# Tag Info

Accepted

### Cross-Entropy or Log Likelihood in Output layer

The negative log likelihood (eq.80) is also known as the multiclass cross-entropy (ref: Pattern Recognition and Machine Learning Section 4.3.4), as they are in fact two different interpretations of ...
• 16.4k
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### Maximum Likelihood Estimators - Multivariate Gaussian

Deriving the Maximum Likelihood Estimators Assume that we have $m$ random vectors, each of size $p$: $\mathbf{X^{(1)}, X^{(2)}, \dotsc, X^{(m)}}$ where each random vectors can be interpreted as an ...
Accepted

### Likelihood - Why multiply?

This is a very basic question, and instead of using formal language and mathematical notation, I will try to answer it at a level at which everybody who can understand the question can also understand ...
• 1,819

### What kind of information is Fisher information?

Let's think in terms of the negative log-likelihood function $\ell$. The negative score is its gradient with respect to the parameter value. At the true parameter, the score is zero. Otherwise, it ...
• 15.3k

### Maximum likelihood method vs. least squares method

I'd like to provide a straightforward answer. What is the main difference between maximum likelihood estimation (MLE) vs. least squares estimation (LSE) ? As @TrynnaDoStat commented, minimizing ...
• 6,646
Accepted

### What kind of information is Fisher information?

Trying to complement the other answers... What kind of information is Fisher information? Start with the loglikelihood function $$\ell (\theta) = \log f(x;\theta)$$ as a function of $\theta$ for ...
• 78.2k
Accepted

### Is there an example where MLE produces a biased estimate of the mean?

Christoph Hanck has not posted the details of his proposed example. I take it he means the uniform distribution on the interval $[0,\theta],$ based on an i.i.d. sample $X_1,\ldots,X_n$ of size more ...
Accepted

### Can you give a simple intuitive explanation of IRLS method to find the MLE of a GLM?

Some years ago I wrote a paper about this for my students (in spanish), so I can try to rewrite those explanations here. I will look at IRLS (iteratively reweighted least squares) through a series of ...
• 78.2k
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### What regression/estimation is not a MLE?

Least squares is indeed maximum likelihood if the errors are iid normal, but if they aren't iid normal, least squares is not maximum likelihood. For example if my errors were logistic, least squares ...
• 283k

### Why is everything based on likelihoods even though likelihoods are so small?

The key lies not in the absolute size of the likelihood values but in their relative comparison and the mathematical principles underlying likelihood-based methods. The smallness of the likelihood is ...
• 731

### Cross-Entropy or Log Likelihood in Output layer

Expanding on @dontloo's answer, consider a classification task with $K$ classes. Let's separately look at the output layer of a network and the cost function. For our purpose here, the output layer ...
• 393
Accepted

### Do we ever use maximum likelihood estimation?

I am wondering if maximum likelihood estimation ever used in statistics. Certainly! Actually quite a lot -- but not always. We learn the concept of it but I wonder when it is actually used. When ...
• 283k

### Maximum Likelihood Estimation for Bernoulli distribution

Its often easier to work with the log-likelihood in these situations than the likelihood. Note that the minimum/maximum of the log-likelihood is exactly the same as the min/max of the likelihood.  \...
• 10.1k

### What kind of information is Fisher information?

Complementary to @NeilG's nice answer (+1) and to address your specific questions: I would say it counts the "precision" rather than the "error" itself. Remember that the Hessian of the log-...
• 44.2k

### MLE convergence errors with statespace SARIMAX

First, mle_retvals should be an attribute of SARIMAXResults if it is constructed using a fit ...
• 1,463

### Is unbiased maximum likelihood estimator always the best unbiased estimator?

But generally, if we have an unbiased MLE, would it also be the best unbiased estimator ? If there is a complete sufficient statistics, yes. Proof: Lehmann–Scheffé theorem: Any unbiased ...
• 8,527
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### Relation between MAP, EM, and MLE

Imagine that you have some data $X$ and probabilistic model parametrized by $\theta$, you are interested in learning about $\theta$ given your data. The relation between data, parameter and model is ...
• 138k

### Maximum likelihood function for mixed type distribution

I admit to puzzling over this question for quite some time earlier in my career. One way I convinced myself of the answer was to take an extremely practical, applied view of the situation, a view ...
• 324k

### Why are the Least-Squares and Maximum-Likelihood methods of regression not equivalent when the errors are not normally distributed?

Short Answer The probability density of a multivariate Gaussian distributed variable $x=(x_1, x_2,...,x_n)$, with mean $\mu=(\mu_1,\mu_2,...,\mu_n)$ is related to the square of the euclidean distance ...
• 78.8k

### Likelihood - Why multiply?

Independence between two events means that the occurrence of one event does not affect the likelihood of the occurrence of the another event . So for any two events $A$ and $B$ in a sample space $S$ ...
• 1,603