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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Can we write the likelihood of a GLM in generality?

So I know we can explicitly write down the likelihood of any specified GLM model, for example the likelihood for the logistic regression model would be ...
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Cauchy MLE Reliability of Asymptotic Results

I have the following regression model $$ p_i = x'_i\beta +s \varepsilon_i $$ with sample size $n \approx 150$ and 4 independent variables. I have reason to believe and $\varepsilon_i$ is distributed ...
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52 views

Construct the likelihood with asymmetric uncertainties

I want to study the correlation between 2 parameters, this is done by fitting a straight line. I have uncertainties on both parameters. I want to solve my problem using the Bayesian approach, i.e. I ...
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14 views

Bayesian Inference for Poisson process 2

How do I calculate Bayesian posterior distribution of Poisson likelihood function with Pareto prior distribution?
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Strange likelihood construction for information borrowing

I recently encountered a strange model while consulting in industry. The goal was to collect information about the parameter of event $A$, say $\theta_A$, but we only observed events $B$ and $C$, ...
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The most general definition of the Likelihood function for continuous data (including truncation and censoring)

How would you rigorously define the likelihood function for censored/truncated observations? Even in most lifetime/reliability literature (where these types of observations are frequently encountered) ...
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60 views

Cauchy distribution (likelihood and Fisher information)

I have a three part question: 1) If I have a $Cauchy(\theta, 1)$ with density: $p(x-\theta) = \frac{1}{\pi\{1+{(x-\theta)^2}\}}$ and $x_1, ..., x_n$ forms i.i.d sample. I see on Wiki that this will ...
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I have problem of my sampling based on secondary data [closed]

I have question and i hope i get the answer I am doing my thesis about the impact of liquidity risk in Islamic banks in GCC countries and I have to choose just two countries one which it ...
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1answer
47 views

How to calculate likelihood of linear regression

This is a pretty basic question, but one I am having a hard time finding an answer to. How do you calculate the likelihood of a simple linear model? Like, say, $$y=\beta_0+\beta_1x+e$$ I am working on ...
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66 views

Construct the likelihood if measurement uncertainties have a Gamma distribution

I want to construct my likelihood. General case: If my data do come from a line of the form $y = mx + b$ and the uncertainties are normally distributed with mean zero and known variance ...
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26 views

likelihood of string of events given a string of probabilities

There are two classes of strings of events. E.g. A: 0,0,1,2,2,3,4,0,3,0,0,0 B: 0,0,0,0,3,3,2,1,5,6,7,0 Both class A and B strings exhibit variability. Many (e.g. ...
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33 views

Finding the posterior distribution

Suppose that an observation $X$ is drawn from the following distribution $f(x|\theta) = \begin{cases} \frac{1}{\theta} & \text{ if } 0 < x < \theta \\ 0 & \text{ if } otherwise ...
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54 views

how to construct the likelihood if my errors are not Gaussian

My aim is to study the correlation between 2 parameters knowing that I have measurement errors in both parameters, i.e. I have uncertainties on the independent and dependent parameters. I want to ...
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21 views

confidence interval comparision to likelihood function

I have question, it is taken from a book. It says: Let $X_1 , X_2$ denote a random sample of size n = 2 from a continuous uniform distribution $U(\theta - 0.5, \theta + 0.5)$ with unknown parameter ...
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6 views

Gray box Model Identification with partial state measurements

I'm trying to estimate the parameters of a gray box linear model, but I have only partial observation for the states. Is it possible to apply maximum likelihood method in this case? I try to explain ...
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2answers
452 views

I am wondering why we use negative (log) likelihood sometimes?

This question has puzzled me for a long time. I understand the use of 'log' in maximizing the likelihood so I am not asking about 'log'. My question is, since maximizing log likelihood is equivalent ...
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55 views

Is a Bayesian estimate with a “flat prior” the same as a maximum likelihood estimate?

In phylogenetics, phylogenetic trees are often constructed using MLE or Bayesian analysis. Oftentimes, a flat prior is used in the Bayesian estimate. As I understand it, a Bayesian estimate is a ...
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22 views

guidance for picking Likelihood in Bayesian analysis

I'm doing a Bayesian analysis for a time series response and wonder whether it is possible to get the Likelihood function without making distributional assumptions. I suppose my response is ...
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30 views

Using MCMC to calculate log likelihood

I have an analytical solution to finding $p(y|\beta)$ and $p(\beta)$. My goal is to find $\log p(y)$. However, the integral $\int p(y|\beta)p(\beta)\,d\beta$ is not analytical. I did manage to use a ...
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Choosing which distribution is most accurate

Let's say we have two samples from different normal distributions and we want to determine which distribution is most accurate relative to an ideal value. How would we evaluate which one is more ...
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422 views

Is the likelihood a true function?

Many books and many posts on this site define the likelihood as a function of model parameters. However, does the output associated with every possible model parameter have to be unique? For example, ...
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35 views

MCMC: The posteriors are too narrow

I have this very simple MCMC there I have ...
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26 views

why is the incomplete log-likelihood difficult to optimize

I am trying to teach myself the expectation-maximization algorithm and the texts say the EM is particularly useful when the incomplete log-likelihood i.e. $P(X|\theta)$ where $\theta$ are the ...
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What is the frequentist take on the voltmeter story?

What is the frequentist take on the voltmeter story and its variations? The idea behind it is that a statistical analysis that appeals to hypothetical events would have to be revised if it was later ...
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319 views

Determine Maximum Likelihood Estimate (MLE) of loglogistic distribution

I am given two data sets containing dates and losses (in some currency). I have to determine the maximum likelihood estimates of the parameters of loglogistic distribution. I googled and found a ...
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101 views

numerical difference between sum of squared residuals and likelihood

I previously asked a question that got labelled as duplicated because I did not explain it correctly. I should not have used the regression model as an example because I can see how, by using that as ...
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Why is type I error not affected by different sample size - hypothesis testing? [duplicate]

I don't understand why the probability of getting a type I error when performing a hypothesis test, isn't affected. Increasing $n$ $\Rightarrow$ decreases standard deviation $\Rightarrow$ make the ...
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Learner in R to predict data from a complex function: if a>b then a*b else a+b

What sort of general purpose learner could learn the data generated by the following function: if a>b then a*b else a+b or something of that sort of complexity. Ideally something general enough to be ...
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Picking a probability distribution for observed intensities

I have an experiment that measures "intensity" (in this case, electron density of a molecule) on a grid. The values it gives are non-negative,. I'd like to write a likelihood for this observation ...
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35 views

Determining a greater than chance cutoff for an assessment's results

I recently collected a set of data, and before I completed an analysis of it, I wanted to remove any "bogus" entries from participants that didn't actually read the stories. The study I conducted had ...
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70 views

Gradient of Log-Likelihood

Considering the following functions I'm having a tough time finding the appropriate gradient function for the log-likelihood as defined below: $a_k(x)=\sum_{i=1}^D w_{ki}\cdot x_i$ $P(y_k|x) = ...
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54 views

dirac delta function in likelihood function

I have tried to understand this myself but what I have found on the internet so far has not helped. I have a likelihood function that for part of it has the following statement: d0 is the Dirac ...
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96 views

Finding sampling distribution of normal MLE and likelihood

I'm reviewing old exams in preparation for a statistics final, and I'm stuck on a particular question: Suppose that you have n independent random variables $Y_i$, with each distributed normal with ...
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Clear explanation of pseudo likehood

In generalized linear mixed model (glimmix) parameters are estimated using pseudo likelihood. I was trying to understand how this type of likelihood calculated. Thanks !!!
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What would be an example of a really simple model with an intractable likelihood?

Approximate Bayesian computation is a really cool technique for fitting basically any stochastic model, intended for models where the likelihood is intractable (say, you can sample from the model if ...
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Beginner learning resources : Pdf and likelihood function for non-Gaussian time series model

I am struggling with exercise problems related to blind system identification where the knowledge about the source input is assumed to be known using maximum likelihood estimation of univariate time ...
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28 views

Bayesian Logistic Regression Likelihood Computation for Binary Data

For computing the likelihood function for a Bayesian Logistic Regression model, I understand the original form to be $$p(y_i|\beta,X) = \prod_{i=1}^{k}\left( \frac{ \text{exp} \{ \beta^{\prime} x_i ...
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25 views

Marginal Likelihoods for Bayes Factors with Multiple Discrete Hypothesis

If I have some data that I believe is Normally distributed and I just want to test the hypotheses that the mean is equal to 1 of 3 values, my understanding is that the Bayes Factor is the ratio of ...
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59 views

Explain log likelihood behaviour

(This question is related to a previous one I made, here) I have a set of 2D observations (measured data) of sample size $N$: $$O = \{(x_1, y_1), (x_2, y_2), ..., (x_N, y_N)\}$$ I also have a model ...
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205 views

Likelihood ratio test for comparing two exponential distributions

I am trying to use a likelihood ratio test to compare the parameters of two exponential distributions. by this thread Likelihood Ratio for two-sample Exponential distribution I found that I can use ...
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86 views

Point estimation MLE and MME

Consider the family of probability mass functions given by f(x;k) = 3(4^(k-x)) x = k + 1, k + 2,.... and indexed by parameter k E Z. For a random sample of size n, derive with justification: a) ...
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Question on Nested Likelihood Ratio Model

Suppose I want to calculate a likelihood test statistic distribution for background only MC using following Likelihood Function with Signal Fraction = $ns$ and $\theta = (a,b,c)$: ...
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Multiplying distributions with different conditioning

I saw this expression in a UBC machine learning class lecture, and I'd like to understand how the math works. Suppose we're trying to predict a class label $y$ given some data $x$. There are prior ...
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432 views

“weight” input in glm and lm functions in R

I am confused with the definition of the weights in glm and lm. Using the McCullagh and Nelder (1989)'s notation, If random variable $y_i$ is from the Generalized Linear Model (GLM), then its density ...
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jump in the sign of loglikelihood

I am trying to find the maximum of a loglikelihood function in a two dimensional parameter space__ e.g. X,Y positions are the free parameters__ by making grids in the parameter space and compute the ...
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Two Gaussian Likelihoods with Two Decision Boundaries , 0/1 loss function

we assume that Y := {1, 2}. Then our decision can be re-written as y ∗ = {1 if p(x|y = 1) > p(x|y = 2) , 2 otherwise} with a decision boundary at p(x|y = 1) = p(x|y = 2). How can we construct an ...
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77 views

Maximizing Log-Likelihood Estimation for Changepoint Detection

I'm trying to code the changepoint detection algo described here: ...
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43 views

Compare Two Logistic Regression Models

I have worked out two models to fit the data (blue) - the first (in green) is the baseline model with the intercept only. The second (red) is the model with the intercept and 2 parameters. Obviously, ...
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Recursive Bayesian Estimation, $p(C_k|\mathbf{x})$ as (discrete) likelihood

I''ve been struggeling with this problem for the last couple of days. The main goal is to use the probabilistic classification output $p(C_k|\mathbf{x})$, from for example a logistic regression, to ...
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Probability to Likelihood

I have a problem on calculating the likelihood of observing a data point x given the predicted lable. My application is on text classification where I have to detect Spam and No Spam documents. I ...