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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Approximating the marginal likelihood in Bayesian Model Comparison

Given some data $y$, my interest centers around a collection of models $\{\mathcal{M}_1,\mathcal{M}_2,\cdots,\mathcal{M}_L\}$ representing competing hypotheses about $y$. Each model $\mathcal{M}_l$ ...
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32 views

Profile likelihood

I am considering a normal distribution with mean $\beta_1 + \beta_2\exp(-\phi x)$ and variance $\sigma^2$, i.e. $y \sim N(\beta_1 + \beta_2\exp(-\phi x), \sigma^2) $. My aim is to calculate the ...
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29 views

Correction for Log normal distribution

Context: I have observed continuous data $\boldsymbol{O}$, for each observation $i$ I have an assumed known $\sigma_i$ for each observation I have an expected model value $E_i$. $E_i$ was produced ...
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37 views

Joint binomial likelihood

Say I have two jars from which I draw items, the first one has A items and the second has B items. I draw a1 and b1 items from each with probability p_a and p_b, respectively. Before my second draw, ...
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36 views

Setting up posterior and likelihood of Bayesian for more than one model

If I have a data-set and I would like to fit a model and determine its two or three free parameters, while I know that I can fit twice or three times the model to my data and obtains the free ...
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37 views

In MCMC simulation, how to deal with very small likelihood values that couldn't be represented by computer? [duplicate]

I am working on a Bayesian project based on Stagnant data from a OpenBugs example, which is a changepoint problem. Basically we assume a model with two straight lines that meet at a certain ...
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1answer
25 views

Log Likelihoods of Exponential Families

How can one derive the log-likelihood of the saturated model of an exponential family in general? Differentiating the log likelihood w.r.t $\theta$ gives $y_i=\hat{\mu_i}$ but I don't think replacing ...
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1answer
25 views

Generalised Likelihood Models

Why would you use a re-scaled binomial distribution rather than just the standard binomial distribution as the distributional assumption in a GLM?
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17 views

likelihood ratio testing

Let $Y_{1}, Y_{2}, \ldots, Y_{n}$ denote a random sample from a $N(\theta,\ \sigma^{2})$ population. Consider testing $H_{o}: \theta\geq\theta_{o}$ versus $H_{a}: \theta<\theta_{o}.$ If ...
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19 views

log-likelihood of multivariate regression

I am struggling with R to calculate log-likelihood of multivariate regression (i.e. more than one response, one or more predictors). I would like to have a function taking object of class "mlm" as an ...
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1answer
82 views

Interpreting prior and posterior

I am bit puzzled on how we can interpret the posterior. Assume a coin which is 0.1 probable to be unfair. So our prior probability on the coin being unfair is 0.1, and being fair is 0.9. Also by ...
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1answer
25 views

Can we write the likelihood of a GLM in generality?

So I know we can explicitly write down the likelihood of any specified GLM model, for example the likelihood for the logistic regression model would be ...
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38 views

Cauchy MLE Reliability of Asymptotic Results

I have the following regression model $$ p_i = x'_i\beta +s \varepsilon_i $$ with sample size $n \approx 150$ and 4 independent variables. I have reason to believe and $\varepsilon_i$ is distributed ...
2
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1answer
62 views

Construct the likelihood with asymmetric uncertainties

I want to study the correlation between 2 parameters, this is done by fitting a straight line. I have uncertainties on both parameters. I want to solve my problem using the Bayesian approach, i.e. I ...
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15 views

Strange likelihood construction for information borrowing

I recently encountered a strange model while consulting in industry. The goal was to collect information about the parameter of event $A$, say $\theta_A$, but we only observed events $B$ and $C$, ...
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20 views

The most general definition of the Likelihood function for continuous data (including truncation and censoring)

How would you rigorously define the likelihood function for censored/truncated observations? Even in most lifetime/reliability literature (where these types of observations are frequently encountered) ...
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1answer
83 views

Cauchy distribution (likelihood and Fisher information)

I have a three part question: 1) If I have a $Cauchy(\theta, 1)$ with density: $p(x-\theta) = \frac{1}{\pi\{1+{(x-\theta)^2}\}}$ and $x_1, ..., x_n$ forms i.i.d sample. I see on Wiki that this will ...
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I have problem of my sampling based on secondary data [closed]

I have question and i hope i get the answer I am doing my thesis about the impact of liquidity risk in Islamic banks in GCC countries and I have to choose just two countries one which it ...
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1answer
52 views

How to calculate likelihood of linear regression

This is a pretty basic question, but one I am having a hard time finding an answer to. How do you calculate the likelihood of a simple linear model? Like, say, $$y=\beta_0+\beta_1x+e$$ I am working on ...
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1answer
72 views

Construct the likelihood if measurement uncertainties have a Gamma distribution

I want to construct my likelihood. General case: If my data do come from a line of the form $y = mx + b$ and the uncertainties are normally distributed with mean zero and known variance ...
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1answer
28 views

likelihood of string of events given a string of probabilities

There are two classes of strings of events. E.g. A: 0,0,1,2,2,3,4,0,3,0,0,0 B: 0,0,0,0,3,3,2,1,5,6,7,0 Both class A and B strings exhibit variability. Many (e.g. ...
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2answers
34 views

Finding the posterior distribution

Suppose that an observation $X$ is drawn from the following distribution $f(x|\theta) = \begin{cases} \frac{1}{\theta} & \text{ if } 0 < x < \theta \\ 0 & \text{ if } otherwise ...
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58 views

how to construct the likelihood if my errors are not Gaussian

My aim is to study the correlation between 2 parameters knowing that I have measurement errors in both parameters, i.e. I have uncertainties on the independent and dependent parameters. I want to ...
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22 views

confidence interval comparision to likelihood function

I have question, it is taken from a book. It says: Let $X_1 , X_2$ denote a random sample of size n = 2 from a continuous uniform distribution $U(\theta - 0.5, \theta + 0.5)$ with unknown parameter ...
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Gray box Model Identification with partial state measurements

I'm trying to estimate the parameters of a gray box linear model, but I have only partial observation for the states. Is it possible to apply maximum likelihood method in this case? I try to explain ...
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779 views

I am wondering why we use negative (log) likelihood sometimes?

This question has puzzled me for a long time. I understand the use of 'log' in maximizing the likelihood so I am not asking about 'log'. My question is, since maximizing log likelihood is equivalent ...
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61 views

Is a Bayesian estimate with a “flat prior” the same as a maximum likelihood estimate?

In phylogenetics, phylogenetic trees are often constructed using MLE or Bayesian analysis. Oftentimes, a flat prior is used in the Bayesian estimate. As I understand it, a Bayesian estimate is a ...
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23 views

guidance for picking Likelihood in Bayesian analysis

I'm doing a Bayesian analysis for a time series response and wonder whether it is possible to get the Likelihood function without making distributional assumptions. I suppose my response is ...
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36 views

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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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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426 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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35 views

MCMC: The posteriors are too narrow

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

why is the incomplete log-likelihood difficult to optimize

I am trying to teach myself the expectation-maximization algorithm and the texts say the EM is particularly useful when the incomplete log-likelihood i.e. $P(X|\theta)$ where $\theta$ are the ...
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60 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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416 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 ...
2
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1answer
126 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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424 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
47 views

Learner in R to predict data from a complex function: if a>b then a*b else a+b

What sort of general purpose learner could learn the data generated by the following function: if a>b then a*b else a+b or something of that sort of complexity. Ideally something general enough to be ...
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1answer
18 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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41 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
74 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
63 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 ...
3
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2answers
106 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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Clear explanation of pseudo likehood

In generalized linear mixed model (glimmix) parameters are estimated using pseudo likelihood. I was trying to understand how this type of likelihood calculated. Thanks !!!
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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 ...
6
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1answer
191 views

Beginner learning resources : Pdf and likelihood function for non-Gaussian time series model

I am struggling with exercise problems related to blind system identification where the knowledge about the source input is assumed to be known using maximum likelihood estimation of univariate time ...
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29 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
26 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 ...
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76 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
290 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 ...