Posterior probability when data consists of $k$ largest of $N$ samples

Question

Given an underlying unknown distribution, I sample $N$ numbers. From those $N$ numbers I take the highest $k$ numbers. How do I model the posterior probability from those $k$ numbers.

I know I can do it if I just get any random $k$ numbers. I would have a prior and update my belief that is posterior based on the observed data, but how do I further exploit the knowledge of top $k$ ?

I have very basic knowledge about prior and posterior. Any resource to guide me through would be heplful as well.

An example of my problem is, I know $N(0, 1)$ to generate 1000 random samples from $N(0, 1)$. I take the top 10 numbers from the 1000. I want to understand how my posterior would be different when I just get any 10 random numbers from the 1000 samples from the estimate when I know that I got top 10 numbers.

Answer 1

Let us take your example of a Normal sample: if you observe the $k$ largest Normal values of a Normal $\mathrm{N}(\theta,1)$ sample of size $N$, the likelihood of this sample is $$\ell(\theta|x_1,\ldots,x_k)=\Phi\left(\min_{1\le i\le k}\{x_i\}-\theta\right)^{N-k}\,\prod_{i=1}^k \exp\left\{-(x_i-\theta)^2/2\right\}$$where $\Phi$ is the Normal cdf. Once you have derived this likelihood, business (or rather bayesness!) proceeds as usual: the posterior distribution is $$\pi(\theta|x_1,\ldots,x_k)\propto\pi(\theta)\Phi\left(\min_{1\le i\le k}\{x_i\}-\theta\right)^{N-k}\,\prod_{i=1}^k \exp\left\{-(x_i-\theta)^2/2\right\}$$ This is not a closed-form likelihood but it can be handled through simulation, either using a simple Metropolis-Hastings algorithm with a numerical computation of the Normal cdf, or using a Gibbs sampler that generates $N-k$ latent Normal variates constrained to be less than $\min_{1\le i\le k}\{x_i\}$.