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 $\theta_i$ ?

  • 1
    $\begingroup$ 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

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 $\beta$.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.