1
$\begingroup$

Suppose we have a known discrete probability distribution $X$ over $\{0,1\}^k$.
Given a sequence of binary values $e = (e_1, ..., e_n)\text{, where } e_i\in \{0,1\}^k$, what is the probability (or the likelihood) that $e$ was drawn from the Distribution $X$?
Is there for example a bound that tells us that such a sequence of samples $e$ is very unlikely to be drawn from $X$, and can thus be ignored?

$\endgroup$
  • $\begingroup$ So you're asking for the density function of all binary sequences of length $n$? Seems like they would all be equiprobable from how you have described the problem. Are you thinking of a particular type of sequences (maybe sequences which have some sort of autocorrelation structure)? $\endgroup$ – Demetri Pananos Nov 13 '18 at 6:14
  • $\begingroup$ @DemetriPananos Actually X is a noise distribution which is not uniform. So the values of $e_i$ are drawn non uniformally. X would have maybe some distribution function, but I'm aiming more at a general case. $\endgroup$ – abdul rahman taleb Nov 13 '18 at 7:29
1
$\begingroup$

At the moment you have not imposed any conditions on the distribution other than saying that it is a distribution on the set of binary vectors of length $k$. Your distribution can therefore be expressed as a probability vector over the $2^k$ possible states of the random variable. If the distribution is known, as you state, then presumably you can see exactly which outcomes are very unlikely.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.