Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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 $θ: \text{L}(θ)=\text{P}(θ;X=x)$

0
votes
1answer
56 views

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}^{...
2
votes
0answers
12 views

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 ...
0
votes
0answers
6 views

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, ...
1
vote
0answers
16 views

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 (...
0
votes
0answers
7 views

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 ...
0
votes
0answers
14 views

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 ...
1
vote
1answer
38 views

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 ...
0
votes
0answers
21 views

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 ...
0
votes
0answers
24 views

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 ...
2
votes
1answer
89 views

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 ...
0
votes
0answers
24 views

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. ...
0
votes
1answer
32 views

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 ...
0
votes
1answer
40 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 ...
0
votes
2answers
86 views

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? ...
0
votes
2answers
14 views

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 ...
1
vote
0answers
21 views

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}...
1
vote
0answers
18 views

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 ...
0
votes
0answers
56 views

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{\...
2
votes
1answer
30 views

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 ...
0
votes
1answer
29 views

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 $\...
0
votes
1answer
57 views

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?
3
votes
1answer
60 views

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 $\...
2
votes
1answer
44 views

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.
1
vote
0answers
30 views

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?
0
votes
0answers
4 views

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 ...
1
vote
1answer
74 views

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 ...
0
votes
1answer
23 views

Plot Scaling Problem

I am trying to implement the following code in R $\theta \sim Beta(a=1,b=1)$ $x|\theta \sim Bin(N=5,\theta)$ $\theta|x \sim Beta(a+\sum x_{i},b+\sum N-\sum x_{i})$ ...
0
votes
0answers
19 views

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 ...
1
vote
1answer
31 views

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 ...
0
votes
0answers
20 views

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 ...
2
votes
0answers
31 views

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 ...
0
votes
1answer
98 views

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 ...
1
vote
1answer
240 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 ...
0
votes
0answers
25 views

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_{...
0
votes
0answers
8 views

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: ...
0
votes
0answers
22 views

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 ...
0
votes
0answers
35 views

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 ...
0
votes
0answers
29 views

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, ...
4
votes
1answer
90 views

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-...
0
votes
0answers
7 views

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,\...
0
votes
0answers
25 views

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 ...
1
vote
0answers
97 views

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 ...
0
votes
0answers
19 views

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 ...
0
votes
0answers
27 views

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 $\...
1
vote
1answer
59 views

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 ...
5
votes
1answer
179 views

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 ...
4
votes
3answers
210 views

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<\...
0
votes
0answers
31 views

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 $y$ ...
3
votes
0answers
32 views

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 ...
1
vote
2answers
44 views

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 $M_1$....