# Questions tagged [log-likelihood]

For questions that use the natural logarithm of a likelihood function.

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### How do I find the expectation value of a log likelihood function?

I am working on a problem where I need to find the fisher scoring update for two maximum likelihood parameters $a_1$ and $a_2$ that are part of a Poisson distribution. I have the derivatives of the ...
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### negative sign of the Log-likelihood gradient

I was watching an explanation about how to derivate the negative log-likelihood using gradient descent, Gradient Descent - THE MATH YOU SHOULD KNOW but at 8:27 says that as this is a loss function we ...
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### Quadratic Approximation for Log-Likelihood Ratio Processes

I'm trying to understand why the quadratic equation can approximate the log likelihood ratio, and how it is derived. enter image description here Is this approximated using Taylor's series or normal ...
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### is the mix of convex and linear functions always convex function?

I want to prove that the following composed function $g \circ L$ is always (strictly) convex : \begin{alignat*}{3} &g&&(t&&) && =-\log(1-e^{-t}) \qquad && \text{...
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### How to find the value that give the global maximum of a function?

I have got the following function: $$L(x,y) = \frac{(16.2x +0.9y+5.2)^{24} e^{-(16.2x +0.9y+5.2)}}{24!} \cdot \frac{(2.1x +4.2y+0.9)^8 e^{-(2.1x +4.2y+0.9)}}{8!} \tag{1}$$ and I am attempting at ...
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### How to find the global maximum of a likelihood-function?

I have an expression for a likelihood function,$L_{A,B}$, with two parameters, $(A,B)$. I want to find the global maximum for this function, how can this be done? A colleague has told me that: "...
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### Simplifying $-\log\left(\left( \frac{1}{2\pi\sigma^2}\right)^{n/2} \exp \left( -\frac{1}{2\sigma^2} \sum_{i=1}^{n}(\mu-y_i)^2\right)\right)$

I'm not the most mathematically minded, but I'm doing my best to learn MLEs. We were provided the following original likelihood function of a Normal Distribution and I am having trouble understanding ...
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### Log-Likelihood Function of Piecewise-Defined Function

I'm a data analyst, no mathematician. Mostly I do 'standard' stuff such as linear mixed-effects regressions or generalized additive mixed models. But now I need to determine the maximum log-likelihood ...
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### Likelihood ratio hypothesis test for the exponential distribution.

I am trying to understand the following logic that I found on the internet that implements a hypothesis test on the exponential distribution using the likelihood ratio test. We want to test the ...
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### How to find the log-likelihood for this density?

We let $(X,Y)$ be stochastic variables with values in $N × R$ determined by $X = x$ being subtracted from one Poisson distribution with mean $\lambda$ after which Y is subtracted from a gamma ...
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### Likelihood values from Sigmoid

There are multiple doubts of mine associated around this theme: In MLE, we try to find the PDF parameters ($\theta$) which maximise the likelihood of the observed data ($L(\theta | data)$). To get ...
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### Hypothesis testing and critical regions

Suppose $X$ follows a chi-squared distribution with $r$ degrees of freedom, where $r \in \mathbb{Z}^+$ is unknown. Let $X_1, X_2, ..., X_n$ be a random sample of $X$. (i) Suppose we have the following ... 1 vote
In inference statistics, one uses the quantity "likelihood" (L), and derivates it with respect to a given parameter $\theta$ (or set of parameter) in order to maximize the likelihood. Thus ...