The likelihood-function tag has no wiki summary.
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How to incorporate costs (into logit model) of false positive, false negative, true positive, true negative if they are different costs?
How to incorporate costs (into logit model) of false positive, false negative, true positive, true negative responses, if they are different costs ? Is it possible to do that on the level of ...
2
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0answers
30 views
Manipulating Binomial Distribution
Recently, I've been reading Yudi Pawitan's book, In All Likelihood.
In the book, there's a section on profile likelihood; the methods explored in this section are subsequently applied to some data on ...
3
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0answers
43 views
Simple question about notation
In the context of likelihood-based inference, I've seen some notation concerning the parameter(s) of interest which I've found a little confusing.
For example, notation such as $p_{\theta}(x)$ and ...
1
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1answer
46 views
Optim result highly dependent on starting value
I want to fit a standardized Student's-t distribution. The log-likelihood is given by:
\begin{align*}
log \mathcal{L}(\nu | l_1,...,l_n)=\sum_{i=1}^n \left( log \left( (\pi ...
0
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1answer
42 views
Multi-parameter log likelihood of Normal distribution for two separate samples
I have been given two sets of independent random variables distributed by two different normal distributions $X_1,...,X_n \sim N(\theta_1,1)$ and $Y_1,...,Y_m \sim N(\theta_2,1)$. And have been asked ...
7
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1answer
114 views
MLE/Likelihood of lognormally distributed interval
I have a variable set of responses that are expressed as an interval such as the sample below.
...
0
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1answer
61 views
Maximum likelihood solution in classification problem
I have a couple of questions regarding the maximum likelihood solution in a classification problem (with only two classes $C_{1}$ and $C_{2}$) Basically, I have the following likelihood function:
...
0
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0answers
73 views
Validity of maximising log-likelihood for maximum likelihood estimation
For reasons owing to mathematical convenience, when finding MLEs (maximum likelihood estimates), it is often the log-likelihood function---as opposed to the standard likelihood function---which is ...
3
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1answer
115 views
What is the name of the estimator that takes the mean of likelihood?
Let $X,Y$ be input and output (observed) continuous variables in $\mathbb{R}$. Let $\{y_1,...,y_n\}$ be the set of $n$ observations. Is there a name for the estimator $\hat x = \int_{x \in X} x ...
3
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1answer
99 views
Calculating the likelihood of time series data when there are missing data
I am trying to calculate the log-likelihood of some time series data given parameter sets estimated in BUGS. I can not figure out how to handle some missing values at random points in time.
For the ...
2
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0answers
110 views
Not quite sure where the log-likelihood function comes from
A Poisson variable $Y_i$ is believed to depend on a covariate $x_i$ and it is propsed that log-linear model with a systematic component $a + bx_i$ is appropriate. In an experiment the following ...
0
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1answer
146 views
Log - likelihood function, why does the summation sign vanish?
I have the log-likelihood function:
$$l(p_i,y_i) = \sum_{i = 1}^n \left( \ln(p_i) + y_i \ln(1 - p_i) \right) $$
And I need to calculate the maximum likelihood estimator of $p_i$. When I do this, ...
0
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1answer
147 views
Kolmogorov-Smirnov test in likelihood function
I want to test how well my data fits a uniform distribution and use this as one factor in a likelihood function I am constructing.
Unfortunately, I have no solid basis in statistics. So far, I ...
5
votes
2answers
250 views
Why does log likelihood function for a model use SSE/n and not SSE/df?
I'm trying to find out how log-likelihood function works for linear regression. I found the formula here and here. Making some experiments with it (see code below), I was quite surprised that the ...
3
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1answer
92 views
KS, AD and loglike results
I'm using R to test some distribution families to my data.
I've done KS, AD tests and determined the loglike.
For one of the data the indications given by KS and AD do not agree with the ones given ...
4
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2answers
239 views
Models for calculating consumer behavior at coffee shop
I have the occasion to sit in a Starbuck's almost every day. I have noticed there are rush hours sometimes. It's like hundred of people decided to buy something at Starbucks at the very same time. ...
5
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3answers
190 views
MCMC to handle flat likelihood issues
I have a quite flat likelihood leading Metropolis-Hastings sampler to move through the parameter space very irregularly, i.e. no convergence can be achieved no matter what the parameters of proposal ...
2
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0answers
87 views
Likelihood for Poisson data
In my book, it says:
Independent random variables $X_1, X_2, \dots, X_n$ are modeled by a Poisson
distribution with mean $\lambda > 0$. The likelihood for $\lambda$ based on data
...
2
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1answer
385 views
Is it okay to compare fitted distributions with the AIC?
suppose I have a data set $x_1, \ldots, x_n$ and I would fit a normal, an exponential and a uniform distribution to them. The fitting function spits out a bunch of goodness-of-fit statistics, e.g. the ...
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2answers
733 views
What is the reason that a likelihood function is not a pdf?
What is the reason that a likelihood function is not a pdf (probability density function)?
14
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3answers
419 views
What are some illustrative applications of empirical likelihood?
I have heard of Owen's empirical likelihood, but until recently paid it no heed until I came across it in a paper of interest (Mengersen et al. 2012).
In my efforts to understand it, I have gleaned ...
4
votes
2answers
124 views
Gaussian Process goodness of fit
Let's say I got a Gaussian Process model $M$ based on some training data. Now I get a stream of sample data of a certain batch size coming in.
The GP does not model a time series, but it's trying to ...
3
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1answer
155 views
Likelihood based model selection
Let's say I got a set of models $M = \{M_1, M_2, \dots M_n\}$.
Now say I got some data $x$ and I would like to know, which model represents the data best.
I know how to calculate the likelihood ...
15
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4answers
593 views
How to rigorously define the likelihood?
The likelihood could be defined by several ways, for instance :
the function $L$ from $\Theta\times{\cal X}$ which maps $(\theta,x)$ to $L(\theta \mid x)$
the random function $L(\cdot \mid X)$
we ...
0
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1answer
170 views
Maximizing: likelihood vs likelihood ratio
Say I have an observed data set ($n_i$) and I want to obtain the best fit out of 10 data sets produced by a model dependent on a single parameter $a$ ($m_i(a)\;a=1..10$).
Suppose I use a Poisson ...
22
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6answers
875 views
Why would someone use a Bayesian approach with a 'noninformative' improper prior instead of the classical approach?
If the interest is merely estimating the parameters of a model (pointwise and/or interval estimation) and the prior information is not reliable, weak, (I know this is a bit vague but I am trying to ...
0
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1answer
157 views
A question about hypothesis testing and maximum likelihood ratio test
Assume there are 2000 students, $m$ boys and the rest are girl. Now we take a sample of 5 students which contains only 1 boy. We claim that there are more girls than boys in the population. So what we ...
6
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3answers
245 views
Finding the MLE for a univariate exponential Hawkes process
The univariate exponential Hawkes process is a self-exciting point process with an event arrival rate of:
$ \lambda(t) = \mu + \sum\limits_{t_i<t}{\alpha e^{-\beta(t-t_i)}}$
where $ t_1,..t_n $ ...
2
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0answers
220 views
Difference between likelihood principle and repeated sampling principle
In statistical inference, there are many fundamental statistical principles, such as likelihood principle and repeated sampling principle. I am wondering whether there are any other principles? And ...
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0answers
300 views
Interpretation of a log likelihood function for PROC NLMIXED in SAS
I have a data set of skewed nutrient intake values, from around 7800 individuals, of whom around 3000 had two measures of daily nutrient intake (the others only had one measure), so this is a repeated ...
2
votes
1answer
149 views
Property of KL-divergence
Let $p_1$ and $p_2$ be two distinct probability distributions. Define
$$
L(q)=D(q||p_1)-D(q||p_2)
$$
where $D$ is the usual Kullback-Leibler divergence. Assume the support of $p_2$ is included in ...
0
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0answers
156 views
Likelihood function of a Linear probability model
What is the Likelihood function of a linear probability model?
I know the likelihood function is the joint probability density, but how to construct the likelihood function when we only have the ...
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0answers
70 views
Is there any stochastic method to approximate the likelihood?
I am looking for a list of stochastic algorithms to approximate likelihoods.
0
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0answers
257 views
Developing the multivariate normal likelihood function (in matrix notation)
I am searching to get a detailed development for the multivariate normal likelihood function in order to enter it to Wikipedia.
Can anyone suggest to me a good reference book (or if you have it ...
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0answers
95 views
Likelihood function for a multiperiod probit with autoregressive latent variable
I'd like to evaluate the likelihood function of a multperiod ordered probit model with an autoregressive random component, but i am having trouble arriving at the likelihood function.
As an example ...
2
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4answers
509 views
How is the bayesian framework better in interpretation when we usually use uninformative or subjective priors?
It is often argued that the bayesian framework has a big advantage in interpretation (over frequentist), because it computes the probability of a parameter given the data - $p(\theta|x)$ instead of ...
1
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1answer
256 views
How can I calculate a probability from a likelihood, e.g. in the Metropolis-Hastings algorithm?
This is a follow-up to my previous question, how can I compute a posterior density estimate from a prior and a likelihood
I am having difficulty understanding how it is possible to calculate the ...
3
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3answers
209 views
What is the correct posterior when data are sufficient statistics?
Say you have N observations that are iid.
$$ \forall i, \quad p(X_i=x_i|\mu,\sigma,I) = \frac{1}{\sqrt{2\pi}\sigma}
\exp\left(-\frac{1}{2\sigma^2}(x_i-\mu)^2\right)$$
then
$$ ...
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0answers
75 views
Help evaluating a posterior probability expression
Consider $\boldsymbol{x}= [x_1,x_2,...x_n]$ and $\boldsymbol{y}= [y_1,y_2,...y_n]$ to be two multivariate Gaussians with an isotropic diagonal variance structure and uninformative priors so that:
...
7
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2answers
241 views
Correlation as a likelihood measure
Various forms of the correlation, e.g.,
$r = \frac{\Sigma_i x_i * y_i}{\sigma_x \sigma_y}$
or
$r = \frac{\Sigma_i (x_i-\bar{x}) * (y_i-\bar{y})}{\sigma_x \sigma_y}$
are popular similarity measures ...
10
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1answer
423 views
What are the disadvantages of the profile likelihood?
Consider a vector of parameters $(\theta_1, \theta_2)$, with $\theta_1$ the parameter of interest, and $\theta_2$ a nuisance parameter.
If $L(\theta_1, \theta_2 ; x)$ is the likelihood constructed ...
1
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1answer
173 views
Expression for conditional density for ARCH processes
I am reading Stephen Taylor's Asset Dynamics book and came across something I didn't fully understand.
For an ARCH process, the return series is modeled as
$r_t = \mu_t + h_t^{1/2}z_t$ where is $z_t$ ...
1
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2answers
1k views
Standardized Student's-t distribution
We know that density for a student-t distribution is given as
$$\frac{\Gamma(\frac{\nu + 1}{2})}{\Gamma(\frac{\nu}{2})} \left(\frac{\lambda}{\pi\nu}\right)^{\frac{1}{2}} ...
3
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1answer
1k views
How to calculate likelihood for a bayesian model?
I am trying to do Bayesian posterior predictive checking, whereby I calculate the DIC for my fitted model, and compare to DIC from data simulated from the fitted model. I can get the DIC out of ...
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7answers
2k views
Why do people use p-values?
Roughly speaking a p-value gives a probability of the observed outcome of an experiment given the hypothesis (model). Having this probability (p-value) we want to judge our hypothesis (how likely it ...
2
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1answer
191 views
How to estimate the likelihood function for random generator of three events?
I have a random events generator. I know in advance the set of event that can be generated (in my case I have only three possible events). The probabilities of the events are not known. I need to ...
2
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1answer
263 views
Likelihood function of DSGE model using Kalman filter
In Frank Schorfheide's class notes on likelihood functions of DSGE models, he expresses the value of the likelihood function for a given vector of parameters $\theta$, and time series $Y^T$ as:
...
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7answers
18k views
What is the difference between “likelihood” and “probability”?
The wikipedia page claims that likelihood and probability are distinct concepts.
In non-technical parlance, "likelihood" is usually a synonym for "probability," but in statistical usage there is a ...
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2answers
112 views
What is the correct likelihood function for an sequential, adaptive data generation process?
Consider the following sequential, adaptive data generating process for $Y_1$, $Y_2$, $Y_3$. (By sequential I mean that we generate $Y_1$, $Y_2$, $Y_3$ in sequence and by adaptive I mean that $Y_3$ is ...
