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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When to use the full and the conditional likelihood

In the context of estimating parameters of a time series model, we may consider either the full likelihood or the conditional likelihood. I was wondering 1) when does one use the full ...
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Is this definition of likelihood presented in an arxived article correct?

In a recently arxived article the following definition of the likelihood is given: The Bayes rule provides the posterior probability $p(h_i|o)$ for each hypothesis given the observation: $$p(...
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How to make sense out of integration over discrete data points?

Looking for a proof of the expected value of the score function equating zero, I came to this document that was recommended in another answer. Considering that we have a sample of n x_i values, I ...
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Total variance on binomial statistic with uncertain probability

Consider an experiment to measure a random occurrence with a probability $p$ to pass some selection criteria. If I have $N$ independent trials and $p$ is known, this is a standard Binomial ...
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How to deal with very small probabilities in likelihood? [duplicate]

Similar as in this thread, I have a problems with small probabilities in likelihoods, but I believe the thread does not apply to me. The likelihood is a product of probabilities. The log-likelihood ...
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Maximum likelihood second derivative test

Can anyone explain what to do if the maximum likelihood second derivative test comes back positive such that the M is a saddle point instead of a maximum value? What do I do after that to figure out ...
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Likelihood Ratio Test for Equality of two proportions

Suppose we have two i.i.d samples from $X_1,...,X_n$ and $Y_1,...,Y_n$ where $$X_1,...,X_n \sim B(n,p_0)$$ $$Y_1,...,Y_n \sim B(n,p_1)$$ I want to show that we get the familiar test of equality of ...
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Likelihood ratio test for GENLINMIXED in SPSS?

I am analyzing some linguistic data using SPSS 25, and have encountered some problems. Please note that I am not a statistician, so excuse me if my question is trivial. Basically, I am using the Mixed ...
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Differentiation with respect to beta j?

I'm not sure how to differentiate the log-likelihood with respect to beta j, so I can find the MLE? Can anyone help me?
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Prove that $T(\textbf{X}) = \hat{\sigma}^{2}$ reaches the Cramer-Rao bound

Let $X_{1},X_{2},\ldots,X_{n}$ be a random sample whose distribution is given by $\mathcal{N}(\mu,\sigma^{2})$, where both parameters are unknown. (a) Prove the normal probability density function ...
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deriving likelihood function for hierarchical bayesian model

I'm struggling with hierarchical bayesian modeling. I need to derive a full likelihood function for the given hierarchical structure of the model. $a_{it}|\lambda_i\sim TN(\lambda_i,\beta)$ $x_{it}|\...
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Likelihood ratio statistic lognormal

I want to determine the LRS lambda of a lognormal distribution under H0: the variances are equal. What I have so far is the following: $H_0: \sigma^2 = \sigma_{o}^{2}$ $H_1: \sigma^2 \neq \sigma_{o}^...
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How to prove or disprove that $T(X_{1},X_{2}) = X_{1} + X_{2}$ is a sufficient statistic

Let $X_{1},X_{2},\ldots,X_{n}$ be random sample from a population whose distribution is given by $X\sim\text{Bernoulli}(\theta)$, $0 < \theta < 1$. a. Show that $T(x) = \displaystyle\sum_{i=1}^{...
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Likelihood ratio tests for quasi- models

I have been playing around with over-dispersion in binomial data and looking into qausi-binomial models as a solution. When comparing binomial models through the change in deviance, I can reproduce ...
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How to calculate Kernel Density for Bootstrap Likelihood

I am attempting to write R code to generate bootstrap likelihood as described in section 3 of this paper https://arxiv.org/pdf/1510.07287.pdf. I am confident that I performed the bootstraps correct, ...
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Homogeneity test in survival meta-analysis with adjusted HR

I was wondering what was the way to measure heterogeneity in a survival meta-analysis. I have for each study, an hazard ratio calculated with a penalized cox model (...
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Trying to find the likelihood of a certain scenario

An experiment has been carried out 5 times and the results are shown below. I am wondering if it is possible to predict the likelihood of Y being greater than X if the experiment were to be carried ...
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SD of a likelihood function: can it replace the Standard error of a sampling distribution

I was wondering if "standard deviation" of a "likelihood function" could ever represent the "Standard error" of a "sampling distribution"? I ask this, because when one follows a Bayesian approach ...
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Targeted Maximum Likelihood Estimation for dummies?

I have tried to get my head around the concept of TMLE, but most references seem to be written by people who despise being understood (or maybe I am just hebetudinous). I have tried to read the paper ...
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Bayesian Inference, Posteriors Priors and Likelihoods [duplicate]

So at the moment I'm reading through my notes on Bayesian Inference and I'm really just not understanding anything. If anyone has any good websites that explain this topic well then I'd be really ...
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Gradient of marginal likelihood of Gaussian Process w.r.t likelihood parameters with Laplace approximation

The derivation of gradient of the marginal likelihood w.r.t covariance function hyperparameters $\theta$ is given in http://www.gaussianprocess.org/gpml/chapters/RW5.pdf, page 125. However, the ...
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Showing the sample mean is a sufficient statistics from an exponential distribution

Suppose that the lifelengths ( in thousands of hours) of light bulbs are distributed Exponential($\theta$), where $\theta>0$ is unknown. If we observe $\overline x = 5.2$ for a sample of $20$ light ...
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1D Bayesian Inference clarification

I'd like some help making sure I understand a 1D Bayesian inference problem. Say I have a data vector which is an array of the number of flu cases reported weekly in California for the past 10 years. ...
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1answer
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What does it mean when the Negative Log-Likelihood returns infinity?

I am using the package mvtnorm and using dmvnorm to calculate the negative log likelihood in R by doing ...
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1answer
42 views

Likelihood - not a pdf and not normalized?

I am reading the book "Patter recognition" by Cristopher Bishop. At Chater 2.3.6 "Bayesian inference for the Gaussian", there is written The likelihood function, that is, the probability of the ...
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What is the likelihood function of having heads 8 times out of 10 toss

Question What is the likelihood function for the event where 8 heads observed after 10 coin tossing? Is below in Python/Scipy using (scipy.stats.binom) correct? ...
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Compare model fit (-2 log likelihood), is my model significantly better?

I have 6 models with -2 log likelihood as an indication of model fit. The fit increases from model 1 to 6, with model 1 having the worst fit (lowest -2 log likelihood) and 6 the best fit. How do I ...
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How can we call likelihood function a joint PDF when the individual terms do not represent Probabilty

My understanding of the Probabilty Density Function is that they evaluate to 0 at a particular point . So if we have some i.i.d points $x_{n}$ from a Normal distribution and we write : \begin{equation}...
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log-likelihood of normal distributed fitted using MLE

Suppose we fit a normal using MLE which means we have parameters $$\mu = \frac{1}{n}\sum_{i=1}^n x_i$$ and $$\sigma^2 = \frac{1}{n}\sum_{i=1}^n(x_i - \mu)^2$$ Then we calculate the log-likelihood as ...
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Plot the approximate the likelihood function around the MLE using Gaussian

Say we are looking to estimate $\theta$ for a variable with distribution $f(x|\theta) = \frac{\theta}{x^{\theta+1}}$, $\theta>0$, $x>1$ The method of moments estimate for ${\theta}$ is $\hat{\...
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1answer
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likelihood of latent state space model

Im trying to calculate the likelihood function of my latent state space model. My model has Poisson observations $p(y_t|\beta_t;x_t) \sim \mathcal{Poiss}(z)$. where $z$ is the rate of the poisson ...
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1answer
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Likelihood of a random variable vs. Likelihood of a sample

As mentioned in the title, I am confused over the difference between $L(\theta|S)$ and $L(\theta|X)$, where $X = (X_1, X_2, ... ,X_n$). From what I understand, $L(\theta|S)$ is the probability of $\...
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what does logLikelihood signifies something in logistic regression?

I am getting logLikelihood as -1500 for one of the logistic regression analysis. What does it mean statistically?
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1answer
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If $X_1, \ldots X_n \sim N(\mu, \sigma^2)$ and we only observe $(X_1 - \bar{X}, \ldots, X_n - \bar{X})$, can we learn about $\mu$ or $\sigma^2$?

Suppose $X_1, \ldots X_n \sim N(\mu, \sigma^2)$ and we only observe $(X_1 - \bar{X}, \ldots, X_n - \bar{X})$, such that each observed point is now mean centered. Can we still learn about $\mu$ or $\...
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1answer
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trouble creating a negative log likelihood for a linear model in R [closed]

I am new to stats and R and I am having trouble figuring out how to calculate a function that gives me the NLL of a linear regression.
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How do I calculate the negative-log likelihood of a linear regression model in r? [closed]

I need to calculate the NLL of the regression and minimize it. How I can do this in R?
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Determining difference in vector of coefficients for a model according to likelihood ratio test in SPSS

I have performed logistic regression on my data set designed to explore significant independent variables (IVs) (Age, Gender, disease severity, co-morbidity, etc) associated with all-cause 30 day ...
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1answer
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Questions about the likelihood in probabilities?

Many define the likelihood of the data something like $\prod_{x} p(x|\theta)$ others like $p(x|\theta)$. Is the likelihood defined for one sample point/data element (like one document from a ...
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1answer
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Plot Scaling Problem

I am trying to implement the following code in R $\theta \sim Beta(a=1,b=1)$ $x|\theta \sim Bin(N=5,\theta)$ $\theta|x \sim Beta(a+\sum x_{i},b+\sum N-\sum x_{i})$ ...
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Behavior of likelihood far from peak

In the neighborhood of the maximum likelihood point the log-likelihood function is often fruitfully expanded as a quadratic function of the parameters. Are there any general results about the shape ...
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1answer
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Ignoring the normalising constant in Bayesian MCMC

This post relates to my original question here, but this time focusing on a more fundamental misunderstanding of how MCMC actually works. When using Bayesian MCMC for parameter inference with a ...
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How to compute a combined likelihood involving multiple datasets?

I have a parametric model that generates $N$ time series, and my goal is to try to find a model where the $N$ predicted time series match $N$ observed time series relatively well. My physics-based ...
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Combining “unbalanced” likelihoods in a “process-based” model

I have a "process-based" water quality model, which is essentially a black-box full of differential equations describing various chemical and hydrological processes. The model is deterministic and ...
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1answer
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What is the Maximum Likelihood Estimator of $f(x;\theta) = \frac{3\theta x^{3\theta -1}}{(1+x^{3})^{\theta +1}} $

$f(x;\theta) = \frac{3\theta x^{3\theta -1}}{(1+x^{3})^{\theta +1}}, x>0, \theta>0 $ I came up with FOC: $ \hat{\theta} = \frac{1}{-3\log(x)+\log(1+x^3)} $ Is this correct? Thanks:-) I took ...
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360 views

Formulating the likelihood in an LRT involving geometric and exp distribution

Here is a question from old exam papers I am having trouble with: Suppose the lifetime of a bulb is distributed exponentially with mean life $\theta$ (in hours). Let $X_i$ be the number of trials ...
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Likelihood Ratio Tests Hogg

Let $n$ independent trials of an experiment be such that $x_{1},x_{2},...,x_{k}$ are the respective numbers of times that the experiment ends in the mutually exclusive and exhaustive events $A_{1}, A_{...
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Likelihood deviation on diseased test

I feel confused with the following problem: the population $n=237$ diseased people are required to perform a 6-successive-day diagnostic test on cancer. And the random variables are given as follows: ...
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MLE of an EGARCH(1,1) Model with Gaussian Innovations

I want to estimate parameters of an exponential GARCH(1,1); namely EGARCH(1,1) model using optimization tools at R. However, I don't want to use a ready package like rugarch or an another package. I ...
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MLE for Overdispersed Poisson

I searched for a while on Google and this website for an answer to this question. I have an overdispersed Poisson distribution and a "hand-wavy" proof is giving me problems. Below is the information ...
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MLE question: summing likelihoods rather than log-likelihoods? [duplicate]

Usually, we are trying to maximize a sum of log-likelihoods in MLE. This is equivalent to maximizing a product of the likelihoods. Given that there's a product, we are assuming independence. However, ...