Questions tagged [likelihood]

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

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How to make the profile likelihood model for estimation?

I tried to make the age estimation model using the chemical compound results from The soil. Initially, I used the multivariable regression model. However, the reviewer highly recommend using the ...
user21268575's user avatar
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To what extent can likelihood methods be used for functional responses?

Let's suppose that we are working with a functional data set, $Y_i(t)$, $Y_i\in L^2[0,1]$, $1\le i\le n$. If we were working with univariate or even multivariate data set, likelihood methods would ...
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Is Pitman-Koopman-Darmois Theorem valid for discrete random variables?

I am interested in the Pitman-Koopman-Darmois theorem. I'm having a hard time finding a simple rigorous version of this theorem as I struggle finding sources. This helpful post provides three sources ...
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Turning a list of cost into categorical probability mass distribution

Background Given a noisy dataset $D$, I have to solve a classification problem where the possible anserwer is $i\in\{1,\dots,N\}$. So far I can get pretty decent result with an algorithm that, based ...
matteogost's user avatar
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Related to consistency of MLE: why $E_{\theta_0}\log\frac{f(X,\theta)}{f(X,\theta_0)}<0$?

Assumptions. $\theta_1\ne\theta_2\Rightarrow F_{\theta_1}\ne F_{\theta_2}$ The set $(x:f(x,\theta)>0)$ does not depend on $\theta$ For a.e. $x$, $f(x,\theta)$ is a differentiable function of $\...
reyna's user avatar
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Negative log likelihood cholesky decomposition

The code in question comes from Spatial Data Science by Pebesma and Bivand. I can pretty much understand what it says except for the last line. How does this give the negative log likelihood? I've ...
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Maximum-LIkelihood Estimation with NLL using parameters in Logscale

Consider the following toy problem: I have a $\mathcal{C}^\infty$ function $f(t,\Theta):[0,t_{\max}] \times (0, \infty)^n \to \mathbb{R}$. I choose some "correct/ground-truth" parameter ...
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Interpretation question about the article "Finite mixture modeling of censored data using the multivariate Student-t distribution"

I am reading this article and I am struggling to understand the following passage from page 6: \begin{align*} L_{i}(\boldsymbol{\theta}|\textbf{V}_{i},\textbf{C}_{i}) = f(\textbf{V}_{i}|\textbf{C}_{i},...
user1234's user avatar
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Likelihood principle and inference

I've been reading Casella and Berger's Statistical Inference. In section 6.3 the author stated the likelihood principle: if the likelihood functions from two samples are proportional, then the ...
INvisibLE's user avatar
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Trading card game poss

A trading card game that is called Flesh and Blood (description and rules here) has two players construct a 60 card deck and the hand limit is 4. Player 2's Deck color combination: 60 cards in deck; ...
Patrick's user avatar
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Understanding the reasoning behind Tobit composite likelihood

The Tobit model defines a likelihood for what amounts to a censored normal distribution - i.e. the tails are clipped - which is meant to be applied to data that has undergone censoring and thus has an ...
anymous.asker's user avatar
10 votes
1 answer
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Reconciling optimisation for log-likelihood and Brier score

Both log-likelihood and Brier score are proper scoring rules. As such, they reach the optimum when the predicted probabilities match the true ones. Since there is only one true probability for each ...
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How to derive marginal likelihood from prior and likelihood

Given the following: X|θ ~ N(θ , 1/k) model M0 stating that θ = θ0 model M1 stating that θ ~ N(u , 1/z) How to show that X|M1 follows a normal distribution with mean u and variance 1/k + 1/z? I ...
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Estimate the likelihood of two continuous samples of unknown distribution

Consider two continuous and unknown distributions $$X : {x_1, x_2, ..., x_n}$$ and $$Y : {y_1, y_2, ..., y_n}$$ both can be tagged as time series with $n > 8000$. I need to estimate the likelihood ...
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Parameterization of Negative Binomial for Dynamical System Model Calibration/Fitting

I'm studying about the applications of bayesian inference to fitting dynamical systems to observations. So the model itself is a deterministic SIR model: $$ f(R_0,D_{inf})=\begin{cases} \frac{dS}{t}&...
Derf's user avatar
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Why is everything based on likelihoods even though likelihoods are so small?

Suppose I generate some random numbers from a specific normal distribution in R: set.seed(123) random_numbers <- rnorm(50, mean = 5, sd = 5) These numbers look ...
Uk rain troll's user avatar
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Correct way to compute denominator in the bayes theorem

Say I have two data points and two Gaussians that these might be coming from. What is my evidence in this case? $P(X) = \prod_{i=1}^{2} (p(xi|\mu1)p(\mu1) + p(xi|\mu2)p(\mu2)) $ Because my samples ...
figs_and_nuts's user avatar
1 vote
1 answer
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Interpreting AIC relative likelihoods ( qpcR::akaike.weights() )

I want to ensure that I am correctly interpreting AIC relative likelihood (RL) scores, specifically those returned by qpcR::akaike.weights$rel.LL. For example, I ...
PhelsumaFL's user avatar
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Likelihood and Probability [duplicate]

So to my understanding, Probabilities are areas under fixed distribution, which can be expressed mathematically probability of data given distribution: $$P(Data|Distribution)$$ Whereas likelihoods ...
Rust32627's user avatar
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Connection between Z-Score and Quasi Likelihood?

Consider the Quasi-Likelihood function (I wrote it in the GEE context): $$ Q(\beta; y) = \sum_{i=1}^{n} (y_i - \mu_i(\beta))^T V_i^{-1}(\beta, \alpha) (y_i - \mu_i(\beta))$$ where: $y_i$ is the ...
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Estimating a time series' likelihood of a missing observation

I have this time series of seasonal monthly data, which sometimes has missing observations. The likelihood of an observation being missing is mostly dependent on the month, but also of the value had ...
BlackNinja's user avatar
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Fitting GLMM using lme4 package, fitting algorithm

I'm a little confused here. So I'm using glmm model to fit user/item interaction. If user #12 liked movie #15 it's 1 otherwise it's 0. Here is my model:  ...
Efecan Bahcivanoglu's user avatar
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Maths Questions - Probability distribution of choices

I need help with a question I am trying to work out: Individuals have 5 choices: Choice 1: utility1 = alpha * R1 + beta * C1 + random_shock_1(mu=0,sig=1) Choice 2: utility2 = alpha * R2 + beta * C2 + ...
Laurence_jj's user avatar
6 votes
2 answers
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Compare failure rates across multiple systems

I work in pharmaceutical manufacturing and one part of a process is a filtration step that uses 'clusters' or 'sets' of single-use (disposable) filters in parallel - the product flows into a manifold ...
ChemEnger's user avatar
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Box-Cox data transformation formulas for one and two-way ANOVA [duplicate]

I want to write my own program with box-cox data transformation for ANOVA. I don't understand how to perform a Maximum Likelihood Estimation of λ for my case. I found out how to find optimal λ for ...
trapped_in_a_corner's user avatar
1 vote
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Understanding the Binomial likelihood notation

Let $X \sim Bin(n,\pi)$. I don't understand why the binomial likelihood is then given by $f(x|\theta)=\binom{n}{x} \theta^x (1-\theta)^{n-x}$. Shouldn't it be $B(x|\pi,n)=P(X=k)=\binom{n}{k} \pi^k (1-\...
BlankerHans's user avatar
2 votes
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60 views

Random effects model and log likelihood

Hello I am struggling a bit understanding the random effects model. What I understood is that we fit a model but allow for sysematic differences of the variance of residuals per group. I would ...
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Why do Likelihood Functions sometimes have Integrals?

Suppose we have a Mixed Effects Regression Model: $$Y = X \beta + z b + \epsilon$$ $$b \sim N(0, \Sigma_{b})$$ $$\epsilon \sim N(0, \sigma^{2}I)$$ $$\Sigma_{b} = \begin{bmatrix} \sigma^{2}_{b1} & ...
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Deriving log-likelihood contribution involving fixed effects (panel model)

I would appreciate some help with the following problem: We have the following panel model: $y_{it} = h(x_{it}\beta + c_i) + \epsilon_{it}, \quad t=1,\ldots,T, \quad i=1,\ldots,N $ where \begin{align*}...
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Marginalising and taking the log of the likelihood

Say you have a likelihood function $L(X|\theta_1,\theta_2)$ and you marginalise out $\theta_1$ and then take the log of the marginalised likelihood function. Would this be the same as taking the log ...
HansKemper's user avatar
1 vote
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Maximum likelihood when the data is a parametric function of fixed data parameterized by the parameter [closed]

Normally in maximum likelihood estimation, we have fixed data $x$, and have a parametric family of models governed by $\theta$. The $\theta$ chosen is the one that maximizes $p(x \mid \theta)$. But my ...
Alojaco's user avatar
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How to deal with extremely smooth (Plateau) log likelihood?

I have a log-likelihood function with four parameters, out of which the variability of the likelihood with two parameters is extremely smooth. It suggests that the derivative is small for large ranges ...
CfourPiO's user avatar
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9 votes
4 answers
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An interesting observation regarding the log transformation of data

I stumbled upon something interesting while attempting to do a log transformation for some data (with zeros) today. It seems that there must be a good reason for this that I'm just not seeing. I'm ...
knrumsey's user avatar
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3 votes
1 answer
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Expectation of the Gaussian likelihood

I'm working on a challenging machine learning problem, where I need to find the expectation of the likelihood of one Gaussian, given the parameters of another. Apologies if any of the notation is ...
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Is the score function in Langevin dynamics related to the informant (score)?

In Langevin dynamics and diffusion models we see the score $\nabla_{\mathbf{x}}\log p(\mathbf{x})$. Here the notation seems to suggest we're taking partial derivatives w.r.t the free $\mathbf{x}$ ...
flinty's user avatar
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3 votes
1 answer
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Analogue of landscape conjecture in likelihood theory or Bayes?

The so-called landscape conjecture in machine learning says that in high dimensions, most critical points of the loss surface are saddle points rather than poor local minima. Out of curiosity I was ...
Durden's user avatar
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1 vote
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AIC score and purpose of logLik to compare residual variance of models

I am comparing the residual variance of two regression models, one fitted with a simple linear regression model and the other fitted with a mixed-effects model using the lme4 package. I am using the ...
four77's user avatar
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Maximum likelihood in linear regression

My understanding is that when we do maximum likelihood we want to choose parameters $\theta$ such that the probability of observing the actual, fixed data is maximized. That's how I understood it ...
AdmiralMunson's user avatar
2 votes
3 answers
155 views

What exactly is likelihood? [duplicate]

My understanding about likelihood, given some reading, is that it is how likely we are to observe the actual data given a certain parameter or parameter values $\theta$. Like with the coin toss ...
AdmiralMunson's user avatar
2 votes
1 answer
91 views

pdf vs probability vs likelihood [duplicate]

How to compute the log likelihood? Let's take a simple example using a normal distribution and scipy to do the work. Assuming X is the data, and the normal distribution as the model (...
brice rebsamen's user avatar
4 votes
4 answers
226 views

Interpretation of Maximum Likelihood Value

I have a question about Maximum Likelihood values, and how to interpret them. In order to explain the question, please see the Figure below. I will add explanation for how this figure has been created ...
user3728501's user avatar
2 votes
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Unbiased estimate of log-likelihood of Markov bridge

Note: I have cross-posted this question to MathSE. I have the following problem I am trying to solve. I have a parametric family of "transition" distributions $p_\theta(x_{i+1}\mid x_i)$ and ...
Daniel Robert-Nicoud's user avatar
1 vote
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65 views

Likelihood ratio as minimal sufficient statistics in infinite parameter space

I just read a question from here (Likelihood ratio minimal sufficient) and have some thoughts. Let me restate the question first: Consider a family of density functions $f(x|\theta)$ where the ...
Cyno Benette's user avatar
1 vote
0 answers
21 views

Likelihood of sum of log-normal and normal distribution [closed]

Given $y(x_t)=e^{f(x_t)}-\varepsilon _t$ with $\varepsilon _t\sim N(0,4e^{f(x_t)})$ and $f\sim GP(\mu,\sum)$. What is the likelihood $p(y|f)$? Is it $p(y|f)\sim N(e^{f(x_t)},4e^{f(x_t)})$? Thanks a ...
manhtr76's user avatar
1 vote
1 answer
31 views

What is the correct way to refer to the Likelihood $L(\theta, \alpha | X)$ where $\theta$ and $\alpha$ are parameters?

I am currently studying some latent variable models. In many works, I found the following equation: $L( \theta, \boldsymbol\pi | x ) = \sum_{c=1}^{C} f(x| \theta, \alpha = \alpha_c) \cdot \pi_c$ where ...
Renato Fernandes's user avatar
4 votes
1 answer
402 views

Where does Quasi-Likelihood formula come from?

In regular likelihood/log likelihood, if there is random variable "$Y$" with pdf (probability distribution functions) $f_Y(y)$... the likelihood of this can be written as: $\mathcal{L}(y_i) =...
stats_noob's user avatar
7 votes
1 answer
144 views

Are these two equivalent forms for the likelihood of a Poisson point process?

I have a Poisson point process in a bounded region $W$. I'm trying to calculate the likelihood of observing a particular set of points within $W$. I'm told that there are two equivalent forms of ...
The Pointer's user avatar
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5 votes
1 answer
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Is likelihood the y axis coordinate on the distribution curve?

Josh Starmer says it in here. I have been searching for a simple way to understand likelihood and it's Bayesian and Frequentist use. Josh's way seems simple to me. Is he correct?
Kirsten's user avatar
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Python statsmodels GLM - log likelihood of null model

I have an issue when calculating log-likelihood for null model to double-check GLMResults.llnull parameter: https://www.statsmodels.org/devel/generated/statsmodels.genmod.generalized_linear_model....
Paweł Orliński's user avatar
1 vote
1 answer
27 views

Why does the score test work for values longer in the tail that have a small log-likelihood derivative?

The score test says that we take the derivative of the log-likelihood at $H_0$ and divide it by the fisher information at $H_0$. $U(\theta )={\frac {\partial \log L(\theta \mid x)}{\partial \theta }}.$...
Estimate the estimators's user avatar

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