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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 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} = \sqrt{\frac{1}{-3\log(x)+\log(1+x^3)} } $ Is this correct? Thanks:-) ...
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Formulating the likelihood in a particular likelihood ratio test

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, ...
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How is it possible for both the likelihood and log-likelihood to be asymptotically normal?

I was trying to understand asymptotic normality of the posterior better, and came across a confusing point. So let's say we have a likelihood, $L(\theta | X) = \Pi_{i=1}^n p(X_i | \theta)$, so the log-...
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Finding Fisher information [duplicate]

Let $X$ distribution belongs for the family $\mathcal{P}\{P_{\theta}, \theta \in \Theta \}$. We need to find Fisher information $I(\theta)$ according $n$ simple sample, when $P_{\theta}$ is $N(\mu,\...
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Terminology Confusion: Probability, Likelihood, AND?

currently I have a terminology issue to accurately write a text. I read up on the definitions on probability and likelihood. Given some continuous random variable(s) as far as I can infer, probability ...
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Modelling startups' funding journey with Brownian Motion

I am trying to implement a "light" version of a paper (Hunter, Saini & Zaman 2017), in which the authors build a model capable of predicting the probability that a startup will exit (either by ...
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Redefining latent variables as observed data

This was just a thought that occurred to me, but technically, is it possible to redefine what I treat as latent variables and what I treat as data? For example, lets assume I have a set of latent ...
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Log Likelihood and parameter space

I'm taking a 3rd year statistical theory class in which we're introducing the likelihood function with a bit more mathematical rigor. Under the discussion for maximizing the likelihood function $\...
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What does the degree of freedom (df) mean in the results of log-likelihood `logLik`

After doing the regression using lm for fixed effect model or lmer for mixed effects model, I pass the results to the ...
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Likelihood modification in Metropolis Hastings ratio for transformed parameter

I want to use MH to get samples from $p(\theta \mid y) \approx p(y \mid \theta) p(\theta)$. Let's assume $\theta$ is heavily constrained and I transform $\theta$ to $f(\theta)$ so I can sample from ...
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MLE of $f(x\vert\theta)=1/\theta$, $x_1 , \cdots , x_n \sim U(0,\theta) \;\;, \theta>0$, [closed]

Original question $x_1 , \cdots , x_n$ are independent random variables, identically distributed as a uniform distribution over $(0,\theta)$. $$ f(x \vert \theta) = \frac{1}{\theta}, \; 0<x<\...
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Estimating likelihood from the Residual Sum of Squares

I'm start studying Bayesian statistics, but I've found that I'm having troubles with the likelihoods. Let's say that I have a vector of observations $y$ and I want to calculate how likely it is $y$ ...
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How can I compare parametric and semiparametric survival models?

On a given dataset, I am running a semiprametric Cox proportional hazards model, together with a series of parametric models (Weibul, gamma, lognormal, exponential, etc.). How can I know which is ...
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Understand a statement about likelihood function

I'm reading Agresti - Categorical Data Analysis and it says Consider two models, $M_0$ with fitted values $\hat{\mu}_0$ and $M_1$ with fitted values $\hat{\mu}_1$ with $M_0$ a special case of $M_1$....
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Likelihood for a test data (sequence of characters) given two unigram models

I would like to find the likelihood of a sequence of characters (the test data), given two unigram models. The sequence (test data) is: A B C B B The models ...
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Do we maximize likelihood or likelihood ratio for ML estimation? [closed]

I was reading link. And I rewrite (3), here, in link to simplify notation as follows $$ \Lambda(X) = \frac{\mathcal{L}(\lambda_S | X)}{\mathcal{L}(X)} $$ Here $\lambda_S$, variance in presence of ...
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Is likelihood also defined as ratio of pdfs

My understanding of likelihood is that it is pdf except that it is a function of parameters rather than observations (as in link). I was reading link. Can likelihood be defined as ratio of ...
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Given a likelihood and a set of possible parameters, determine probability a sample point was generated by a particular parameter?

Suppose I have some probability density function $f(x | \theta)$ depending on a parameter $\theta$. I also know that $\theta$ must assume one of some finite number of values, for simplicity, let's let ...
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How to convert IRT theta score to a percentage score

I am trying to implement an adaptive test using 3PL IRT model. We need to screen the candidates and label their expertise as ...
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Likelihood is “proportional to a probability”. Which one? [duplicate]

In various places (see quotes below) it says that the likelihood is "proportional to a probablility". Which probability is it proportional to? In the context of Bayes theorem, it is not proportional ...
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Likelihood is not “proportional to” a single probability density?

In various places it says that the likelihood (e.g. in the Bayes formula) is "proportional to a probablility". For example https://alexanderetz.com/2015/04/15/understanding-bayes-a-look-at-the-...
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Maximum Likelihood Estimator (MLE) for $2 \theta^2 x^{-3}$

I'm having a bit of trouble solving this. $$ f(x_i; \theta) = 2 \theta^2 x_i^{-3}, 0 \le \theta \le x_i \lt \infty $$ I start by finding $f(\textbf{x}; \theta)$: $$ f(\textbf{x}; \theta) = \prod{f(...
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Measure of accuracy for a Bayesian model

I am reading Statistical Rethinking (Section 6.2.1.2). The topic of this section is measuring accuracy for a Bayesian model, i.e. accuracy of the model of predicting correctly an outcome. The ...
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what does non-nested copulas mean?

I read from "Pair-copula constructions of multiple dependence", the following statement: "The likelihood of the Clayton copula is lower than that of the Student copula (39.72 vs. 47.81). However, ...
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Likelihood function vs probability distribution function [duplicate]

I've been reading about Bayesian statistics and data analysis, and constantly see that $\text{posterior} \propto \text{prior} \ \times \text{likelihood}$. I'm familiar with fundamental probability and ...
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Bayesian- How to determine the distribution for likelihood function [closed]

I have a question about likelihood model. Given I have a set of data, how do i find out what type of distribution is suitable for my likelihood function? (eg. Poisson, exponential etc.)
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Likelihood raised to a power; how to set the power?

Suppose ${\bf{\theta}} = (\theta_1 , \ldots, \theta_d)$ and you have a posterior as below: $$\pi(\theta | D ) \propto L(\theta |D ) \pi(\theta)$$ Suppose we are in active learning setting and need ...
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Likelihood function for linear regression

For linear regression, the likelihood function can be found with: However if your data points are multi-dimensional such that x,...
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41 views

How to interprete likelihood equal to 1

I'm trying to interprete the Example 1 from the wikipedia page: the likelihood function of a coin flip with a single parameter p expressing how likely a head will come up. The likelihood is defined ...
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Deriving likelihood function of binomial distribution, confusion over exponents

This question focuses on a specific aspect of this one: How to derive the likelihood function for binomial distribution for parameter estimation? In my own derivation, I start with: $$f(x\mid p) = ...
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Is the Quadratic Approximation of Log-Likelihood Equivalent to the Normal Approximation of the MLE?

Let $X_1, X_2, ..., X_n \sim \text{IID N}(\theta, \sigma^2)$ with $\sigma^2$ known, and let $\hat{\theta}$ be the MLE of the mean. (1) How can I show that in this case, the following is true? $$\...
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Likelihood function when only $\max_{1\le i\le N}X_i$ is observed and $N$ is parameter

Let $X_1,X_2,\ldots,X_N$ be i.i.d random variables having $\text{Exp}(1)$ distribution where $N$ is unknown. Suppose only $T=\max\{X_1,X_2,\ldots,X_N\}$ is observed. I have to derive a most ...
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Likelihood-Based Confidence Intervals for ratio of risk ratios

Ratio of Risk Ratio (RRR), along with Relative Excess Risk Due to Interaction (RERI), has been used to quantify the joint effects of 2 exposures in epidemiology. Quoting Joshua N. Pritikin CIs ...
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Computing the Hessian of maximum log likelihood function

I am trying to find the Hessian matrix for the maximum log likelihood function given training data {(xi, yi)} for i=1:N with yi ∈ {+1, −1} for each i = 1, . . . , N for the function: When I try to ...
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HMMs: How to Interpret The Average Likelihood Of My Data

I have recently trained an HMM using R's depmixS4 package, and am evaluating its performance via the average likelihood of my data. The equation is provided below: However, I noticed the average ...
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40 views

Unbiased estimator for Theta of a Normal Distribution

If $X_1,\ldots,X_n\sim \operatorname{iid} \operatorname N(\theta, \sigma^2)$, then verify that $\bar{X}_n$ is unbiased estimator for $\theta$ and that Cramer Rao bound is met? I am facing difficulty ...
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114 views

Why is the deviance defined with a factor 2 (or likelihood ratio squared)?

Deviance is defined as I see the motivation in why we would define the deviance as a difference of logLikelihoods or just the log(Likelihood Ratio), but why the factor 2? Why square the ratio? Does ...
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1answer
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Calculation of the likelihood in the Boxcox transformation

I have an outcome which may be transformed in cases of egregious departures from normality to its Box-Cox optimal normal transformation in an unconditional model. How is the likelihood calculated for ...
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Infill likelihood for a continuously observed continuous-time process

Consider a continuous-time stochastic process $y(t)$ having the following linear (Gaussian) state-space representation for $t \geq 0$ $$ \left\{ \begin{array}{c c l} \text{d}{\boldsymbol{\...
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An example where the likelihood principle *really* matters?

Is there an example where two different defensible tests with proportional likelihoods would lead one to markedly different (and equally defensible) inferences, for instance, where the p-values are ...
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1answer
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From likelihood P(Data|H) to P(H|Data)

If there are four possible hypotheses and I calculate the likelihood of the data given each of these hypotheses, can I calculate the probability of one of the hypotheses as ...
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1answer
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Is it ever convenient to maximize different functions of the likelihood than the logarithm?

We all know that it's often much more convenient to maximize the log-likelihood rather than the likelihood to get a parameter estimate, since it amounts to the same thing by the fact that the ...
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210 views

The likelihood function: Why is it no pdf? [duplicate]

I know that there have already been a lot of questions about why the likelihood is no probability density function and I ve read most of the answers. However, to me the point is still not clear yet ...
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Likelihood of one datapoint given $k$ models

Introduction I'm currently facing a problem where I'm constraining a set of (physical) parameters $\theta_k$ with $k\in [1,2,...,K]$ via several independent datasets. One of those datasets, however, ...
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Writing Likelihood of Poisson in R

Here is my attempt to make the likelihood function for Poisson distribution for data x and parameter theta in R: ...