# Tagged Questions

A likelihood function gives the probability of observing the given data as a function of a parameter $\theta$.

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### why minimize loss function instead of maximizing reward function?

Why is the "de-facto" in statistics to minimize the sum of squared errors cost function instead of maximizing some reward function like the likelihood function?
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### Logistic mixed model

In the logistic mixed model ${\rm logit}(P(Y_i=1))= α + βX_i + u_i + ε_i , i=1,...,m$, when we know $u_i\sim \mathcal N(0,σu^2)$, and $ε_i\sim\mathcal N(0,σi^2)$, and if we know $σi^2$ in each area ...
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### Unique or multiple maxima of log-likelihood function?

How can I find out if the log-likelihood function has only one global maximum or if it has multiple local maxima?
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### how to find out the likelihood of a model given data

If i have a non-stochastic model that predicts the following dataset: [.2, .2] and the actual dataset found empirically (averaged over participants) is [.3, .3] How would I determine the ...
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### Model selection for nonlinear regression of a Gaussian CDF mixture distribution

I have a number of distributions which I want to fit to a CDF that is comprised of one or more Gaussian CDFs. I was able to use weighted least squares regression to find the best fit parameters for ...
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### Solving a difficult equation for a variable?

I'm trying to obtain the maximum likelihood estimate of the parameters for a model I'm building. I have constants $\sigma$, $\mu$, and $q_0$; a boolean matrix $\alpha$; and vectors $A, \beta, r, d,$ ...
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### 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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### 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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### LogLikelihood Parameter Estimation for Linear Gaussian Kalman Filter

I have written some code that can do Kalman filtering (using a number of different Kalman-type filters [Information Filter et al.]) for Linear Gaussian State Space Analysis for an n-dimensional state ...
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### Normalization of circularly-symmetric complex Gaussian distribution

I have a hard time describing my problem, but I'll try my best. It's all about the well-known zero-mean, circularly-symmetric, multivariate complex Gaussian distribution ...
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### 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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### Likelihood of a set of ranks

I'm sorry if I cannot formulate the problem precisely. Let us consider an ordered set $Q$ with $n_q$ elements, and a number of its (possibly overlapping) subsets $B_1, \dots, B_k$. For a given $i$, ...
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### 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* ...
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### How to interpret Hidden Markov Model parameters (transition matrix, emission matrix, and pi values)?

I am working on channel modeling for cognitive radio using HMM. I've written a MATLAB program for forward, backward and Baum-Welch algorithm for multiple sequences. After given some random input and ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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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### 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)?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...