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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45 views

How do we analyse likelihood in a dataset? [on hold]

I am working to analyze poverty rate using census data. I have a huge dataset. I want to extract the likelihood from this dataset in order to create patterns for energy consumption. What is the ...
3
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1answer
56 views

Likelihood vs. Probability

I have difficulties with Likelihoods. I do understand Bayes' Theorem $$p(A|B, \mathcal{H}) = \frac{p(B|A, \mathcal{H}) p(A|\mathcal{H})}{p(B|\mathcal{H})}$$ which can be directly deduced from ...
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15 views

Empty asymptotic confidence interval?

I have a sample $x=(4, 3, 1, 2, 2, 2, 2, 5, 7, 3, 1, 2, 3, 4, 3, 2, 3, 3, 3, 4)$ of size $n=20$. from a binomial distribution with 10 trials and probability of success $p$. I am asked to construct the ...
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37 views

How do I think about conditional probability in this situation?

I am trying to understand some data relating the likelihood of a positive stock return following a certain signal. The frequency of positive returns differ across datasets (over a particular time ...
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59 views

Can likelihood be changed when the prior changes?

I have a data which follows gamma distribution and want to know the uncertainty of the parameters of this data. $\text{Data} \sim \text{Gamma} (\alpha, \beta)$ Parameters $\alpha \sim \text{Gamma} ...
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17 views

Marginal Likelihood Prior

I have a model with probability matrix for a distribution of $x$, $y=\{0,1\}$, $p(x,y|w)$ where $w=[w_1,w_2,w_3,w_4]$ $p(x=0,y=0)=w_1$, $p(x=0,y=1)=w_2$ $p(x=1,y=0)=w_3$, ...
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38 views

The probability that one bernoulli process has a higher p than another?

I have two data generating processes that are independent Bernoulli processes with probabilities of success $p_A$ and $p_B$. I am taking repeated samples from these two data generating processes, so ...
3
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133 views

How to decide on the MLE when pmf is 0?

Suppose you have $\theta=\{1,2\}$ and the sample of (0,1,2) with the task of finding MLE: \begin{array} {|c|c|c|} \hline x & p(x|\theta=1) & p(x|\theta=2) \\ \hline 0 & 1/2 & 1/4 \\ ...
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21 views

likelihood of a model

the likelihood of a model is defined as the probability of data given model: Likelihood(Model) = p(DATApoints | Model) which is equivalent to the product of all p(datapoint | Model) for each ...
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36 views

how to choose from Poisson and Poisson random effect model?

i have fitted both a Poisson random effect and Poisson model to my data,however some of the estimates are of different signs and SE are quite similar for both of the models. The log likelihood for the ...
3
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1answer
55 views

When is Likelihood Function Positive Semidefinite

This may be a very misinformed question, but I cant figure out why its not true. Here goes: According to Wikipedia and this post, the hessian of a likelihood function equals the information matrix, ...
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1answer
65 views

relation between confidence interval and likelihood function

I once meet the following question,which is also listed by book written by Cosma Rohilla Shalizi ...
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1answer
53 views

Likelihood score function 101

I have some trouble with score functions in likelihood calculation. I'm not good at statistics or probability, so I'm still confused on formalism and mathematical-probabilistic language. Some ...
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0answers
26 views

Can I split this likelihood term

I have a likelihood that is modelled using the IID distributed noise assumption. Now the likelihood at a 3D location $i$ is normally distributed with 0 mean and some precision $\sigma$. So, I can ...
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3answers
152 views

Posterior very different to prior and likelihood

If the prior and the likelihood are very different from each other, then sometimes a situation occurs where the posterior is similar to neither of them. See for example this picture, which uses normal ...
2
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0answers
51 views

Expectation of density ratio of two iid variables

Let $X \sim N(0,1)$ and $Y \sim N(0,1)$ be independent RVs and let $f$ be their density function. I'd like to compute the expectation of the density ratio \begin{align} ...
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22 views

Fitting of bivariate data to a self-defined probability density function

I have a bivariate set of data points which I want to fit to a self-defined distribution (i.e. not standard normal or chi-square or like that, a different, let's say "new" density function). I would ...
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1answer
46 views

Posterior and Likelihood probabilities meaning [duplicate]

I am a computer scientist, so I have a background at maths (however limited). I am reading about posterior distribution from here http://en.wikipedia.org/wiki/Posterior_distribution . It says there: ...
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58 views

Fisher information of a statistic

I have a random sample $(X_1, X_2,...,X_n)$ and I have an estimator $\bar{X_n}=\sum_{i=1}^{n} X_i$ I need to compute the Fisher information of $\bar{X_n}$. The Fisher information is defined as ...
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62 views

Likelihood principle: difference between weak and strong version

Does anyone understand the difference between weak likelihood principle and strong likelihood principle?
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1answer
24 views

Log-likelihood via posterior sampling

I wish to evaluate the following quantity $\log\int_\mathcal{R}p(y|\beta)p(\beta)d\beta$ where $p(y|\beta)=exp(-|y-\beta|)$ and $p(\beta)=\mathcal{N}(0,1000)$. The $y$'s are known. You could probably ...
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1answer
80 views

Likelihood of censored data

Let $X_1,X_2,\ldots, X_{n_1}$ be IID with PDF $f(x-\theta) $, for $-\infty<x<\infty$ and $-\infty<\theta<\infty$. Denote the CDF of $X_i$ by $F(x-\theta)$. Let $Z_1,Z_2, \ldots, Z_{n_2}$ ...
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1answer
100 views

Proof that the log-likelihood is asymptotically quadratic

I was reading this article, where the author says that Maximum Likelihood (ML) estimates are asymptotically normal if the log-likelihood is asymptotically quadratic. I have heard or read other ...
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43 views

Multiparameter log-likelihood function in R and power analysis (by using simulations)

I would like to estimate power of the following problem. I am interested in comparing two groups that both follow e.g. Weibull distribution. So, group A has two parameters (...
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2answers
149 views

Most suitable algorithm for optimizing Maximum likelihood function

What is the most suitable optimization algorithm for optimizing maximum likelihood estimator? In excel I used GRG non linear optimization algorithm, is that good enough? I want to write my own code ...
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63 views

Lizards probability

I am trying to learn basic probability and I would like to get the answer of the questions below. I honestly have no idea on how to even start solving it. Could someone show me or give me some advice ...
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164 views

Minimizing relative error (or mean square error) and maximizing likelihood

I'm not a statistician, so I would appreciate an answer in the simplest possible words. I've read that, in some sense, when we minimize the mean square error, we are maximizing the likelihood. This ...
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28 views

Testing time homogeneous Markov chains

I am working with different transition diagrams and want to calculate the likelihood ratio statistic for testing time-homogeneous. I saw that there are already some comparable questions, but I still ...
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34 views

Formal statistical test for comparing likelihood distributions obtained via MCMC

I am trying to formally compare the distribution of the likelihood values generated using two different models with marginal posterior values of the parameters obtained using MCMC in order to assess ...
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1answer
101 views

Doubt in derivative of logarithm

This seem to silly but I wanted to confirm if the derivative of the log-likelihood $\hskip 2 pt l(x_i)$. The derivative of $$\frac{d (\sum_{i=1}^{M} log(x_i))}{dx} = \frac{1}{x_i} \sum_{i=1}^{M} ...
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1answer
62 views

How can I use likelihoods to compare these three groups? Should I want to do this?

EDIT: Perhaps I should also note that my initial attempt at analyzing this data used a hierarchical model in rJAGS sampling the means from uniform distributions and variances from gamma distributions ...
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68 views

How to use profile likelihood?

I am new to profile likelihood and do not really understand what advantages it may have. Lets say I have the following results estimating the means of three groups. What can I say about them? R ...
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177 views

Multivariate gaussian log-likelihood

Let's say that we have a 2x5 matrix A where rows correspond to observations and columns correspond to variables from a p-variate Gaussian distribution, and we want to learn the inverse covariance ...
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108 views

Likelihood function of truncated data

I am having a little trouble understanding the concept and derivation of the likelihood of truncated data. For example, if I want to find the likelihood function based on a sample from a ...
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19 views

Likelihood of Conditional Grouped Continuous Model

I would like to find MLE of the likelihood above by using optim function in $R$. However, I couldn't understand the terms. I couldn't write the likelihood in $R$. I have the data given, some of ...
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28 views

How to find the Likelihood function of superposed processes for MCMC?

i'm trying to understand how MCMC works. In my special case I have to stochastic processes which are superposed. I observe data $Y=\{Y_1,...Y_T\}$ and assume, that there are some latent variables ...
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91 views

Measure the goodness-of-fit in boosted regression tree

What is the apropriate statistic to measure the goodness-of-fit in Boosted Regression Tree (or Gradient Boosting Regression) with continuous response? How can I calculate the coefficient of ...
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27 views

Likelihood clarification

In a 20 station assembly line, each station has 0.5% chance of defect. at the end of the assembly line, quality control detects 80% of the defectives. they will be sent for rework. 95% of the rework ...
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1answer
99 views

How do I combine multiple prior components and a likelihood?

Lets imagine I am comparing two groups of animals (treatment/control). There is previous data from cell cultures indicating the treatment should have a positive effect. This gives me "prior component ...
4
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1answer
187 views

When does it make sense to reject/accept an hypothesis?

In To P or not to P: on the evidential nature of P-values and their place in scientic inference, Michael Lew has shown that, at least for the t-test, the one-sided p-value and sample size can be ...
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66 views

Log Likelihood and Score Vector of Conditional Grouped Continuous Model (CGCM) when the data is given

I am interested in the relationship between weight Y_i at mating and the number Z_i of lambs born in a flock of n=25 female sheep.Assume Z_i can be 0(no lambs born) 1 and 2, then I know that Y_i* ...
2
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1answer
210 views

Recalculate log-likelihood from a simple R lm model

I'm simply trying to recalculate with dnorm() the log-likelihood provided by the logLik function from a lm model (in R). It works (almost perfectly) for high number of data (eg n=1000) : ...
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1answer
131 views

How can I prevent Overflow for the Gamma function?

I have a Dirichlet distribution from which I'm maximizing $\alpha$ by calculating the log likelihood with the following equation $p\left(s|\alpha\right) = ...
3
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1answer
453 views

How to interpret and compare models in Cox regression?

I am trying to interpret the results of a Cox regression; I am doing a PhD in medicine. I love statistics but my question is still pretty basic, I think, and I did not find an answer in previous ...
1
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1answer
149 views

Treating the log-likelihood as a probability distribution and normalizing?

I have modeled a distribution, $f$, over a r.v. $x \in \mathbb{R}^3$. At inference a set of measuring points, $X$, of the r.v. variables show up. I want to form a distribution over this sample set so ...
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1answer
69 views

How to approximate 0 in transition probability matrix without loss of generality?

In trying to implement Mixture Markov Model, (see question here), I have extreme cases ( e.g. 0's in the Transition Probability Matrix). I have approached this with replacing 0 with 1e-17. However, I ...
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1answer
200 views

Log-likelihood distance measure validity for clustering

I have calculated log-likelihood distances between 50 sequences according to the Formula (1): $$ D(X_i,X_j)= 1/2(\log p(X_i|Mod_j)+\log p(X_j|Mod_i)), $$ where $ p(X_i|Mod_j) $ is the likelihood ...
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2answers
114 views

E-step in EM-algorithm using MAP estimate (mixed Markov models), what does it calculate?

I am trying to grasp what exactly is "estimated" in the E-step of the algorithm. According to all definitions, in E-step the "conditional expectation values , or posterior probabilities of the ...
3
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1answer
118 views

Likelihood of multiple event times modeled as independent Poisson processes

I am modeling three events A, B, and C as Poisson processes with rates $\lambda_A$, $\lambda_B$, and $\lambda_C$ and I would like to calculate the likelihood of observing some data given my model. A ...