# Questions tagged [maximum-likelihood]

a method of estimating parameters of a statistical model by choosing the parameter value that optimizes the probability of observing the given sample.

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### What does it mean in Deep Learning, that L2 loss or L2 regularization induce a gaussian prior?

A prior for what? Let's take a toy example of a Deep Neural Network with weights w being fit on a dataset D with an L2/Mean Square Error loss. I looked up Maximum Likelihood Estimation (MLE) and ...
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### Estimating the parameter $\beta$

The lifetime of computer monitors has a exponential distribution where the expected value can be written as: $\mu(s) = \frac{\beta}{s}$ Where $s$ is how bright the monitor is and both $s$ and $\beta$ ...
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### Nakagami Distribution Shape Parameter

I've problem to solve the parameter estimation using the Maximum Likelihood Function for Nakagami distribution. I've tried to derive the likelihood function with shape and scale parameters. then, I ...
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### Why is the solution to the Quasi Exponential MLE a minimum of the log-likelihood function?

In the paper "Quasi-maximum exponential likelihood estimator and portmanteau test of double AR(P) model based on Laplace(a,b)" (Xuan, Song, Cig, Sun, Dai, 2018), they want to estimate the ...
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### Asymptotics of MLE without closed form solutions

Suppose $L(\theta;X)$ denotes the likelihood of a model where $\theta$ is the parameter and $X$ is the data. $L(\theta;X)$ doesn't have a closed-form solution for MLE. I use a numerical procedure to ...
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### Find the Fisher Information for geometric distribution

Given $X1,\dotsc,Xn \sim \mathcal{Geo}(p)$ , and I need to find the MLE and the CI for the MLE. I found the MLE for this distribution, using the maximum likelihood function: $L(p;X) = (1-p)^(Xi-1) * p$...
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### Fitting Gumbel distribution based the maximal observation

Assume that we only consider $$G(x)=\exp(-\exp(\frac{x-\mu}{\sigma}))$$ is the Gumbel distribution. Question: Suppose we have a set of maximum values $\{Y_i\}_{i=1}^m$, why can the article directly (...
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### Alternative-specific slope ordered probit

Consider the following ordered response model: $Pr[Y=j|X]=Pr[\alpha_{j-1}-x'\beta_{j-1}<u<\alpha_{j}-x'\beta_{j}]\quad j=1,2,3,4$ i.e. the coefficients of x is alternative-specific. Can we ...
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### The Monte Carlo of the mean square error of the maximum likelihood estimates

I try to get mean square error of the maximum likelihood estimators in R (using Monte Carlo). I can write the calculation for the MLE that is repeated once, but I need to repeat the Monte Carlo ...
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### How can I compare the MLE by standard error?

In the paper Extreme value theory based on the r largest annual events (page 32), the idea is that he wants to fit the Gumbel distribution using a dataset. In this dataset, we have the largest ten ...
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### Likelihood in Discrete distribution vs Continuous distribution in Bayes' Theorem [duplicate]

I would like to have a clear understanding of what exactly is a likelihood function in Bayes' Theorem, and why isn't considered a probability. As well as the distinction between the likelihood in ...
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### MLE for a mixture of betas without using EM algorithm

Suppose I have a mixture of two Beta densities say $f_1 = \text{Beta}(1,1)$ and $f_2= \text{Beta}(1,\beta)$ where $\beta$ is unknown. The sample $X_1,....,X_n$ is observed based on latent Bernoulli ...
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### Can we fit extreme value distribution by build-in package?

I try to find a package in R to fit Gumbel distribution by Block Maxima Approach using maximal likelihood function (see here) $$G(x; \mu , \sigma)=\exp[-e^{-\frac{x-\mu}{\sigma}}].$$ The block ...
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### Can we get a Global Maxima by using EM algorithm? [duplicate]

Actually, I read many articles and blogs that EM converges to local maxima can someone help me that it converges to local maxima always? Can we get a global maximum?
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### EM algorithm for Bivariate Normal

Consider a random sample $X_i = (U_i,V_i)$ where $i=1,2,...,n$ from a bivariate normal population with mean $(\mu_1,\mu_2)$ and variances $(\sigma_1 ^2, \sigma_2 ^2)$ and correlation $\rho$. Let's ...
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### Estimating parameters for a set of related random variables

Suppose I have some random variables $$X_i \sim Dist(\theta_i)$$ for $i = 1, ..., n$ where $Dist$ is some known probability distribution family and $\theta_i$ are some parameters which may vary ...
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### How do entropy-based estimators relate to more conventional ML, least square, and GMM estimators

Over the years i have done a lot of analysis, mostly of parameters of linear approximations to the data or a forecast, and I have used linear and nonlinear least squares, maximum likelihood, and GMM ...
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### Maximum likelihood estimation with censored data that include repeated measures

I have received the review for my latest journal article in which I present an experimental study with human subjects. In this study, we have loaded our subjects with impact loads created by a ...
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### do probabilistic neural networks learn the mean values before the variances?

Hi i trained a simple probabilistic MLP with the maximum likelihood approach. I have assumed my target values normally distributed around the true value and i have plotted the values that the NLL and ...
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My attempt is like this: Let $X \sim \text{Weibull}(\alpha, \lambda)$ be a random variable following a Weibull distribution with pdf $fx(x; \lambda) = \begin{cases} {\alpha}{\lambda^{-\alpha}} x^{\... 2 votes 0 answers 30 views ### How to prove the MLE for theta? I started off with finding L(Xij; θ), but I'm still struggling to get to this equation in the problem. Any help would be appreciated. Thank you! 7 votes 4 answers 456 views ### Why the variance of Maximum Likelihood Estimator(MLE) will be less than Cramer-Rao Lower Bound(CRLB)? Consider this example. Suppose we have three events to happen with probability$p_1=p_2=\frac{1}{2}\sin ^2\theta ,p_3=\cos ^2\theta $respectively. And we suppose the true value$\theta _0=\frac{\pi}{...
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$$f(x) = \frac{\pi}{2b}\sin\left(\pi\frac{x-a}{b}\right),~~x\in[a,a+b]$$ So it's a pretty simple distribution, which is just a scaled sine wave. As an experiment, I was trying to find the MLE ...