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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57 views

Are we frequentists really just implicit/unwitting Bayesians?

For a given inference problem, we know that a Bayesian approach usually differ in both form and results from a fequentist approach. Frequentists (usually includes me) often point out that their ...
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15 views

Information matrix for a Student's T distribution

I'm reading a paper from Creal, Koopman, Lucas "Univariate Generalized Autoregressive Score Volatility Models" and I'm stuck with this computation. After considering the log likelihood, the score ...
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9 views

Application of Barndorf-Nielsen Formula for Maximum Likliehood Inference and Confidence Intervals

I've seen various forms of what is called the $p^*$ or Barndorff-Nielsen formula for the conditional distribution of the MLE. The most general form I've found is here. I'll reproduce it below: $$ f(\...
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1answer
17 views

drop1 LRT is zero in R

So for my current binomial model I am dropping some components and I found out that for one variable the results look a bit different. For 'hurseason' (class factor with two levels Y/N), the LRT is ...
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5answers
292 views

Wikipedia entry on likelihood seems ambiguous

I have a simple question regarding "conditional probability" and "Likelihood". (I have already surveyed this question here but to no avail.) It starts from the Wikipedia page on likelihood. They say ...
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1answer
29 views

METROPOLIS-HASTINGS with likelihood

I am trying to set up a Metropolis-Hastings algorithm in Matlab in order to estimate the parameters ${\theta}$ (it is a vector of 5 elements) to fit a curve to a set of data $D={X_i,Y_i,\delta_i}$. $X$...
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40 views

likelihood for binomial proportion

A man must evaluate the proportion $\pi \ (0 \leq \pi \leq 1)$ of diesel vehicles crossing a road. He notes the first 10 cars arriving and records 3 diesel vehicles. I know that $f_Y(y;\pi)\sim Bin(\...
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1answer
25 views

Can a likelihood function be integrated to find the CDF and probabilities?

Likelihood analysis uses the likelihood function: $L(\Theta | data) = P(data | \Theta)$ to determine how likely it is that some value is the true population parameter ($\Theta$) compared to some ...
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17 views

Expected ratio given a list of observed ratios

This might be a strange question (or not). Given a case where: I have a box of objects. Within the box are 3 types of objects (A, B, C) I have several observations of the contents of the box made ...
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1answer
32 views

Learning just a decoder (autoencoder without encoder)

I am trying to do something quite unusual: learning a latent representation of some data just by optimizing a decoder. Basically, a probabilistic model of a neural network autoencoder without the ...
2
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1answer
28 views

Need help understand likelihood in survival analysis

I really appreciate it if you could help me understand the likelihood in a survival model with a time varying covariate. To be more clear, let's first start with a survival analysis with fixed ...
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0answers
23 views

How to measure clustering algorithm performance? [duplicate]

For supervised learning, both regression and classification have ground truth. The model performance can be measured against ground truth. For example, $R^2$ in regression or accuracy (0-1) in ...
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53 views

Transforming observed Fisher information

Let $x_1$,$x_2$,....$x_n$ be a random sample from the exponential distribution with p.d.f.: $$f(x;\lambda)=\lambda e^{-\lambda x}\;\;\;x\geq0\;\;\;\lambda>0$$ The log-likelihood is: $$l(\lambda)=...
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22 views

Appropriate logistic regression model and log- likelihood

Are two simple independent causal samples from two distributions of bernoulli medium respectively: $i=1,...,10$ $ π_1 = e^{B_1} / (1 + e^{B_1})$ $i=1,...,10$ $ π_2 = e^{B_1 + B_2} / (1 + e^{B_1 +...
7
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1answer
125 views

Strange likelihood trace from MCMC chain

I've got a model that goes: Single parameter -> Complex likelihood function -> Log-likelihood. I executed an MCMC chain (using pymc) and plotted the trace of ...
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0answers
10 views

Proportional Hazards Model With Multiple Treatments?

Is there some way to do a proportional hazards model where the "treatment" occurs multiple times? Here's an example: Let's say you are running a Ford dealership where you sell only one car... The ...
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0answers
52 views

Recalculate log-likelihood from a R lmer model with one random effect

I'm trying to recalculate the REML log-likelihood given by the logLik function from a linear mixed model with one random effect. I use the ...
2
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2answers
179 views

How to evaluate how well some samples come from some given distributions?

I have a complicated procedure that, given an input $x$, outputs several random samples $S_i(x)$ for $i\in\{1,\dots,k\}$ (each sample consists of $n_i$ points). Each sample $S_i(x)$ follows an unknown ...
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1answer
81 views

Likelihood of observations for HMM with continuous state and observation distributions

I have a "real" and estimated HMM model given as $(\pi,\mu, \nu)$ and $(\pi^{\text{est}},\mu^{\text{est}}, \nu^{\text{est}})$, where $\pi$ is initial state distribution of Markov chain, $\mu$ is state ...
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1answer
14 views

inferential structure determination

I'm trying to get my head around Bayesian inference and the difference between the posterior and likelihood. Going off the back of these answers, I'm under the impression that the posterior is $P(data|...
2
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1answer
19 views

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 $$ p(X)=\...
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36 views

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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21 views

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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1answer
33 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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1answer
86 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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1answer
35 views

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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1answer
13 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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27 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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42 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 : $(n+1/2)^m\...
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36 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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38 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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18 views

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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1answer
287 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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28 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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2answers
40 views

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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1answer
24 views

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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33 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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14 views

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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25 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 ...
0
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0answers
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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1answer
89 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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34 views

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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1answer
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) &= p(\...
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1answer
84 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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24 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. $N(\mu, 1)$, 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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30 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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19 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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11 views

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 $\theta_j=\beta$ ...