What is the likelihood function?

Question

When a coin with bias θ for flipping heads is flipped thrice the following result is recorded:
$1st$ Trial: $1$
$2nd$ Trial: $0$
$3rd$ Trial: $1$
($1 =$ heads, $0 =$ tails). Let this data be represented by x. Would the likelihood function (x|θ) be $$θ^2(1-θ)$$ or $${3 \choose 2}θ^2(1-θ)?$$

Thanks!

Answer 1

You know the order of occurrence.

So The likelihood is the first one $\theta(1-\theta)\theta$

---

But for optimization there is no difference since $3 \choose 2$ is just a constant.

Answer 2

In general, the likelihood is the probability of the observed data given the specified parameter values. But that probability is always defined in terms of a particular event space a.k.a. sample space for the data. In your case, there are two possible ways of conceiving of the event space.

One way is think of the space is that there are two possible outcomes, namely $y=1$ (head) or $y=0$ (tail). Then your data consist of $N=3$ data values, $y_1,...,y_3$, and the likelihood of your set of data is $\Pi_i^N \theta^{y_i} (1-\theta)^{y_i}$.

A different way to think of the space is the set of possible total heads $z$ in $N$ flips. When $N=3$, this space has four possible outcomes, namely $z=0$, $z=1$, $z=2$, and $z=3$. Thought of this way, your data is a single datum $z$, and its likelihood is ${N \choose z} \theta^z (1-\theta)^{N-z}$.

When you put either one into Bayes rule, you get the same answer because ${N \choose z}$ cancels out.