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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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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120 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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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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Likelihood-weighted sampling and resampling

Suppose we have a Hidden Markov model with continuous state space $X$ and continuous observation space $E$. I'm currently at the very beginning trying to write a code that performs so called ...
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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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20 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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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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37 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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26 views

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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56 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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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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26 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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17 views

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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49 views

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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72 views

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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79 views

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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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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27 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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37 views

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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22 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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57 views

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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31 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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29 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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56 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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31 views

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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40 views

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 ...
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143 views

Do you have to adhere to the likelihood principle to be a Bayesian?

This question is spurred from the question: When (if ever) is a frequentist approach substantively better than a Bayesian? As I posted in my solution to that question, in my opinion, if you are a ...
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Likelihood Ratio Test statistic for the exponential distribution

I need to test null hypothesis $\lambda = \frac12$ against the alternative hypothesis $\lambda \neq \frac12$ based on data $x_1, x_2, ..., x_n$ that follow the exponential distribution with parameter ...
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28 views

Plot of likelihood Function for the Uniform Density. $ (\theta-1,\theta+1)$

Let the random variables $X_1,X_2,...,X_n$ iid $U[\theta-1\,,\theta+1]$. So the likelihood function is $L(\theta|X)=\prod_{i=1}^nf(X_i|\theta)=\frac{1}{2^n}I(X_1, . . . , X_n \in ...
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how to measure the likelihood of a sample?

I am selecting a sample of features from a large pool with each feature having a different frequency and hence probability of being selected. The probability of selecting any given combination of ...
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1answer
26 views

Is this the correct likelihood equation?

Consider a 2x2 contingency table with X having two categories {men,women} and Y having categories {yes,no}. For each category of X the observations was fixed (so the rows had fixed totals).The ...
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In layman's terms, what is the difference between likelihood and probability? (discrete mathematics)

Also, what is the difference between with and without replacement when solving for both likelihood and probability. I have seen that when it is without replacement, there is a denominator, why?
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20 views

Likelihood for dependent data above a threshold

Let $(Y_t)$ a real-valued stationary Markov chain and $u$ some positive threshold. We assume that for $y>u$, $$Y_{t+1}|\{Y_t=y\}\sim\mathcal{N}(\alpha y+\mu y^\beta,\sigma^2 y^{2\beta})$$ I want ...
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18 views

Average likelihood in coin flipping game

Alice plays Bob in a coin flipping game where both are handed possibly unfair coins (coin $i$ has $p_i$ probability of landing as a head, $i = \text{A}, \text{B}$). Both Alice and Bob are going to ...
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38 views

Relationship between the probability and likelihood functions

I've read a number of different explanations trying to understand the likelihood function, and I understand the purpose of it, but some statements sound contradictory. Consider observed data X, model ...
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73 views

What is the difference between partial likelihood and maximum likelihood?

I don't understand these terms. They both have "likelihood". How are they different? Can someone provide an intuitive explanation of them?
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Reestimating prior probabilities after making assumptions about them

Imagine that I have $M$ observations, and each of them can be classified in two different ways: it belongs to one of the classes $A = \{A_1, A_2\}$ and to one of the classes $B = \{B_1, ..., B_n\}$. ...