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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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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48 views

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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24 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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30 views

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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24 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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3 views

How to plot a function involving summation and logs in R [migrated]

How to plot in R the following function $l(\theta) = ln(\theta)*\sum{y_i} -n*\theta -n*ln(1-e^{-\theta})-\sum{ln(y_i!)}$ where the summations are from $i=1$ to $n$ I have the data set, but I know ...
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18 views

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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9 views

What is integration of empirical likelihood(EL) with respect to random variable? [closed]

Given i.i.d sample $X_{1},...X_{n}$ with size $n$, the empirical likelihood of population mean case is given by (see, Owen 1988) $EL(\underline{X},\theta)=max_{0<p_{1},...p_{n}<1}\left \{ ...
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1answer
17 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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15 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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32 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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1answer
38 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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19 views

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 $B = \{B_1, ..., B_n\}$. ...
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13 views

Sampling from an Isometric Multivariate Gaussian

I implemented a Multivariate Gaussian Distribution. I know that it will be isometric, so the covariance matrix will contain solely diagonal values. As standard way to sample from a Multivariate ...
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1answer
12 views

Basic question regarding construction of likelihood function from a Cox PH model

I have a simple question (which I sense may have a complicated answer) regarding the fundamental logic concerning how the likelihood function from a Cox PH model is derived. Assuming the $i^{th}$ ...
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34 views

Calculating the Deviance Information Criterion for a Hierarchical Regression Model

I'm not entirely sure how to phrase this question but maybe some background information might help. I am using MATLAB to perform hierarchical bayesian regressions and so I really need to understand ...
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22 views

Stochastic Gradient on the Simplex

I have a probability density function defined over $x_1,...,x_K$ with a simplex constraint $\sum_{k=1}^K x_k = 1$. I'm trying to perform stochastic gradient descent on this density. I know I can keep ...
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24 views

Importance Sampling of a Student-t with mean 0, $\sigma^2$ and degrees of freedom unknown

I want to calculate the posterior parameters of a Student-t by importance sampling. In order to tune the importance density I sample from the prior first. My data is demeaned, the degrees of freedom ...
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14 views

Likelihood ratio test in R for Generalized Pareto Distribution (GPD)

I am trying to use a likelihood-ratio test to compare between two Generalized Pareto Distribution (GPD) models. All the functions and packages I found are for Linear Models (LM) or Generalized Linear ...
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41 views

Using confidence interval to learn about likelihood?

likelihood and 95% CI What is the likelihood that there will be 70 murders, 2500 assaults and 5000 Burglaries next year? (Hint: Construct a 95 percent Confidence Interval using the data.) I ...
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5 views

Similarities between 2 groups from different data set

I have 2 groups, A and B. I got 2 independent experiment data that stated these data are A and B. Now, I want to compare from experiment 1 and experiment 2 the similarities of A and B. I have filtered ...
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1answer
99 views

How to write down the log-likelihood expression for this moving average model

Question is based on the paper Maximum likelihood for blind separation and deconvolution of noisy signals using mixture models pdf download link Assume the model is a FIR system of order 2 expressed ...
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61 views

How exactly does one marginalize over parameters in an N-dimensional likelihood?

I see no equations for the following, so I'm not sure exactly what they are talking about: "For each model, we determine the best fit parameters from the peak of the N-dimensional likelihood surface. ...
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30 views

Prior pdf decay in Recursive Bayesian Estimation

I'm doing Recursive Bayesian Estimation numerically. I have a state vector, x, that I'm trying to estimate by regularly taking noisy measurements, z. I use Posterior = Likelihood x Prior / ...
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51 views

What does “PDF overlap” mean? “To see whether a probability density function overlaps”

I ran across the following sentence in a journal: "To see whether a probability density function overlaps" What does this word mean in the statistics literature, "overlaps"?
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30 views

How to compute the likelihood distribution?

I am not an expert in statistics, but I need the bayesian inference to resolve a problem in artificial intelligence. The problem is not about estimation, no. The problem is about computing the ...
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1answer
16 views

Likelihood Ratio Test on Multiple Samples

I have 32 samples and I can calculate the likelihood of each data point individually, but how would I calculate a total likelihood for the model (i.e. how well it fits all data points?) Thanks!
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21 views

Likelihood over a arbitrary set of hypotheses

I have the following issue. I have finite, large, enumerated set $H$ of hypotheses that maps a time series to an integer, let's say days to integers i.e. $H \subseteq (\mathrm{Day} \to \mathbb{N})$. ...
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16 views

Likelihood ratio test: multiple observations

I have 32 samples and I can go through and calculate the likelihood of my observed data under different models and do a LRT for a given point under multiple models. However, is there a way that I ...
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17 views

Maximum likelihood estimator for a variable and its reconstruction

I have a spatial variable $X_1$ observed at time $t_1$ and its original form $X_0$ observed at time $t_0$ (I don't have any info between these two stages). I want to reconstruct $X_1$ on the basis of ...
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60 views

Posterior probability of the max of uniform distribution

Let $y$ be a uniformly distributed random variable over the interval $(0, \theta)$, whereby Pr$(\theta = 1)$ = Pr$(\theta = 2) = \frac{1}{2}$ are the prior probabilities of the parameter $\theta$. If ...
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1answer
41 views

Bayesian Information Criterion and Basic Marginal likelihood identity

Given the basic marginal likelihood identity: $$ ln \; m(y)=ln \; p(y|\theta^*)+ln\; \pi(\theta^*)-ln \pi(\theta^*|y) $$ is there a way to derive from this the Bayesian Information Criterion? ...
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53 views

Likelihood overflow in Metropolis-Hastings acceptance probability

Consider a Bayesian framework where we have priors for some parameters and a likelihood based on the data. Consider the likelihood (and its parametric format) to be very sensitive to the choice of the ...
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1answer
80 views

Plotting PMFs and PDFs

This is probably a silly question, but I was reading Computational Statistics with Python and there are a few plots describing prior, likelihood and posterior distribution in the context of ...
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1answer
27 views

How does R calculates convergence tolerance and what does it stands for?

I'm using R for the first time for estimating parameters of a vector function. In the summary of the method I used (nls to be precise), the information on "achieved convergence tolerance" appeared. ...
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212 views

What would be the reason that the posterior distribution looks like the prior using MCMC

I am trying to use MCMC to obtain the posterior probabilities of the free parameters of a model. I have tried first to leave two free parameters for my model and I was able to estimate the posterior ...
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33 views

What is the logic behind the MLE [duplicate]

I just needed some help understanding the concept behind MLE; and why when we have the observed values and a likelihood function, does maximizing this likelihood function with certain parameter values ...
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1answer
40 views

Why need frequentist considered “a distribution of possible data sets D” in likelihood function p(D|w)?

There some words in PRML: In a frequentist setting, w is considered to be a fixed parameter, whose value is determined by some form of ‘estimator’, and error bars on this estimate are obtained by ...
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299 views

What is the correct definition of the Likelihood function?

I am doing the CS229:Machine Learning of Stanford Engineering Everywhere. All trhough the first chapter he uses $$L(\theta) = P(Y | X; \theta)$$ i.e. the likelihood of the parameter $\theta$ is ...
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75 views

Maximize *custom* likelihood function for logistic regression

My dataset contains n observations $X_i$ of n individuals and I want to predict a binary outcome $Y_i$. logistic regression model It is fair to assume that this $Y_i$ is the realization of a ...
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32 views

calculation of the log-likelihood evolution for my MCMC simulation resulted in infinite values

i want to plot the log-likelihood evolution for my MCMC simulation results. i was estimated a Dirichlet mixture model parameters using the Gibbs sampling method in R environment and utilizing the ...
2
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1answer
94 views

Confusion about concept of likelihood vs. probability

I've been recently trying to wrap my head around the concept of likelihood, and have made some good progress, but there is one thing that is bugging me, and I think this issue is what makes the ...
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27 views

Likelihood function and plot Pareto distribution for posterior distribution

I implemented two ways to obtain the likelihood of a Pareto distribution with unknown $\alpha$, to find the posterior distribution for $\alpha$. ...
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39 views

Posterior of alpha parameter (Shape) of Pareto Distribution

Im trying to generate the posterior distribution of $\alpha$ parameter of Pareto Distribution. I did all the job correctly on paper, but when i go to implement in R i have some problems.I have a ...
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26 views

Negative binomial log-likelihood in penalized regression

I am trying to understand how penalized logistic regression works and I got stuck with negative binomial log-likelihood. I understand the the first two formulas and the penalization part in the ...
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20 views

Question about log likelihood and Bayes estimator

The question is the picture. Thanks for helping.
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1answer
37 views

Likelihood-ratio test for three models?

The likelihood-ratio test is the optimal test for comparing the goodness-of-fit of two models. Is there some similar test that would allow me to test three models, or should I just compare models (A, ...
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22 views

plot log likelihood function evolution in mcmc simulations

Is it possible to plot log likelihood function evolution in mcmc simulations? I have a mixture model and its parameters are estimated using the gibbs sampling method in r environment and using the ...
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19 views

Finding a likelihood function for similarities

in order to compare human action sequences and computer modeled predictions for action sequences I use a similarity measure for these sequences. All similarities can have a value between 0 (terrible ...
0
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1answer
9 views

Clarification on marginal, posterior and likelihood distributions?

I would like to clarify a basic question about how to use the concepts of marginal, conditional, and likelihood distributions. When we observe data from phenomena $x$. We are looking at realizations ...