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In the extreme case, possibly you could find a function $\delta(\theta)$ that makes the likelihood function constant in the entire range. … That would require you to solve for a function $\delta(\theta)$ that makes the likelihood constant. …

The posterior is prior$\,\times\,$likelihood$\,\times\,$constant; the uniform density is simply a constant and gets absorbed in the other constant term. … Take as an explicit example the prior $\mathrm{uniform}(0,1)$; then, since the prior pdf is $f(\theta) = 1$, prior$\,\times\,$likelihood = 1$\,\times\,$likelihood = likelihood. …

From the Details section we have: The log-likelihood and hence the AIC is only defined up to an additive constant. …

While we commonly refer to "the" likelihood function, this is actually a class of functions defined up to a positive multiplicative constant. … Moreover, multiplication by a constant does not even change the fact that you are still using a valid likelihood function. …

First, likelihood cannot be generally equal to a the probability of the data given the parameter value, as likelihood is only defined up to a proportionality constant. … If you choose a different model you get a different likelihood function, and you can get a different unknown proportionality constant. …

Why does the constant go away? … More philosophically, a likelihood is only meaningful for inference up to a multiplying constant, such that if we have two likelihood functions $L_1,L_2$ and $L_1=kL_2$, then they are inferentially equivalent …

We often don’t care about the likelihood, just the value for which the likelihood is maximized. … When you use the likelihood function to find a maximum likelihood estimator, you get the same point giving the maximum whether you include constants out front or not. …

When you multiply the likelihood by the prior, the resulting function may no longer integrate to $1$, hence why you need to know the normalising constant to analytically solve the posterior. … Hence why you don't need to know the normalising constant when emperically estimating the posterior by MCMC. So theoretically, multiplying the likelihood by some constant should not affect MCMC. …

What happens is that boxcox transform maximizes a likelihood function constructed from a constant variance normal model. … What happens is simply that the contribution to the likelihood function from constant variance assumption greatly overshadows changes to the likelihood by small changes to the form of the basic density …

model likelihood $\pi(\theta)$ is the prior density is a misnomer in that it is not a likelihood function [as a function of the parameter], since the parameter is integrated out (i.e., the likelihood … See also Normalizing constant in Bayes theorem Normalizing constant irrelevant in Bayes theorem? Intuition of Bayesian normalizing constant …

No, since for linear regression log likelihood is a sum of squared residuals plus some other terms, log likelihood is scale dependent. … So for the same model multiplying the regressors by some constant will change log likelihood but R squared will remain the same. …

With prior $\pi$ and likelihood $L$, you can write the marginal likelihood of a model as $$m(y)=\int \pi(\theta) L(\theta;x)d\theta.$$ If your prior is improper, nothing stops you from replacing $\pi … (\theta)$ by $K\cdot \pi(\theta)$, thus multiplying the marginal likelihood by an arbitrary constant $K$. …

For example, if you want to estimate parameter $p$ of binomial distribution by maximizing it's likelihood, then the only the only thing you need is the binomial density known up to a normalizing constant … The constant is $ \binom n k$ and since it does not change anything about finding maximum of likelihood function $L(p)$, it is not needed. You can ask: So what? …

Reproducing an answer of mine from here: Technically, a probability cannot be >1, so a log-likelihood cannot be >0, so a negative log-likelihood cannot be negative. … If we had a discrete distribution (so that the likelihood was really a probability and had to be <= 1) and the normalization constant was sufficiently small, dropping the normalization constant could in …

It is called the marginal likelihood because: this constant value $p(D)$ is what you obtain when you integrate over all possible values of $H$, leaving you with the probability of observing $D$ under your … It is called the normalizing constant because: this constant value $p(D)$ normalizes $p(D|H)p(H)$, making it a proper distribution that integrates to one. …