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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Likelihood estimator for JMP decision trees

In the documentation for JMP decision trees- it details that the calculation for the "Entropy RSquared" (Actually McFadden's $R^2$) is calculated thus: $R^2 = -2\log(L_m/L_0)$ Where $L_m$ and $L_0$ ...
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Likelihood of 10000:1 probability happening exactly once in 10,000 tries

I am interested in understanding the difference between "likelihood" of a random event with a particular probability actually occurring the exact probability it is said to be likely. i.e. if an event ...
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Question about the calculation of likelihood function

I am looking at the answer on this thread: Why likelihood is not always a density function? Here as I understand that the likelihood function is given by: $$ L(\theta) = \frac{1}{\theta} \quad ...
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+50

Parameter estimation using bayesian estimates in $R^k$ Euclidean space?

I am facing difficulty in identifying how the formula given by Eq(2) in the paper ...
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47 views

Understanding Likelihood Function

Just learning Bayesian techniques through Insua et.al.'s "Bayesian Analysis of Stochastic Process Models." On page 18 they give an example of a gambler estimating the parameter $p$ in a binomial ...
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Find a likelihood function from varying data for Bayes Theorem

I'm not sure how to model a likelihood function for the following problem: Assume I've got a sensor producing the following raw values (normalized to an interval $\pm$1): In reality there are more ...
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How to use a normal log-likelihood function to estimate the variance?

I have an array of data that is normally distributed, i.e. we're dealing with a multivariate Gaussian. We write the data as $X = \{x_1, x_2, \ldots , x_N\}$ So, there are unknown parameters $\mu$ and ...
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1answer
23 views

What is the dimension of the Gaussian log-likelihood function?

I am having trouble comprehending the log-likelihood of a multivariate normal distribution. For an n-dimensional vector $\mathbf{r}$ of N i.i.d. data points $\mathbf{r}=(r_1,...,r_N)$, the ...
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How to understand a Gaussian likelihood function vs. the variance of data points? [duplicate]

This is an elementary question, but I find myself very confused visualizing this (if there are errors in anything below, please correct me): The likelihood function describes the probability density ...
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1answer
25 views

How to form likelihood function of a variable that follows a quadratic function

I am having difficulty understanding likelihood functions. If we have a probability density function of a random variable $X$ like this: $f_{X} (x)=ax^2 + bx + c$ (i.e a simple quadratic polynomial), ...
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79 views

Bayes theorem: normalisation denominator and likelihood

I have been racking my brains trying to understand Bayes theorem. So, the way I have understood is that the likelihood is the probability of observing the particular outcome given a set of parameter ...
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What does this terminology mean in this introduction to likelihood ratios?

I am currently reading In All Likelihood by Yudi Pawitan (2013 edition) and I am making my way through the second chapter on the likelihood function. In part 2.4 which is where likelihood functions ...
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log-likelihood of regression model with exponential distribution as response

I would like to calculate the log-likelihood of a simple regression model, where the response variable $y$ is exponentially distributed. I thought I could just use least-squares (LS) to find the best ...
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1answer
83 views

Likelihood-based hypothesis testing

$N_A$ and $N_B$ are variables of the counts of the number of events 'A' and events 'B' respectively. Those variables follow Poisson distributions with parameters $\lambda_A$ and $\lambda_B$. In ...
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Combined likelihood of multiple counting experiments

I have two counting experiments, each experiment is an independent detector (A and B) which detects some events (photons). Detector A has frequency resolution - it measures that a count occurred at a ...
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13 views

Using Likelihood Ratio Test to deal with heteroscedastic data results in unreliable results

Suppose $Y$ and $x$ are not related. Therefore the linear regression analysis should not reject the null hypothesis ($H_0: b=0$) in $E(Y) = a+bx$. Suppose the variance in $Y$ increase with $x$ (i.e., ...
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Marginal Likelihood of a Non-Linear Mixed Effects Model

The marginal likelihood of a non-linear mixed effects model does not admit a closed-form expression, unless data is normally distributed... or so I was told. Does anyone know of any literature, ...
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40 views

Imposing a model on a pdf

(This question is an attempt to zoom in on the key issue in this question using as little information as possible.) Lets say I want to derive the likelihood function of $\beta$ given $x$ and $y$ for ...
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41 views

Derivation of likelihood function for latent variable model made explicit

I am trying to make the steps deriving the likelihood function for the following latent variable model as explicit as possible: $$Y^0=X\beta + u$$ where $$u \sim NID(0,\sigma^2).$$ The observed data ...
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38 views

Approximating the marginal likelihood in Bayesian Model Comparison

Given some data $y$, my interest centers around a collection of models $\{\mathcal{M}_1,\mathcal{M}_2,\cdots,\mathcal{M}_L\}$ representing competing hypotheses about $y$. Each model $\mathcal{M}_l$ ...
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38 views

Profile likelihood

I am considering a normal distribution with mean $\beta_1 + \beta_2\exp(-\phi x)$ and variance $\sigma^2$, i.e. $y \sim N(\beta_1 + \beta_2\exp(-\phi x), \sigma^2) $. My aim is to calculate the ...
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30 views

Correction for Log normal distribution

Context: I have observed continuous data $\boldsymbol{O}$, for each observation $i$ I have an assumed known $\sigma_i$ for each observation I have an expected model value $E_i$. $E_i$ was produced ...
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In MCMC simulation, how to deal with very small likelihood values that couldn't be represented by computer? [duplicate]

I am working on a Bayesian project based on Stagnant data from a OpenBugs example, which is a changepoint problem. Basically we assume a model with two straight lines that meet at a certain ...
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1answer
26 views

Log Likelihoods of Exponential Families

How can one derive the log-likelihood of the saturated model of an exponential family in general? Differentiating the log likelihood w.r.t $\theta$ gives $y_i=\hat{\mu_i}$ but I don't think replacing ...
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25 views

Generalised Likelihood Models

Why would you use a re-scaled binomial distribution rather than just the standard binomial distribution as the distributional assumption in a GLM?
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likelihood ratio testing

Let $Y_{1}, Y_{2}, \ldots, Y_{n}$ denote a random sample from a $N(\theta,\ \sigma^{2})$ population. Consider testing $H_{o}: \theta\geq\theta_{o}$ versus $H_{a}: \theta<\theta_{o}.$ If ...
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log-likelihood of multivariate regression

I am struggling with R to calculate log-likelihood of multivariate regression (i.e. more than one response, one or more predictors). I would like to have a function taking object of class "mlm" as an ...
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1answer
85 views

Interpreting prior and posterior

I am bit puzzled on how we can interpret the posterior. Assume a coin which is 0.1 probable to be unfair. So our prior probability on the coin being unfair is 0.1, and being fair is 0.9. Also by ...
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42 views

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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1answer
68 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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15 views

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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106 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
57 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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1answer
76 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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1answer
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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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34 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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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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22 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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7 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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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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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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24 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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39 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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428 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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MCMC: The posteriors are too narrow

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