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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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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10 views

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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396 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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29 views

MCMC: The posteriors are too narrow

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

What is the frequentist take on the voltmeter story?

What is the frequentist take on the voltmeter story and its variations? The idea behind it is that a statistical analysis that appeals to hypothetical events would have to be revised if it was later ...
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2answers
159 views

Determine Maximum Likelihood Estimate (MLE) of loglogistic distribution

I am given two data sets containing dates and losses (in some currency). I have to determine the maximum likelihood estimates of the parameters of loglogistic distribution. I googled and found a ...
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59 views

numerical difference between sum of squared residuals and likelihood

I previously asked a question that got labelled as duplicated because I did not explain it correctly. I should not have used the regression model as an example because I can see how, by using that as ...
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3answers
396 views

Why is type I error not affected by different sample size - hypothesis testing? [duplicate]

I don't understand why the probability of getting a type I error when performing a hypothesis test, isn't affected. Increasing $n$ $\Rightarrow$ decreases standard deviation $\Rightarrow$ make the ...
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1answer
16 views

Picking a probability distribution for observed intensities

I have an experiment that measures "intensity" (in this case, electron density of a molecule) on a grid. The values it gives are non-negative,. I'd like to write a likelihood for this observation ...
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32 views

Determining a greater than chance cutoff for an assessment's results

I recently collected a set of data, and before I completed an analysis of it, I wanted to remove any "bogus" entries from participants that didn't actually read the stories. The study I conducted had ...
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1answer
62 views

Gradient of Log-Likelihood

Considering the following functions I'm having a tough time finding the appropriate gradient function for the log-likelihood as defined below: $a_k(x)=\sum_{i=1}^D w_{ki}\cdot x_i$ $P(y_k|x) = ...
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1answer
49 views

dirac delta function in likelihood function

I have tried to understand this myself but what I have found on the internet so far has not helped. I have a likelihood function that for part of it has the following statement: d0 is the Dirac ...
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2answers
67 views

Finding sampling distribution of normal MLE and likelihood

I'm reviewing old exams in preparation for a statistics final, and I'm stuck on a particular question: Suppose that you have n independent random variables $Y_i$, with each distributed normal with ...
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222 views

What would be an example of a really simple model with an intractable likelihood?

Approximate Bayesian computation is a really cool technique for fitting basically any stochastic model, intended for models where the likelihood is intractable (say, you can sample from the model if ...
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27 views

Bayesian Logistic Regression Likelihood Computation for Binary Data

For computing the likelihood function for a Bayesian Logistic Regression model, I understand the original form to be $$p(y_i|\beta,X) = \prod_{i=1}^{k}\left( \frac{ \text{exp} \{ \beta^{\prime} x_i ...
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1answer
22 views

Marginal Likelihoods for Bayes Factors with Multiple Discrete Hypothesis

If I have some data that I believe is Normally distributed and I just want to test the hypotheses that the mean is equal to 1 of 3 values, my understanding is that the Bayes Factor is the ratio of ...
3
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52 views

Explain log likelihood behaviour

(This question is related to a previous one I made, here) I have a set of 2D observations (measured data) of sample size $N$: $$O = \{(x_1, y_1), (x_2, y_2), ..., (x_N, y_N)\}$$ I also have a model ...
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1answer
106 views

Likelihood ratio test for comparing two exponential distributions

I am trying to use a likelihood ratio test to compare the parameters of two exponential distributions. by this thread Likelihood Ratio for two-sample Exponential distribution I found that I can use ...
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1answer
50 views

Point estimation MLE and MME

Consider the family of probability mass functions given by f(x;k) = 3(4^(k-x)) x = k + 1, k + 2,.... and indexed by parameter k E Z. For a random sample of size n, derive with justification: a) ...
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41 views

Question on Nested Likelihood Ratio Model

Suppose I want to calculate a likelihood test statistic distribution for background only MC using following Likelihood Function with Signal Fraction = $ns$ and $\theta = (a,b,c)$: ...
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5 views

Multiplying distributions with different conditioning

I saw this expression in a UBC machine learning class lecture, and I'd like to understand how the math works. Suppose we're trying to predict a class label $y$ given some data $x$. There are prior ...
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1answer
235 views

“weight” input in glm and lm functions in R

I am confused with the definition of the weights in glm and lm. Using the McCullagh and Nelder (1989)'s notation, If random variable $y_i$ is from the Generalized Linear Model (GLM), then its density ...
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37 views

Two Gaussian Likelihoods with Two Decision Boundaries , 0/1 loss function

we assume that Y := {1, 2}. Then our decision can be re-written as y ∗ = {1 if p(x|y = 1) > p(x|y = 2) , 2 otherwise} with a decision boundary at p(x|y = 1) = p(x|y = 2). How can we construct an ...
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66 views

Maximizing Log-Likelihood Estimation for Changepoint Detection

I'm trying to code the changepoint detection algo described here: ...
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29 views

Recursive Bayesian Estimation, $p(C_k|\mathbf{x})$ as (discrete) likelihood

I''ve been struggeling with this problem for the last couple of days. The main goal is to use the probabilistic classification output $p(C_k|\mathbf{x})$, from for example a logistic regression, to ...
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1answer
53 views

Probability to Likelihood

I have a problem on calculating the likelihood of observing a data point x given the predicted lable. My application is on text classification where I have to detect Spam and No Spam documents. I ...
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1answer
71 views

What would be the likelihood function of a pdf, $p(n)=1-|n|$ for $|n|<1$?

This might seem like a basic question to some but I am utterly confused by the fact that the given pdfs are not Gaussian or any other distribution commonly seen in examples. I have two hypotheses ...
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94 views

Gaussian Process Regression/ Classification

How do we estimate parameters of the model while performing Gaussian Process Regression or Classification? While performing regression, we estimate parameters such that the model is the best fit to ...
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41 views

Cross validation with unequal sample size for the left out sets

I am trying to do cross validation on several (20) subsets of samples, which all have unequal sample size. I cannot subsample so that sizes are equal. Example: batch 1: 500 samples batch 2: 400 ...
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1answer
117 views

R - MLE of modified Champernowne density

I've come across an article (http://papers.ssrn.com/sol3/papers.cfm?abstract_id=704903), in which author wrote about maximum likelihood estimates of parameters in the so called modified Champernowne ...
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111 views

Observed Fisher information under a transformation

From "In All Likelihood: Statistical Modeling and Inference Using Likelihood" by Y. Pawitan, the likelihood of a re-parameterization $\theta\mapsto g(\theta)=\psi$ is defined as $$ ...
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1answer
42 views

Copula estimation

I want to fit a copula distribution. My question is: Is it equivalent to estimate the marginal distributions using marginal samples and later estimate the parameters of a copula to estimating all the ...
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21 views

Marginal pseudo-likelihood and consistency?

For a given set of random variables, $X_1,...,X_n$ we know that in many cases finding the maximum of the pseudo likelihood: $$PL(x_1,\ldots,x_n) = \prod_{i=1}^n p(x_i | ...
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1answer
48 views

How is it meaningful to give fitted values for random effects if they don't appear in the likelihood function?

Software for fitting random effects models tends to output values for the random effects. For example, suppose the model is $$y_i = \alpha_i + \varepsilon_i, \qquad 1 \le i \le n$$ where $$\alpha_i ...
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Segregation Analysis for predicting age-specific cancer risk

I am relatively new to the worlds of bioinformatics and genetics research. I have been tasked with presenting to my lab the potential value of a paper that uses Complex Segregation Analysis for a risk ...
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20 views

How adapt MCMC when (constant) weights for oberservations are introduced?

I have the following problem: I have already set up a model and MCMC sampler for a mixed model without weights, i.e., every observation contributes the same amount of information. Now I would like to ...
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1answer
31 views

Distribution of log-likelihood gradient

My question is simple: Is there any results regarding the distribution of log-likelihood function gradient? It may be asymptotic results as well.
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31 views

likelihood problem with exponential distribution using R

How can I solve this problem using likelihood in R? The data below are the survival time (in years) for 10 patients over 65 years after the diagnosis of a particular type of cancer: 10.5 0.2 7.4 0.8 ...
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14 views

In-sample likelihood ratio test for losses

Probably a very simple question for practitioners but I am doing this the very first time: I have written a programm for the estimation of a conditional variance model (HEAVY) of return data and an ...
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60 views

Problem with Finding Likelihood: Bayesian

I am really unfamiliar with Bayesian methods particularly parameter estimation. Suppose I have a test to find a parameter, theta which is the number of packaged bag for retail sale that could contain ...
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11 views

Trying to reproduce predictive likelihood

I'm trying to reproduce the average predictive likelihood from page 13 (Eq 4.1) from the following paper and I'm struggling to find the function f_M,k(...). It details more about it on page 14 but I ...
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50 views

Distribution of a MA(1) process

Suppose I have this MA(1) model: $y_t = \mu + \epsilon_t + \theta \epsilon_{t-1}$ with $\epsilon_t \sim \mathcal{N}(0,\sigma^2)$ The marginal distribution of $y_t$ for all $t$ is ...
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Log likelihood of a Markov network

This is the plot of the log likelihood function from Daphne Koller's course. Two axes are $\theta_{a_1, b_1}$ and $\theta_{b_0, c_1}$. Can someone please explain for me the shape of this plot? From ...
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75 views

log-likelihood for a beta-binomial in R

Although I may ask a question that is already solved (but I found none that would explicitly refer to this), I would like to know, if (and I am no statistician) I am right with programming the ...
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Marginal Likelihood Latent Variable Model

I am trying to apply the method proposed by Chib in Marginal Likelihood from the Metropolis Hastings output to calculate the marginal likelihood of a logit model the includes latent variables. ...
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22 views

Can we use log-likelihood to cluster classes?

I have an SVM classifier for m classes and n data points (somewhat evenly distributed across each class). Could I use the resulting MxN log likelihood matrix to merge classes that are similar?
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1answer
67 views

Loglikelihood and unusual transformations with binary response models

I was just doing some thought on the transformation for likelihoods to log likelihoods for binary response models and I realized what I am sure many people have realized before that the transformation ...
2
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2answers
104 views

Log-likelihood (and AIC) of robust nlrob model differs from standard nls model

Comparing models generated by nlrob to ones generated by nls, I've noticed that even though the models might be nearly identical, the log-likelihood of the models is sometimes significantly different, ...
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49 views

Estimation of log-likelihood via importance sampling

I am looking at a model trained with stochastic gradient variational Bayes. In this paper an importance sampler is proposed to estimate the likelihood: $$p(x) \approx {1 \over S} \sum_{s=1}^S ...
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Simulated MLE does not exist, when trying to Bootstrap likelihood combinant

Consider this simple logistic model: We have ten $0/1$ observations $y_1,...,y_{10}.$ We model with an intercept and a predictor variable.The ten first observations have predictor value $X_i=0$, ...