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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How to correctly address “ALERT: Iterations finished, maximum likelihood not found” in poLCA?

I intend to use Latent Class Analysis on a large dataset with 12 response categories and approximately 50,000 observations. I am getting an "ALERT: Iterations finished, maximum likelihood not ...
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Consistency of likelihood importance sampling estimator

In a lecture recently our lecturer described a method for approximating the expectation of a function over a posterior distribution using likelihood importance sampling. That is: $$ \mathbb{E}_{p(x|D)}...
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Is sequential reduction approximation of likelihood/importance sampling more accurate than adaptive gaussian quadrature or monte-carlo?

https://cran.r-project.org/web/packages/glmmsr/vignettes/glmmsr-vignette.pdf Is the approximation of the likelihood in glmm by sequential reduction approximation or importance sampling more accurate ...
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Bayes Factor and Likelihood Function Estimate

I am so stumped on this too: Imagine that a researcher computes a Bayes Factor for a point-null hypothesis against an alternative with a weakly informative prior centered on zero. The resulting Bayes ...
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Maximum Likelihood Estimator for a given density function

I have the following problem: Assume you observe $Y_1,...,Y_N$ independently from the distribution $f_y$: $$ f_{Y}(y)=\frac{12}{12-\theta}\left\{\begin{array}{ll}-\theta(y-0.5)^{2}+1 & \text { if ...
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What is the log-likelihood of the saturated model in Gaussian family GLM with identity link? Is it not defined?

This question is related to the unanswered one here: Estimate the variance for log likelihood of gaussian saturated model Consider the log-likelihood function in Gaussian family GLM with identity link:...
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Likelihood function and sampling distribution symbol

I read a introductory stat book that the likelihood function has symbol like , e.g., L(theta, x); while the sampling distribution has the symbol like f(x,theta). May I ask does anyone know whether the ...
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$t$-test and likelihood ratio test for testing the regression coefficient

I am studying hypothesis testing for the regression coefficient, it is given that The hypotheses for testing the significance of any individual regression coefficient, such as $\beta_{j},$ are $$ H_{0}...
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Analyzing the Sum of Two Independent Normal Distributions [closed]

Let's say I have two independent variables: X∼N(0,1) and Y∼N(0,2). Suppose X represents the starting position of a moving object on a straight path and Y represents it's displacement. If Z is the ...
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Running accelerated gradient descent on $\prod_{i=1}^{n}\alpha \beta y_{i}^{\beta - 1}exp(-\alpha y_{i}^\beta)$

Running accelerated gradient descent on $\prod_{i=1}^{n}\alpha \beta y_{i}^{\beta - 1}exp(-\alpha y_{i}^\beta)$ I have this code for running AGD on the above function in MatLab ...
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Is it meaningful to regularise a GEV log-likelihood?

Situation/Data: I'd like to start with an example from climate science. Suppose you have a univariate time series $\vec{z} = (z_1, z_2, ..., z_n)^T$, where $z_t$ are block maxima of time step $t\in1,.....
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Maximum Likelihood Estimator : special case of uniform which exclude the upper limit [duplicate]

I would like to understand why we can't get a MLE in this special case of PDF : "Sometimes it is not so easy to find the maximum of the likelihood function as in the examples above and one might ...
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How to plot $x^{1700}(1-x)^{300}$?

I'm trying to plot a Bernoulli likelihood function on R: $$x^{1700}(1-x)^{300}$$ But when I try to plot this function on R it looks like this: I think the maximum should be at 0.85, but it shows me a ...
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Likelihood of Student's t distribution

The integral form of student's t distribution is given as follows [1]: $p(x|\mu,\lambda,\nu)=\int_0^\infty \textrm{Normal}(x|\mu,(\lambda\eta)^{-1})\textrm{Gamma}(\eta|\nu/2,\nu/2)d\eta~~~~~~~~~~~~$ (...
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What is “Likelihood Principle”?

While I was studying "Bayesian Inference", I happen to encounter the term, "Likelihood Principle" but I don't really get the meaning of it. I assume it is connected to "...
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calculating multinomial likelihood

I have the counts for four nucleotides: a,g,c,t at each position and the corresponding frequencies in a string. I want to compute multinomial likelihood of observing a given count e.g. 'a' given the ...
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Probability that the sample comes from a certain distribution

Assume we have a data sample: $x_{1}, \dots, x_{n}$ from $n$ i.i.d. continuous random variables. Then, for simplicity, let us consider two distributions, $f(x)$ and $g(x)$. Is there any statistical ...
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Shouldn't -2*loglike of an lm object = deviance of that lm object in R?

I want to make sure my R code below is accurate. Because generally -2*logLik(lm_object) should equal deviance(lm_object). But in ...
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Can we derive cross entropy formula as maximum likelihood estimation for SOFT LABELS?

For hard integer labels {0,1}, the cross entropy simplifies to the log loss. In this case, it is easy to show that minimizing the cross entropy is equivalent to maximizing the log likelihood, see e.g. ...
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Batch posterior vs Recursive posterior

Consider for a unknown parameter $\Theta\in\mathbb{R}^n$ the following posterior density conditioned on a dataset $d=\{y_1, y_2\}$ of two generic measures $y_1,y_2 \in \mathbb{R}^p$ \begin{equation} ...
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How is the Fisherinformation derived for log odds ratio

In my biostatics book Im working with there is one derivation in an example of the fisher information of the log odds ratio which I do not understand. There are two independently, binomially ...
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Log Likelihood Converting from Sum Notation to Matrix Notation

I have found the log likelihood of a poisson GLM with the canonical link function to be $$\sum_{i=1}^n y_i \eta_i - exp(\eta_i) - log(y_i!) $$ Where $\eta = X \beta $. My question is how can I ...
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Particle filter for likelihood evaluation

I struggle to implement a particle filter to evaluate the likelihood of a textbook example. I got the following process: $x_t = \alpha + \beta x_{t-1}/(1+x_{t-1}^2) + w_t$ where $w_t \sim \mathcal{N}(...
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What is the relation between Minimum description length, Model evidence and Shannon's source coding theorem?

Given samples $(x_i,y_i)$ drawn independently from $P(x,y)$, we usually have in supervised learning framework the objective of minimizing\maximizing an approximation : $$min_{w} \{ \sum_iL(y_i,f_{w}(...
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Need help comparing whether one value is more likely than another in two different probability density functions

I have two histograms plotted from a simulation in Python, each of which has been normalized and fitted with a skew normal probability distribution (I'll call these functions A(t) and B(t) - the first ...
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What are some common prior/likelihood choices for Bayesian logistic regression?

I'm not really clear on the Bayesian approach to logistic regression. From everything I've read, the prior and likelihood can be can be whatever you want them to be. Well, I've a couple things; namely,...
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Finding partial likelihood for non-proportional model

When the hazard rates for treatment groups are not proportional, then we can use a model with two time-dependent covariates $$Z_1(t)=\begin{cases} 1 & \text{if chemotherapy only and} ~~t\le ...
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Are maximizing the log probability and assigning the ground-truth token the highest rank the same?

I have been reading this paper titled Neural Text Generation with Unlikelihood Training. It is about the maximum likelihood function used to train generative models. Anyway, it says that a major flaw ...
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The similarity between Mallows Cp and AIC?

It is possible to compute the log-likelihood used for AIC as $n /log(RSS/n) + const$ or even as $RSS/\sigma^2 + n\log(\sigma) + const$ considering the least-square or MLE scenario for linear and non-...
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Log-likelihood of a exponential distribution

I have an exercise that I don't quite understand: The life of 100 lamps has been measured. Each lamp has been used with a intensity between 0 and 1, where 0 is off and 1 is the maximum intensity. It ...
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Understanding censored likelihoods

I'm a little confused about how to interpret the maximum likelihood procedure for censored data. I'll write out an example, and then ask my question -- it's sort of a soft question, so my apologies if ...
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Measuring predictive uncertainty with Negative Log Likelihood (NLL)?

I see that in many papers about prediction uncertainty and calibration of neural networks, methods are compared in terms of the negative log-likelihood. What does it represent in this context? And why ...
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Can we maximize the log of the odds instead of log of the probability?

As above. This question relates to all optimization formulations in statistics.
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How can I derive the distribution of parameters from data?

For example, I know that my data comes from some normal distribution. I have data - measures (5.5, 4.9, 4.4, 5.3). Is it possible to tell the probability that the mean is less than 5? More generally, ...
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Help lme4. The p-value, obtained from the stargazer-table, where from is it computed? What does it say?

I am using the stargazer R package to produce a table to present my result from a linear mixed model. Have two questions: What does the log likelihood tell me? I ...
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Bayesian Likelihood function range

I have given a prior which has a range of data between[0.5,1.5] and a normal distribution N(0.9,1) which should truncate at the above range and normalize. also I have the likelihood distribution in a ...
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How to interpreting the log-likelihood values of Hidden Markov Model?

I have working with Hidden Markov Models for prediction purposes. I have used 8-HMMs each with 3,4,5 and 6 hidden states for a dataset. So, I have 4 sets: ...
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Reweigh a sample obtained through genetic algorithms to emulate rejection sampling

I have a function $f(X) \to \text{true/false}$ where $X$ is a parameter vector of large size (say 100 elements). What I'd like to do is to sample from the posterior of $X|f(X)=\text{true}$. Let's ...
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Do we ever take the log of a probability like we do with likelihoods?

I am trying to learn about naive Bayes by implementing a simple naive Bayes model to classify the titanic dataset (so a binary classification). To keep it simple for now I am just including the two ...
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How to describe an “incomplete” prior?

I would like to know how to describe sources of uncertainty neglected when I approximate a prior distribution $p(x)$ by a marginal distribution. Specifically, let's say that I have a marginal ...
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Log-Likelihood Computation for AIC & BIC

Considering $n$ observations that an be modelled by a Gaussian error model and two nested motion models with $p = 4$ and $p = 7$ parameters, I want to compute the log likelihoods $L$ given the Maximum ...
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If Y has an exponential family distribution show that $E(\frac{\partial L}{\partial \theta}) = 0$

I'm working in a self study fashion preparing for a course I'm going to take this semester in generalised linear models. The question is, given that the Y random variable belongs to the exponential ...
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Given that $X\mid Z=j \sim \mathrm N(\mu_j, \sigma_j) \ j=1,2$ derive the likelihood function of $X$ [duplicate]

Consider the concrete example where we flip a coin, with probabilities $\tau$ and $1- \tau$ for heads and tails respectively, and then let $X$ be the random outcome of a normal distribution whose ...
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Covariance matrix of regularized likelihood

My question is how to estimate the covariance matrix of parameters in a regularized likelihood maximization. Lets assume we have constructed some negative log-likelihood with a set of parameters and a ...
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Likelihood ratio test for mixed effects model

I'm currently struggling with how to assess the type I error of a permutation test for significance of variance term in R. The idea that I want to follow is outlined below: Suppose we simulate data ...
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How do I create a beta likelihood that will work for MCMC?

I am attempting to compare two beta distributions under a Bayesian framework. (My data are survival rates, so they fall between 0 and 1 and are best fit with a beta distribution. After estimating ...
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Impossible to get about 70% of height of Likelihood when I project the edge 1 sigma joint distribution on the 1D Likelihood

This post has been initially asked on maths.exchange but I didn't get any help, so I try to transfer it on this forum hoping someone could help me (I am going to delete the post on maths.exchange to ...
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Can we directly use the likelihood function for Bayesian inference? [duplicate]

I'm new to statistics, so this might be obvious. When doing full Bayesian inference, we first compute the parameter posterior $P(\theta | \text{data})$ using Bayes' rule. Then, to make a prediction, ...
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Create log likelihood function for three classes for SEIR modelling

I am working on SEIR model for measles data which has the following variables and sample observations. ...
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Is the likelihood term in Baye's rule a conditional probability? [duplicate]

Baye's rule allows us to estimate the distribution of model parameters $\theta$ given the data that we have seen. where $$p(\theta |D) = \frac{p(D|\theta)p(\theta)}{\int p(D|\theta)p(\theta)d\theta}$$ ...

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