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13 votes
Accepted

How to obtain $p(x)$ given samples from $p(y|x)$ and $p(y)$?

This problem is equivalent to solving a Fredholm integral equation of the first kind. This is, solving for $p(x)$ such that: $$ p(y) = \int_{\text{supp}(X)} p(y\mid x)\, p(x)\, \text{d}x,\quad \forall ...
Johan de Aguas's user avatar
8 votes
Accepted

Why are some sampling algorithms better than others?

Both rejection sampling and Metropolis-Hastings are really classes of algorithms rather than single algorithms. However, one key difference between rejection sampling and Metropolis-Hastings ...
Thomas Lumley's user avatar
5 votes

Distribution of medians of triplicate samples taken from Gaussian distribution

This must be known somewhere because Mathematica (and likely MATLAB and Maple, too) solves this easily. ...
JimB's user avatar
  • 3,849
5 votes

How to obtain $p(x)$ given samples from $p(y|x)$ and $p(y)$?

Depending on what we mean by specifying the conditional distribution, one possible answer is that we can't in general. If we specify the conditional distribution for $Y$, without specifying what the $...
Glen_b's user avatar
  • 284k
4 votes
Accepted

Does uniform sampling from a sample set preserve its distribution

Yes, they are. The key assumption here is that the subset selection is done independently of the sample. If this would not be the case, then one could think of many cases where this does not hold. For ...
Stan's user avatar
  • 385
3 votes

Why are some sampling algorithms better than others?

There are (at least) four different types of sampling methods: (i) inversion (ii) rejection (iii) random walks (iv) physics Inversion: This an an algorithm which lets you sample from a random variable ...
Nicolas Bourbaki's user avatar
3 votes
Accepted

Cluster sample or stratified random sample?

This kind of questions requires precise definitions of what's being sampled, and often a single survey can have multiple populations studied. In your context, I believe your textbook gives the correct ...
Alex J's user avatar
  • 2,456
3 votes
Accepted

Why is a frame needed for a simple random sample?

Here is an example of "a situation where a cluster sample can be done, but a simple random sample cannot be done." I want to do a survey of N=1,000 that is representative of all college ...
Graham Wright's user avatar
2 votes

Distribution of medians of triplicate samples taken from Gaussian distribution

Suppose $X_1, X_2, X_3 \text{ i.i.d. } \sim N(0, 1)$, then the density of the median $M := X_{(2)}$ is given by: \begin{align*} f_M(x) = 6\Phi(x)(1 - \Phi(x))\phi(x), \; x \in \mathbb{R}, \tag{1}\...
Zhanxiong's user avatar
  • 19.7k
2 votes

Sample a random subgraph from an undirected, unweighted graph, what's the probability of "every two nodes's distance is at least 3 in the subgraph"?

Suppose there is a bound $b$ on $$\frac{|S|}{\sqrt{|G|/\ln|G|}}$$ and a bound $k$ on the number of vertices within distance 3 of a vertex. Then as the graphs get larger, the probability goes to $1$ ...
Matt F.'s user avatar
  • 5,032
2 votes

Why are some sampling algorithms better than others?

I would say that Acceptance-Rejection method is easier to implement in principle, so I would love to be able to use that, however as soon as you have more than 2-3 dimensions, the former becomes very ...
Cryo's user avatar
  • 656
1 vote
Accepted

What is the advantage of using bootstrapping to estimate variance?

You don't need the bootstrap here. There's no real advantage to a simple bootstrap for the mean (though second-order bootstraps might be more accurate when the distribution of the mean is a bit ...
Thomas Lumley's user avatar

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