26 votes

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 ...
ADAM's user avatar
  • 601
9 votes

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

First, as others have mentioned, we usually work with the logarithm of the likelihood function, for various mathematical and computational reasons. Second, since the likelihood function depends on the ...
Durden's user avatar
  • 1,255
5 votes

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

I can think of two things that might help you. First, likelihoods are defined only to a proportionality factor and their utility comes from their use in a ratio and while they are proportional to the ...
Michael Lew's user avatar
  • 14.8k
5 votes
Accepted

Finding the MLE for a piecewise function

$\require{cancel}$ $$ L(\theta) = f(x\mid\theta) = \xcancel{ \prod_{i=1}^n \frac{x_i^\alpha} {\beta^{n\alpha}}I\{0<x<\beta\} \cdot \prod_{i=1}^n 1 I\{x>\beta\}}. $$ First you have the density ...
Michael Hardy's user avatar
3 votes

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

If you flip a coin which is known to be weighted $100$ times and it comes up heads $80$ times, then you probably have a guess as to what the weight might be. One way to formalize this intuition is to ...
Steven Gubkin's user avatar
2 votes

Bias of an estimator depends on whether you take expectation of the estimator or its inverse

Answered in comments, copied below: It is not the same parameter. Bias is indeed not invariant under re-parametrization, this is a well known property that follows from the fact that $E[f(x)] \ne f(E[...
2 votes

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

likelihood $\neq$ probability The likelihood function is not the same as a probability distribution and it can be defined up to a constant. Seperating likelihood from probability has always been ...
Sextus Empiricus's user avatar
1 vote

SEM: Multivariate normality of the residuals?

Most SEM experts probably agree that violations of multivariate normality are not as problematic nowadays given that appropriate correction methods for the standard errors and test statistics (which ...
Christian Geiser's user avatar
1 vote

How to select the "best" distribution of the errors in linear data?

The first thing to understand about issues like this one is that statistical models live in the land of mathematics rather than in real life, and in real life no statistical model is ever fulfilled (...
Christian Hennig's user avatar
1 vote

How to select the "best" distribution of the errors in linear data?

You can always try different distributions and compare (penalized) likelihoods to get a measure of what fits best. This provides a solution, but can be criticized for being purely data-based as ...
BigBendRegion's user avatar
1 vote

Bayesian estimates for Deming regression coinciding with least-squares estimates

I have a running version of Bayesian Deming regression. I used a fixed variance ratio approach starting from Linnet's work on Weighted Deming, 1988. Thus, sampling a single error term. I also added a ...
Giorgio's user avatar
  • 11
1 vote

MLE for triangle distribution?

MLE's for triangle distributions are now implemented in R's triangle package. There is also a short discussion of the methodology here in the vignette. The ...
R Carnell's user avatar
  • 5,093

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