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 interpret marginal likelihood definition?

Say we have a Beta-Bernoulli model where $X_i$ are i.i.d. Bernoulli variables, $p(X_i=1)=\theta$, and $\theta\sim\operatorname{Beta}(\alpha,\beta)$. The marginal likelihood is defined as $$ ...
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assessing confidence of simple likelihood ratio test

I have a Gaussian mixture model of two normal distributions, which are scaled relatively to each other by a factor $\alpha$. I observe the data vector $x$ with values from $x_1$ to $x_n$, which is ...
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7 views

Adjust data to some distributions [on hold]

I have some data that looks like this: > str(nidd) 'data.frame': 154 obs. of 1 variable: $ Nidd: num 32.24 124.02 3.84 7.21 12.26 ... What should I do to ...
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Composite-likelihood ratio test

I am interested in a number of articles (such as Kim and Stephan 2002 and follow-up articles) that use composite likelihood ratio tests to infer selection pressure on linked (phased) genetic data. I ...
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32 views

How do u find the log likelihood function of Y^1/2 = XB + u?

Let $Y_i>0$ for all $Y_i$ and $u_i \sim N(0,\sigma^2)$. Where the $u_i$ are iid. How do you find the log-likelihood function of $Y_i ^{1/2} = B X_i + u_i$? I am confused because the dependent ...
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What is the effect of using the partial or maximum likelihood upon estimation sample size?

I'm running a series of different models on some longitudinal data with repeated waves. I've noticed that a Cox proportional hazards with time-varying covariates drops entire participants even where ...
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80 views

Distribution of posterior mean from different datasets

This question has originated from this question. Suppose we have the following simple setup, for $i = 1, \dots, n$ $$y_i \mid \mu \sim N( \mu, 1) \text{ and } \mu \sim N(0,1). $$ Then due to the ...
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The likelihood function of related samples

I know how to get the likelihood function for two independent samples from normal distribution. What will happen to the likelihood function if the samples are related? What will be its form?
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11 views

Bayesian estimation for the distribution of the results of an experiment when the cardinality of the result set is unknown

Suppose I have an experiment X with mutually exclusive outcomes from a set S. My goal is to determine the probability distribution for S. The problem is that I do not know how many elements are in S ...
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17 views

Log-likelihood function for Multinomial Logistic Regression

Can you help me to calculate log-likelihood function in R code for multinomial logistic regression, if I have X as design matrix(N x K+1), Y is dependent variable matrix(N x J-1), and B as coefficient ...
3
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41 views

Maximum Likelihood of n

Let $X=\{x_1,x_2,...x_m\}$ is a rv with density $f(x)=cx^{2n}$ if $-1<x<1$, $0$ therwise and $n \in \{ 1,2,3,4\}$. Find fe mle of n. I tried to find the Likelihood function : ...
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27 views

First Derivative of Log-Likelihood of Multinomial Logistic Regression

I want to create my own Multinomial Logistic Regression function in R and get stuck on the first derivative of Log likelihood matrix. The matrix have (J-1)(K+1) rows and (J-1)(K+1) columns, where K is ...
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32 views

Is there any point to Reverse Engineering the Fisher Information Matrix from an Inverse Covariance Matrix?

Would there be any advantage in deriving a Fisher Information Matrix backwards from an inverse covariance matrix? I've discovered that this is much easier to do on the SQL Server platform I use than ...
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Using likelihood for event correlation in Bayesian network

Assume we have a simple Bayesian network, each event (alarm propagation) is a binary vector. Each node can trigger an alarm independently, but alarm propagates in the network with the probabilities of ...
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178 views

Computation of the marginal likelihood from MCMC samples

This is a recurring question (see this post, this post and this post), but I have a different spin. Suppose I have a bunch of samples from a generic MCMC sampler. For each sample $\theta$, I know the ...
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16 views

Using likelihood to compare model accuracy on two different data sets

I have a model that produces conditional probabilities, $p(Y|X)$, where $Y$ is either 0 or 1, and $X$ is just some random variable. I have two different data sets $Z_1, Z_2$ consisting of pairs of ...
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What is the difference and relationship between posterior distribution function and likelihood function in MCMC?

I am learning MCMC in class, and I encounter one question about the relationship between posterior probability and likelihood function. In our lecture, the professor asked us to take samples from ...
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How to estimate the likelihood that a distribution is not random

Not sure I phrase the title right, but here's by question in a nutshell. Say you have a box of balls with different colors. Let's say there are $N$ balls of $k$ different colors, and that the ...
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30 views

Calculating likelihood in R

I want to "by hand" calculate a Bayes factor in a simple case. I'm sure I'm off somewhere in my calculation of likelihood of data under H1. I think I don't "scale" my likelihood under H1 correctly. ...
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likelihood equation for a simple model

Consider a simple model $y_{i}=\alpha_{i}+e_{i}$, for $i=1,...,T$ with $\alpha_{i} \sim \mathcal{N}(0,\tau^2)$ and $e_{i} \sim \mathcal{N}(0,\sigma^2)$. Could anyone please help in writing a ...
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21 views

compute likelihood

How can I compute the likelihood or posterior probability on data when the distribution function is continuous. For example the dist. function is a Gaussian and I have one hundred data points. Now how ...
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28 views

Likelihood ratio test of two normals

If $(X_1, . . .,X_m)$ is a sample of $\mathcal{N}(\mu_1,\sigma_1^2)$ random variables and $(Y_1,. . .Y_n)$ is a sample of $\mathcal{N}(\mu_2,\sigma_2^2)$ random variables. We have no known parameters, ...
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46 views

Log likelihood in EM Algorithm

I try understand the log likelihood in weka. I read about that is a probabilistic metric, but i cant understand, if is better when have low value or high value? How i can get the likelihood value, ...
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Maximum likelihood of coin toss of different type?

I was self-studying EM (Expectation Maximization) algorithm, where I came across this example given by the paper. In this paper, there are two types of coins A, B with unknown parameters $θ_A$ and ...
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Jeffreys' invariance principle - A doubt on the one-dim case

In this wiki link concerning a change of parameterization of a likelihood $L$ from $\theta$ to $\varphi$, why can the $d\theta/d \varphi$ go inside the expectation? $$\eqalign{ p(\varphi) &= ...
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61 views

Is likelihood ratio test a nonparametric test?

I have a few doubts about the Likelihood ratio test. I understand that we compute a p-value based on the ratio of likelihoods between two models. I am wondering: Is the likelihood ratio test, a ...
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17 views

Likelihood based on sample mean from Gaussian

If we know that mean of sample is 4, and that sample size is 3, and that samples are taken from Gaussian distribution with variance 1, i.e. N(u, 1), what can we tell about distribution's mean?
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Is it safe to assume that $\ln{L} = -\frac{\chi^2}{2}$?

I have written a code that fits a 2D model image to some observed 2D image. One of the optimization libraries I use (MultiNest) in order to do that requires to be ...
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27 views

Which one is more robust ? CI with maximum likelihood method or by bootstrap?

I have a question about confidence interval calculation. I am wondering is calculatin the CI with the function confint is more robust with the likelihood method or the boostrap method ? thanks !
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Which likelihood function for ML estimate of continuous variable?

Let's say my data consists of the counts of the following words in a text. I can then estimate the probabilities of occurrences of these words using MLE: ...
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Profile Likelihood Algorithm Assumptions

Given a statistical model $$\{p_\theta(y) : \theta \in \Theta \}$$ and same data $y$. The log likelihood function is then $l(\theta)=\log(p_\theta(y))$. For a one parameter hypothesis $\theta_j=\beta$ ...
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39 views

insurance exponential

i would like to ask for help with following example, i know how to derive next steps that I am not even showing here, but cant derive the loglike function, thank you.
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Help explaining (offering intuition/examples of) the score function and fisher's information to students

Next week I will teach my students the score function and its variance (i.e.: fisher information). I am looking for way(s) to illustrate these concepts so to help my students understand them (and not ...
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Properties of Average Multinomial Likelihood

I am trying to understand the Kullback-Leibler Information: I read in http://arxiv.org/pdf/1404.2000v1.pdf the following: Ideally, we want the probability to be invariant to the number of ...
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Posterior distribution of parameter

I am looking at pg. 257 of the textbook "Statistical computing with R" by Maria L. Rizzo, where an example is given of a simple model of stock returns: We have 5 stocks, and at the end of a year of ...
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maximum likelihood two unknown parameters

I have a problem. I have to find the mle's (maximum likelihood estimators) for $a$ and $b$ whose density is $f(x|a,b)$, when I derive the $l(b|x)$ (The log-likelihood function) to find the maximum, ...
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53 views

Likelihood function and MLE

Please let me know how to find the likelihood function and MLE when the data have density $f(x; \theta ) = ( \theta +1) x^\theta$. I have tried using the general formula however not sure how to ...
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1answer
44 views

EM algorithm with dependent observations

I am trying to implement an EM algorithm for dependent observations. Specifically, I am dealing with families where the hidden variables $Z$ of the children are dependent on the hidden variables of ...
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28 views

Generalized likelihood test question

This example appears in Rice's stats book. In the first rectangle, why do we have to maximize the denominator? and why do we use MLE as the denominator to maximize it? I know that MLE best reduces ...
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Cumulative distribution function :Maximum Likelihood

We have distribution given by (sample of size n from a rv) $F(x)=\begin{cases} 0 & x<0\\ (\frac{x}{\beta})^{\alpha} & 0<x<\beta\\ 1 & x>\beta \end{cases}$ Find the mle of ...
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Compare two sets of probabilities to outcome data

Suppose there are two predictive models that both output the probability that the home team wins a given match. Then suppose there is data for thousands of matches, in the format: ...
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24 views

Likelihood function for non indipendent values

We all well know that if $x_i$ are independent and identically distributed a likelihood function will be something like that. ${L}(\theta|\{x_i\}^n_{i=1})=\prod\limits_{i=1}^n {f}(x_i|\theta)$ But ...
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Marginalizing over a parameter: integrate the total joint likelihood, or the each individual likelihood?

This is a general question about a model-fitting task. Suppose you have IID data $Y_1, ..., Y_n$ arising from a data generating modeling indexed by a parameter $(\theta, \lambda)$, where you are only ...
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35 views

Interpretation of the log likelihood in clustering techniques

Can Someone explain me how to interpret the log likelihood measure when evaluating clustering techniques? Let's say I am using Gaussian Mixture with Expectation Maximization, and I want to choose ...
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1answer
32 views

How to derive likelihood of an event from probability values? What's the difference/relation between odds and likelihood?

I am trying to understand the relationship between probability, odds, logit values and likelihood. Is there any?
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62 views

Likelihood Ratio Test for Exponential Distribution with a Limited Parameter Space

Suppose that we are given an exponential distribution model with a pdf $f(x,\theta) = \theta^{-1}\exp(-x/\theta)$ with an iid sample $X_1, ..., X_n$, and we would like to test hypothesis $H_0 : \theta ...
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negative LRT statistic value

I am still working on an item response theory (IRT) analysis to find differential item functioning (DIF) in any of the items of a 17-item graded response likert scale. To briefly summarize what ...
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Likelihood ratio test applied to the marginal likelihood of a serial model

There's a serial statistical model I'm working with. In it, you give some parameters and a initial time and it returns a produced time. In this model, some internal variables are used, I managed to ...
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Does the EM algorithm converge exactly where a grid search on the marginalized likelihood converges?

I have successfully implemented a grid search algorithm to estimate two parameters of a likelihood. I computed the likelihood $l(X;\theta)$ of observed data $X$ by integrating out the discrete ...