# Solving Sampling Problem: How to Sample from Q(x,y) with Rejection Method?

• Alamino
In summary, the conversation discusses a sampling problem where the distribution of fields x and y (W(x,y)) needs to be sampled according to a second distribution Q(x,y), which is related to W(x,y) by a function F(y). The individual is looking for ideas on how to use the rejection method to sample from Q(x,y) using the knowledge of F(y). One suggestion is to pick a vector from W(x,y), calculate F(y) and use it as a probability to determine whether to keep the value or reject it. This would result in a distribution following Q(x,y).
Alamino
I would appreciate if someone could give me some idea about how to solve this sampling problem. I have a distribution W(x,y) of fields x and y characterized by a population of N vectors (x,y). I have to sample x and y fields according to a second distribution Q(x,y) which is related to W by:

Q(x,y) = W(x,y) F(y),

where given some y, I know how to calculate F(y). Now, I can sample from W simply by choosing a vector from my population and then, I can calculate F(y). I am hoping I can use something similar to the rejection method to sample from Q.

Does anyone has some idea about that?

Cheers,
Roberto.

Uhm... I'm not sure I understood what actually IS your problem. Do you have just a bunch of values, or an analytical shape for W(x,y)? Anyway, I think you could do like this:

1) Pick a vector from W(x,y)
2) Calculate F(y)
3) Use it as a probability and make a random sort; if it's "yes", then keep the value; otherwise, reject it and pick another one. Like, if F(y) (normalized) is 0.7, then you have a 70% probability you'll keep the vector.In this way, final values should be distributed following Q(x,y). And it's some sort of rejection method.

## 1. What is the "rejection method" in sampling?

The rejection method is a technique used in sampling to generate a sample from a probability distribution, Q(x,y), when the distribution is difficult to sample from directly. It involves generating a larger sample from a simpler distribution, P(x,y), and then rejecting some of the points based on a comparison with the target distribution.

## 2. How does the rejection method work?

The rejection method works by generating a sample of points from a simpler distribution, P(x,y), that is easy to sample from. These points are then evaluated against the target distribution, Q(x,y), and any points that fall outside the desired range are rejected. The remaining points are then used as the sample from the target distribution.

## 3. When is the rejection method useful for sampling?

The rejection method is useful for sampling when the target distribution, Q(x,y), is complex or difficult to sample from directly. It is also helpful when the target distribution is only known up to a constant factor, as the rejection method does not require knowledge of the normalization constant.

## 4. What are the limitations of the rejection method?

The rejection method can be inefficient for generating samples, particularly when the target distribution has a low acceptance rate. This means that a large number of points may need to be generated and rejected in order to obtain a representative sample. Additionally, the rejection method may not be applicable for distributions with infinite support.

## 5. Are there any alternatives to the rejection method for sampling?

Yes, there are several alternatives to the rejection method for sampling from complex distributions. These include Markov Chain Monte Carlo methods, importance sampling, and sequential Monte Carlo methods. Each of these techniques has its own advantages and limitations, and the choice of method will depend on the specific application and distribution being sampled from.

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