votes
0answers
211 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 ...
1
vote
0answers
303 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(...
2
votes
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 $d$...
0
votes
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 ...
2
votes
0answers
130 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 ...
3
votes
1answer
275 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 ...