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For an assignment that asked "Compute the likelihood function when $Y=10$", I had written:

$$ L(\theta|n=30, y=10) = {30 \choose 10} \theta^{10} (1-\theta)^{20} ; \theta \in \left[0, \tfrac{1}{5}, \tfrac{2}{5}, \ldots, \tfrac{5}{5}\right] $$

I will omit the context of the problem since my question is not about that. My professor then suggested to me that I should make a change in notation, specifically to change $\theta$ into $\theta_i$, but I don't see why it should be the case. As I understand, the likelihood function takes as an input the parameter and outputs a likelihood value. What I interpret from the expression that I wrote above is "the likelihood for a specific value of $\theta$ given the data is ..., and $\theta$ can only acquire these possible values". Is there anything wrong with this/why should I using $\theta_i$ ?

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    $\begingroup$ I even think your notation is desired, since writing $\theta_i$ may lead one to thinking the parameter $\theta = [\theta_1, \dots, \theta_n]$ is a vector $\endgroup$ – Łukasz Grad Mar 4 '17 at 12:58
  • $\begingroup$ The likelihood function is a function of the parameter $\theta$ as in your notation. The value(s) that maximise(s) the likelihood is (are) one particluar values $\hat{\theta}$ for $\theta$. $\endgroup$ – user83346 Mar 4 '17 at 14:34
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You normally use $\theta_i$ to indicate that the probability of "success" depends on each observation, i.e. it is not constant across your sample. Logistic regression is a famous example of such practice. There you model each probability in terms of one or more explanatory variables:

$$\theta_i = \alpha + \beta x_i$$

Having said that, I don't see anything wrong with your notation. Beware, however, the likelihood can be a function of more than one parameter. In the above model, for example, the parameters you want to estimate are both $\alpha$ and $\beta$.

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