I'm struggling to work out why in the likelihood function we have (1-p)^n-x and not (1-p)^(n-sum of x). In previous questions, like the example in chapter 2, the question says 1 observation. How can I tell whether to use x or the sum of x?
Why take "sum of x"? We've observed only one value "x" from Bin(n,p). So likelihood should be " nCx * p^x * q^(n-x) "
We would use the sum of x when we have made more than 1 observation. If there is only one observation, then there is nothing to sum! The prior is what we think before. The posterior is what we think after. Why would we change what we think? Well, from observations. We would want to include everything we observe when we decide what we're going to think after. So, if we've made lots of observations then we would want to include all of them in the likelihood function. If we've only made 1 observation then, again, there is nothing to sum. Hope this helps, John
