Questions tagged [log-likelihood]

For questions that use the natural logarithm of a likelihood function.

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11 views

Simplification and implementation of 2D log likelihood Bernoulli model

I'm trying to implement a paper titled "The spatial scan statistic: A new method for spatial aggregation of categorical raster maps" that uses the "log form" of a likelihood ...
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60 views

Show that expression is Gamma distribution with given parameters

I had the expression $$\frac{1}{p(y)}(n\lambda)^k e^{-n \lambda}$$ for $k \in \{0,1,2,...\}$ where we here have that $k=\sum_{i=1}^n y_i$ where I think I can ignore $p(y)$, but if not it is given by $...
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34 views

Negative Log likelihood and Derivative of Gaussian Naive Bayes

I am trying to derive negative log likelihood of Gaussian Naive Bayes classifier and the derivatives of the parameters. So there are class labels $y \in {1, ..., k}$, and real valued vector of $d$ ...
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13 views

Finding out the generative log-likelihood $\log(p(x|t))$

The question ask to build classifiers to label images of handwritten digits. Each image is $8$ by $8$ pixels and is represented as a vector of dimension $64$ by listing all the pixel values in raster ...
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23 views

Calculate likelihood odds ratio with t-score

I'm researching in NIFTY and got struck on the calculation of log likelihood ratio. I'm having $2$ t-score $t_1$ and $t_2$ generated from binary hypothesis. The first test has null hypothesis is $|...
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3answers
55 views

All solutions to $f(a,b)+f(b,a)=0$

I am looking for all functions $f:(0,1)^2\to\mathbb{R}$ that satisfy $f(a,b)+f(b,a)=0$ for every $(a,b)\in(0,1)^2$. I know that $f\equiv 0$, $f(a,b)=c(a-b)$, and $f(a,b)=\pm\log(a/b)$ are such ...
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48 views

Likelihood of p and q given Y

Suppose that Jon has a coin with a probability of landing on heads p and Ann has a coin with a probability of q landing on heads. Each time they flip a coin and they write the sum of the outcome ...
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1answer
83 views

MLE + Cramér-Rao bound of a discrete random variable

Can someone explain how to compute the maximum likelihood estimator, the Fisher information and the Cramér-Rao bound of a discrete random variable please? I came across this exercise while reviewing ...
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31 views

Which optimization method should I use?

I am trying to maximize a likelihood function using the Newton-Raphson method; however, it seems that the Hessian matrix is badly scaled by definition of my optimization problem. Now, I do not ...
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2answers
62 views

Roots of log binomial likelihood equation

I have been wondering if there is a way to approximate the roots of the log binomial likelihood equation. To be clear the equation is $$a \cdot \log\left(x\right) + b \cdot \log\left(1 - x\right) = t$$...
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14 views

Formulate a likelihood function for an uncertain discrete "n", where y is binom(n,p), and we know y and p

The problem goes as followed: multiple balls have been tossed into a box, but we do not know how many that were tossed. However, we know that 5 of the balls that were tossed had a blue colour ($y=5$), ...
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71 views

When does sup and function commute?

For $f,g$ real-valued functions, $f$ weakly increasing and continuous, $A\subseteq \mathbb{R}$, can we say \begin{align*} \sup_{x\in A}f(g(x))=f(\sup_{x\in A}g(x)) \end{align*} I ask because I notice ...
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22 views

Log likelihood from multi-variate Normal distribution?

If we consider a linear mixed model that consider fixed effect and random effect: $$y = X\beta + W_uu + e,$$ where $n$ is the sample size $y \in \mathbb{R}^{n \times 1}$ is the response vector $X \in ...
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22 views

How could I estimate the population mean parameter using empirical likelihood in case the distribution of data is unknown?

$F$ - cumulative distribution function. Likelihood function: $L(G) = \Pi_i p_i$ and constrains are $p_i \ge 0, \sum_i p_i = 1, \sum_i p_i g(X_i, \theta) = 0. $ To estimate the mean we choose $$g(x,\...
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15 views

What is the Bayesian classifier given two bivariate Gaussians with same covariance matrix?

Given two bivariate Gaussian distributions, and conditional distributions of feature vector $X=(X_1,X_2)$, given label $Y = y$, given by \begin{equation} (X_1, X_2) \sim \left\{ \begin{array}{...
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43 views

Differentiating log likelihood with respect to data

I'm working with the probabilistic model derived from the KL NMF, in which we sample each entry of a data matrix X from the distribution $\forall_{i,j},X_{ij}\sim\text{Poisson}\left(\left[WH\right]_{...
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37 views

Log-Likelihood of Piecewise-Defined Function

This is a follow-up to this question, which might be too difficult. So I break the problem down. What I need to find is the log-likelihood of the following function: $$f(x, a_{sg}, a_{pl}, p) = \begin{...
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78 views

Log-Likelihood Function of Piecewise-Defined Function

I'm a data analyst, no mathematician. Mostly I do 'standard' stuff such as linear mixed-effects regressions or generalized additive mixed models. But now I need to determine the maximum log-likelihood ...
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55 views

Likelihood ratio hypothesis test for the exponential distribution.

I am trying to understand the following logic that I found on the internet that implements a hypothesis test on the exponential distribution using the likelihood ratio test. We want to test the ...
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1answer
33 views

How to find the log-likelihood for this density?

We let $(X,Y)$ be stochastic variables with values in $N × R$ determined by $ X = x $ being subtracted from one Poisson distribution with mean $\lambda$ after which Y is subtracted from a gamma ...
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44 views

Likelihood values from Sigmoid

There are multiple doubts of mine associated around this theme: In MLE, we try to find the PDF parameters ($\theta$) which maximise the likelihood of the observed data ($L(\theta | data)$). To get ...
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1answer
39 views

Hypothesis testing and critical regions

Suppose $X$ follows a chi-squared distribution with $r$ degrees of freedom, where $r \in \mathbb{Z}^+$ is unknown. Let $X_1, X_2, ..., X_n$ be a random sample of $X$. (i) Suppose we have the following ...
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46 views

Is there a risk, in some special cases of likelihood form, that using logarithm in likelihood derivation, drives to losing some good solutions?

In inference statistics, one uses the quantity "likelihood" (L), and derivates it with respect to a given parameter $\theta$ (or set of parameter) in order to maximize the likelihood. Thus ...
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56 views

Likelihood ratio test for distribution with two parameters

We let $X$ and $Y$ be independent and exponentially distributed random variable with $E(X)=\lambda_1$ and $E(Y)=\lambda_2$ where $\lambda=(\lambda_1,\lambda_2) \in {R}_+^2$ and we let $(X_1,Y_1),...,(...
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96 views

Fisher Information for Beta Distribution

I am trying to find the Fisher Information for $\operatorname{Beta}(\alpha,2)$. I used the following approach: $$f_X(\alpha,2)=\alpha(\alpha+1)x^{(\alpha-1)}(1-x),\alpha>0$$ $$ L(\alpha|x_i)=\...
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176 views

Maximum likelihood estimator doesn't exist

I was reading a paper, and the last paragraph it says For example, consider the density function $$p_{(\theta,\sigma)}(x)=\frac{1}{2\sqrt{2\pi}}e^{\large{-\frac{1}{2}(x-\theta)^2}} + \frac{1}{2\sqrt{...
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35 views

Demonstration of Fisher element expressions under 2 forms

A Fisher element $F_{ij}$ is defined by : $$ \mathbf{F}_{i j} \equiv-\left\langle\frac{\partial^{2} \ln f}{\partial \theta_{i} \partial \theta_{j}}\right\rangle\tag{1} $$ and its inverse $\mathbf{F}^{-...
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74 views

Maximum Likelihood Estimation with Poisson distribution

We have two independent random variables $X$ and $Y$ with $X\sim Poisson(\Phi)$ and $Y\sim Poisson(2\Phi)$, and the observations $x=2$ and $y=4$ of these. Show that the expression for the log-...
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17 views

Show that the likelihood ratio test for the null hypothesis can be written in the form

Show that the likelihood ratio test for the null hypothesis $λ=1.1$ versus the alternate $λ=2.1$ with $α=0.05$ can be written in the form: reject $H_0$ when $S≤C$ for some constant $C$, where $S=X_1+⋯+...
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30 views

Multiplying by 0 ( zero ) when finding max-likelihood estimation

I am kind of confused about the rules in math when you are solving equations I think. I am solving the following equation: $\frac{15}\lambda - \sum_{i=1}^{15} X_i = 0 $ According to maple and symbolab,...
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26 views

How to derive the $\frac{\partial logL}{\partial \beta_j}$ of binomial Generalized Linear Model?

For the response Y = (event, no event). Let $Y_i$~$B(n_i,p_i)$, and $(x_{i2}, x_{i3}, ..., x_{ip})$ is sampled with $Y_i, i=1,2,...,n$ To model the effects of $X_{1} ≡ 1, X_{2},...,X_{p}$ on the ...
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87 views

Hypothesis testing in elections

The following is a problem I am working on for my stats class regarding hypothesis testing: I have a random sample $X_1,...,X_n$ of voters who either voted for Candidate A ($X_1=1$) or for Candidate B ...
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21 views

log-likelihood for Caesar encryption key

During a practical class on breaking the Caesar cypher with frequency analysis, the teacher presented us with a formula for the log-likelihood of $k$ being the key of the code: $$ \sum_{i=0}^{26}{h((i+...
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69 views

Get covariance from log-density function

Problem Given a following log-density function $$ \ln p(y| a, b) = a \cdot g(y) + b \cdot h(y) + k(a,b)$$ where $g(y), h(y), k(a,b)$ are difined function and $a,b$ are parameters. Find $\Bbb Cov( g(Y),...
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19 views

Deriving a Likelihood Ratio Test, based on Null and Alternate Hypotheses?

LRT is just an abbreviation for Likelihood Ratio Test. I have been working on this problem for a while now. I currently constructed by Null Hypothesis as (lambda 1 = lambda 2 = lambda 3) and my ...
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102 views

Find covariance of estimator and derivative of the log-likelihood function

Problem: Given an estimator $\hat k$. The estimation method is either max likelihood or other method. We know that it's unbiased. Let $L$ be the likelihood function and $\ell = ln L$. Find $\Bbb Cov( \...
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86 views

Let 𝑌1 ,… , 𝑌𝑛 be independent and identically distributed (i.i.d.) random variables with distribution function

Apologies for the formatting. I didn't really know how to format this. Let $𝑌_1,\ldots, 𝑌_n$ be independent and identically distributed (i.i.d.) random variables with distribution function : $$P(Y_i ...
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82 views

Maximum likelihood of log-normal distribution

I am trying to find the maximum likelihood function of log-normal distribution for both parameters $\mu$ and $\sigma^2$. I've gotten the derivative of the log-likelihood for $\mu$ to be $$\frac{\...
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22 views

Fisher element calculation : How to compute covariance term in the expectation of 2 variables

I know that $$E[XY] = E[X]E[Y] - \text{Covar}(X,Y)\quad(1)$$ Now , I would like to apply it to the definition of Fisher element : $$F_{\alpha \beta}=\left\langle\frac{-\partial^{2} \ln \mathcal{L}}{\...
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16 views

Likelihood of of a sum of densities

I am trying to find the likelihood, and then the log likelihood for the following function: $f(x)= cf_1(x) + (1-c)f_2(x)$. I am stuck on how to find this likelihood because the two densities are being ...
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19 views

Cannot replicate the BIC, CAIC and AWE values from a textbook example

Dear StackExchange users, I am trying to replicate the estimation of model fit measures from a textbook example. Masyn (2013:568) provides the following formulas: $$BIC = -2LL + d\log(n)$$ Where d is ...
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30 views

Find the maximum likelihood estimate of probabilities of ordered pairs of random variables

$P((X_k,Y_k)=(i,j))=\pi_{ij}$ where $\sum \pi_{ij}=1$ and the frequency of $(i,j)$ is $n_{ij}$. I'm now asked to find the MLE of $\pi_{ij}$ if $X$ and $Y$ are independent and again if they are not. I ...
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67 views

Finding the MLE estimate from a log function

If I have the following function: $$\ln L(\mu,\sigma^2)= -n \ln (\sqrt{2 \pi \sigma^2})+\sum_{i=1}^n \frac{-(x_i-\mu)^2}{2\sigma^2}$$ How can I find the MLE estimate $σ^2_\text{MLE}$? I have already ...
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40 views

Finding log likelihood [closed]

Consider $X_1,X_2,…,X_n\sim N(\mu,\sigma^2)$. The density of a single $X_i\sim N(\mu,\sigma^2)$ is $$f_{X_i}(x_i;\mu,\sigma^2)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp\left(\frac{−(x_i−\mu)^2}{2σ^2}\right).$$...
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20 views

Log likelihood from density [closed]

Suppose $X_1,…,X_n$ are iid with density $f(x|α)=(α+1)x^αI(x∈[0,1])$, where $α>−1$ parameterizes the family. Suppose $x_i∈[0,1]$ for all $i$. I am trying to find the natural log of the density (log ...
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How to write the likelihood function for an inexact matching design?

How to write the likelihood function for an inexact matching design? For a typical matched case-control design, we select one control for a case based on similar values of matching variable $X_1$ (e.g....
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84 views

Confusion about Fisher information and Cramer-Rao lower bound.

I recently learned about Fisher information and the Cramér-Rao lower bound, but there is something that is bothering me. Take for example a poission distribution, the likelihood function is defined as,...
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43 views

If $\hat{m}$ is the MLE of $m=e^\mu,$ how do I show that the $i$-th element of vector $(\hat{m}- m)$ is asymptotically normal?

Let $\mu=\log_e(m)$ be a column vector space of $X$. Leverage of the ith case is $\hat{a_i}$ And, $$\frac{\hat{m}_i\hat{a}_i-\hat{m}_i^2}{N}.$$ If $\hat{m}$ is the MLE of $m=e^μ,$ how do I show that ...
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198 views

On the notation of the likelihood function

Let $X$ be a random variable realized as the event $(X=x)$. The corresponding likelihood function is given by $$\mathcal{L}_x:\Theta\rightarrow[0,1]$$ $$\theta\mapsto P(X=x|\theta)$$ for a space $\...
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21 views

What is the negative log likelihood of Huber function?

I was reading a paper which gave the negative log-likelihood of a function similar to Huber. I was wondering, what exactly is a "negative" log-likelihood function, and what would it be for ...

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