I am stuck on a very simple topic in Bayesian statistics: estimating the bias, $\theta$, of a coin based on data, $d$, from repeatedly flipping it.
In particular, various texts describe the likelihood, $p(d|\theta)$, as: $$p(d|\theta)=\theta^H(1-\theta)^{T}$$
where $\theta$ is the probability of heads, and $H,T$ are the number of heads and tails, respectively, in the data. In other words the likelihood is the product of various random variables, characterising each coin flip, drawn independently from a Bernoulli distribution with parameter $\theta$.
My question is: why is the likelihood function not Binomial? $$p(d|\theta)=\binom{H+T}{H}\theta^H(1-\theta)^T$$
In his introductory text on Bayesian Data Analysis, Kruschke explains that "the reason is that here we are considering each flip of the coin to be a distinct event, whereby each observation has just two possible values... Therefore the likelihood is Bernoulli... If we considered instead [the data] to be a single 'event'... then we would need a Binomial..." Can anyone flesh this out, or provide an alternative explanation?