I am not able to understand what is $x_0$?
Likelihood for a model
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$\begingroup$
probability
statistical-inference
maximum-likelihood
log-likelihood
2 Answers
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The model apparently has some observables. In this case, you've made an observation and gotten the specific value $x_0$. Now it's asking about the conditional distribution $f(\theta|x_0)$ given the information in the observation.
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0Can you help me with it? I have been trying to do it but have no idea on how to proceed further. My biggest problem is : we don't know it's distribution type so how can we do it? – 2017-02-07
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0The question is really only about normalization. I guess $L$ is the likelihood function? (That notation was probably defined somewhere else.) Is the distribution $f(\theta|x_0)$ normalized for any $x_0$? What about the likelihood? – 2017-02-07
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Regarding the second question - the answer is no. Take for instance $U[0,\theta]$ where $\theta \in [0,1]$, then $$ \int_{[0,1]}L(\theta|x_0) d\theta =\int_{[0,1]}\frac{1}{\theta^n} d\theta, $$ is not even integrable over $[0,1]$ (Actually, $L$ is basically not defined for $\theta=0$, so you can take $(0,1]$ for better ollustration. In this case, the integral is simply diverge).
The likelihood function is a function defined over the parametric space $\Theta$, i.e., it is not a density w.r.t to $\theta$, hence, integration w.r.t the parameters can equal whatever you want or can be even non-integrable at all.