Bernoulli parameter in the following partitioning schemes

Question

Suppose I have a Bernoulli process: $2n$ Bernoulli observations whose locations are distributed over an interval, say $(0,1]$. Conditional on the locations, $x_k\stackrel{iid}{\sim}Bern(\theta)$, $k=1,...,2n$. $n$ is known, but $\theta$ is not. The likelihood is then $$ \mathcal{L}(\theta|x_k's)= \prod_{x_k=\{0,1\},k=1,...,2n}\theta^{x_k}(1-\theta)^{1-x_k}=\theta^{\sum_k x_k}(1-\theta)^{2n-\sum_k x_k} $$ I have two partitioning schemes:

1) Partition interval $(0,1]$ into equal-length segments, $(0,1/2]$ and $(1/2,1]$ with potentially unequal number of $x_k$'s in each of the two equal-length segments.

2) Partition interval $(0,1]$ into segments of potentially unequal length, but $n$ $x_k$'s in each segment.

Let's say $X_1=\sum_{x_k\in\text{segment 1}}x_k$ and $X_2=\sum_{x_k\in\text{segment 2}}x_k$ are two random variables. Given my data generating process, is there any reason to think that my parameter $\theta$ should be different for $X_1$ and $X_2$ when I choose one partition scheme over the other?

In other words, for partition scheme 2, I expect $X_1$ and $X_2$ to follow a binomial distribution with size $n$, but can the parameter be the same because they came from $iid$ Bernoulli trials? For partition scheme 1, my gut tells me that the parameters won't be equal.

Answer 1

The probabilistic model you describe is such that the Bernoulli variates $X_1,\ldots,X_{2n}$ are iid conditional on the locations. Since you do not describe the distribution of those locations, it is impossible to know whether or not the distribution depends on the parameter $\theta$. If not, that is, if the locations are governed by an independent process that does not involve $\theta$, or if the locations are deterministic rather than random, the term "conditional" in your description is irrelevant. And we can assume we have $2n$ iid Bernoulli variates $X_1,\ldots,X_{2n}$.

Since the partition of the index set $\{1,2,\ldots,2n\}$ into two groups is unrelated with $\theta$ or the realisations of $X_1,X_2,\ldots,X_{2n}$, the distribution of the variables within the resulting groups of random variables remains the same, iid Bernoulli variates with parameter $\theta$. However, in the second case, the number of variables is fixed and you obtain for the sums, two iid Binomial $\text{B}(n,\theta)$ variables, while for the first case, the number of variables $N$ in the first group is random and hence $$X_1\sim\sum_{i=1}^N X_i\qquad X_2\sim\sum_{i=N+1}^{2n} X_i$$ implies that

• $X_1$ and $X_2$ are not Binomial variates, but instead compound Binomial variates;
• $X_1$ and $X_2$ are not independent.