Questions tagged [likelihood]

Given a random variable $X$ which arise from a parameterized distribution $F(X;θ)$, the likelihood is defined as the probability of observed data as a function of $θ: \text{L}(θ)=\text{P}(θ;X=x)$

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What is the analog of the PDF and CDF for the likelihood function?

In probability, we can find the cdf using the pdf and vise-versa. Integrating pdf yields the cdf. Does integrating the likelihood function yield any important thing? In statistics, $\mathcal{L} (M\...
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Do likelihood functions require sigma-algebras as probability spaces do?

In statistics, the likelihood function (often simply called the likelihood) is formed from the joint probability of a sample of data given a set of model parameter values; it is viewed and used as a ...
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Why are p-values probabilities rather than likelihoods?

The p-value is the probability of obtaining test results at least as extreme as the results actually observed during the test, assuming that the null hypothesis is correct. Why is the p-value a ...
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What problems happen when $L(\theta|x_1,x_2,…,x_n)$ be a function of differently distributed random variables?

Given $L(\theta|x_1,x_2,...,x_n)$, may $x_1,x_2,...,x_n$ not all be identically distributed? What problems happen in the not identically distributed case compared to the iid case?
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If Maximum Likelihood estimation finds the best set of parameters for a regression model, then are likelihood ratio tests unnecessary? [closed]

My logic is this. Models are sufficiently defined by their parameters. Since MLE picks the n parameters that maximize the log-likelihood, then MLE picks the best model given n degrees of freedom of ...
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Statistical method for estimation of Penetrance

I was thinking on a problem: there is a rare disease, associated with some ultra-rare genomic variant (which means ~ 1 out of millions), and there are estimations of penetrance (how many individuals ...
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Why can't regression via Maximum Likelihood shrink coefficients to zero?

Why can't regression via Maximum Likelihood shrink regression coefficients to zero as in LASSO? Does shrinking coefficients to zero not give higher L-likelihood? Does the answer to my question have to ...
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How can I calculate the likelihood of my data given that two variables in my data are dependent?

I have a data set D that has 3 variables X, Y, Z, where each variable has 100 samples and have a Normal distribution. I am interested in calculating $p(X \not\!\perp\!\!\!\perp Y | D)$ and $p((X \not\!...
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Does the newton-raphson either find the maximum of the loglikelihood function or estimate the MLE and likelihood function? [closed]

Does using newton-raphson method or some other optimization method used in nlme package or mixed effects models actually find the maximum of the log-likelihood function's height or does it simply find ...
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Difference Between $L(θ | X=x)$ and $P(θ | X=x) $? [closed]

Although statistically different, I feel like they both say the same thing? $L(θ | X=x)$ = the probability that a sample provides support for particular values of a parameter in a parametric model. ...
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Choosing between two normal distributions

I have two normal distributions with different means and variances: N(u1, s1) N(u2, s2) And I have some data points (X) that were sampled from each of them. For each data point, I want to calculate ...
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How does likelihood differ from probability? [duplicate]

i.e. Tossing a fair coin $p_H =$ probability of heads $= 0.5$ $L(p_H | HH) = P(X = HH | p_H) $ $= 0.25$ Okay, right-hand of equation I understand, $P(X = HH | p_H) = 0.25 =$ probability of "$X = ...
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Likelihood of Poisson R.V with Bayesian Inference

In Bayesian Inference. Say I have a Poisson distributed R.V. K which denotes the observed number of some event over a specified time period. The rate of this ...
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Concavity of negative binomial GLM

I need to estimate the log-likelihood of the negative binomial regression. I mean full log-likelihood, including the dispersion parameter. The problem is: when I start LBFGS, BFGS, or gradient ...
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Log-likelihood using the link identity for poisson?

I understood the Log-likelihood using the link “log” for poisson, λ=exp(α+βx). But I can’t get the Log-likelihood in the case of “identity”, λ=α+βx. How do I get it?. The example is the following data....
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Expected value and Maximum Likelihood Estimation

I'm doing this exercise about Poisson distribution and maximum likelihood estimation: I have had no problem with points a and b, but I'm struggling with the correct answers of the C part. From my ...
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Choice of prior and combination with likelihood, sample from exp distribution

I am taking a course in Statistical Theory of Science. We have an exercise where we are comparing "Classical" and Bayesian parameter estimation. We have a sample $(x_1,...,x_5) = (0.28,0.30,0.94,0.42,...
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How to transform/convert likelihoods to scores?

I have the probability of loan default for a labeled dataset where the distribution of probabilities is heavily skewed. Labels are defined as "good/0" for no default and "bad/1" for defaults. My goal ...
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On likelihood functions and characteristic functions

Let me preface this by saying that if someone manages to provide a solution to my problem, I will forever be indebted to them, as this problem has driven me crazy. Let us first assume that the ...
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Extend likelihood equation to P(Y>=y) in R

This question involves both math and coding in R. Apologies if this should be on Stack Overflow, but I decided statisticians would ultimately provide better support....
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Is it not necessarily to put observed data at the right of the bar symbol $\mid$ when computing likelihood?

This Machine Learning course with timestamp is talking about PERFORMANCE MEASURE. Assume there are 3 loan officers A, B and C (correspond to you, blank, your "Friend" on the blackboard) , assigned ...
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Likelihood for aggregated survival data?

Most literature on survival models assumes that the data is either a collection of individual survival times or right-censored individual survival times (so you know when some subjects failed but for ...
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Using Likelihod Ratio vs Hypothesis Testing to test claims?

I don't know if I am missing something obvious but it seems likelihood ratio and hypothesis test serve the same function, to test claims about population parameters using sample data ( correct?). I ...
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Delimit the area in that parameter space that contains 95.4% confidence

Given the equation $y = fc + fe\times \sin(2\pi(x-t_0)/12)$ Considering the two parameters of amplitude fc and fe simultaneously, delimit (using the χ2 variation method) the area in that parameter ...
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Post-hoc t-tests for ANOVA

My sociology professor said that when performing multiple t-tests between two groups after the ANOVA f-test, the likelihood to make at least one type-1 error adds up by 5% with each t-test. So for one ...
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Likelihood ratio test for language modeling

I am trying to use a likelihood ratio test to evaluate whether one language model is significantly better than another. (Note: In the example below, the language models should not be very different, ...
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Invariance of maximum likelihood estimates to rearrangements of parameters/constants in the model?

I know that maximum likelihood estimates are invariant to re-parametrization (https://stats.stackexchange.com/a/335368/267430). Is the MLE also invariant to rearrangements of the constants and ...
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How to calculate 'times likely' for a percentage and whole number data set?

I'm analyzing speech data for single people vs those in group. I have two types of datasets for which I want to calculate "times likely". ...
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Likelihood of linear mixed effects model

Consider the following model $$\left \{ \begin{array}{l} y_i = x_i\beta + z_ib + \varepsilon_i,\\\\ b_i \sim \mathcal N(0, \Sigma), \quad \varepsilon_i \sim \mathcal N(0, \sigma^2), \end{array} \right....
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What is the “direct likelihood” point of view in statistics?

I am reading a Springer title from 1997 called Applied Generalized Linear Models by James K. Lindsey. In the preface, Lindsey writes For this text, the reader is assumed to have knowledge of basic ...
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In-sample likelihood cannot be lower for student-t model compared to Gaussian model

I am looking at the solution of a question that asks us to briefly comment on how we can compare two linear models with different distributed $\epsilon_i$. For the first linear model $\epsilon_i$ is t-...
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Bayesian update vs optimization

Say I have a multivariate normal vector $$ r \sim N(\mu , \Sigma ) \Rightarrow Pr \sim N(P\mu , P'\Sigma P ) $$ and I observe that $$ Pr = Q $$ Now I can use Bayes rule to calculate the ...
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Why do we sometimes define likelihood as $p(\textbf{T}|\textbf{X},w)$ and sometimes as $p(\textbf{X}, \textbf{T}|w)$?

Let's suppose we have a dataset $\mathcal{D}=(\textbf{X}, \textbf{T})$ where $\textbf{X}$ are the samples and $\textbf{T}$ are the targets. We want to find $w$ such that the likelihood is maximized. ...
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Derivation of score vector

Can anyone explain the process of this derivation, step by step? This derivation is from Joint Models for Longitudinal and Time-to Event Data by Dimitris Rizopoulos. \begin{equation} \begin{aligned} ...
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Likelihood function for a hyperplane

I have a hyperplane defined by $y = b_1x_1 + b_2x_2 + b_3x_3$. I have a list of $n$ data points $(y,x_1,x_2,x_3)_i, i=1\ldots n$. These data points have been measured with errors drawn from a ...
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Use of term 'likely' without log regression in corporate survey data

I work in the corporate research space producing white papers from survey data. When writing up results of these investigations, our writers are inclined to use the term 'likely' when referring to ...
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How to find manually the value of the likelihood function?

I have a statistic homework and this is my question: Outcome (binary)=f(age, number of books) And that I have four observations in my dataset: Observation 1: Outcome=1, age=0.5, number of books=5 ...
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Power of uniformly most powerful test

Let $X_{1}, \dots, X_{n}$ be independent random sample with cumulative distribution function $F_{\theta}(x) = 1 - 2^{-(x-\theta)}$ for $x > \theta$ and $0$ elsewhere, where $\theta$ is unknown ...
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Maximum Likelihood - Normal Errors - When is the Jacobian needed?

I am considering the following non-linear model $$h(z) - \lambda_0 - \lambda_1 z - \lambda_2x = v$$ where $v \sim \mathcal N(0,\sigma^2)$ unobserved error and where $\lambda_j$ are unknown ...
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How to integrate the marginal likelihood numerically?

Consider a log-likelihood function $\ell(\theta,b)$, where $b\sim F$. I want to calculate the marginal log-likelihood $$\ell(\theta) = \int\exp\left(\ell(\theta,b) \right) dF(b).$$ However, $\exp(\...
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How to determine likelihood of each observation from a fitted model in R?

Suppose you have the following data and model: ...
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Sampling distribution of the Score statistics of a GLM MODEL

In the context of a GLM (with a distribution that belongs to the exponential family), we often compute the score statistics $$ U = \frac{\partial LogLike(\boldsymbol{\beta};\mathbf{y})}{\partial\...
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How to compute statistical significance using a likelihood ratio?

The title really says it all. Suppose I have a change in log-likelihood (i.e., $\Delta LL = LL_{fitted} - LL_{null}$), and I would like to compute the $1\sigma$, $2\sigma$, etc. confidence region from ...
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Maximum likelihood estimator when $\sum_{j} \theta_j = 1$. How to impose this condition? [duplicate]

I have a sample $x_1,\dots,x_n \stackrel{iid}{\sim}f(;\boldsymbol{\theta})$, where $\boldsymbol{\theta} = (\theta_1,\dots,\theta_d)$, and , $0<\theta_j<1$, $\sum_{j=1}^d\theta_j = 1$. I can ...
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Explanation of Equation 5.3 from Gaussian Processes for Machine Learning

I am currently reading through C. E. Rasmussen & C. K. I. Williams' Gaussian Processes for Machine Learning and was going through chapter 5. I could not exactly understand the derivation of ...
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likelihood function of a line regression equation

I am a bit confused at how can i find the likelihood function and the solutions of likelihood function for a line equation, for example y=3x+15
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likelihood functions

Assuming that a linear regression model is fitted to 5 distinct datasets and that in all cases the regression line is y = ax+b+e . Will The likelihood functions are the same for the two different ...
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Calculating Likelihood from misfit?

I am performing a geophysical inversion where the result is an ensemble of models (~400000) with their misfits. I want to calculate the likelihood of each model from the misfit values which is given ...
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Are they cheating?

I'm trying to determine the probability that three students cheated on a recent exam. In a nutshell: What are the odds that three students sitting next to each other in a class of 22 could select (...
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Do adaptive lasso, group exponential lasso and composite MCP methods have the same optimization algorithm of the likelihood function?

As part of survival analysis and variable selection, I would like to compare adaptive lasso, group exponential lasso (gel) and composite MCP (cMCP) methods. In terms of cross-validation, do these ...