Take N examples sampled from a multinomial distribution (p1,p2,⋯,pm) for m outcomes, with Ni being the number of examples taking the outcome i. Here I assume p1,p2,⋯,pm are listed in the descending order; if not, we can reorder them to make them descending.
My objective is to make the most frequently occuring outcome, i.e., argmax_i{N_i} as close to 1 as possible.
So my question is what size of N we should take to make the most frequently occuring outcome is less than some predetermined number m0 (1
When k is 2 and n is 1, the multinomial distribution is the Bernoulli distribution. When k is 2 and n is bigger than 1, it is the binomial distribution. When n is 1, it is the categorical distribution (k is the probability of counts and n is an independent trial)
It is not clear to me what you mean by the most frequently occuring outcome.
The most probable seqence of outcomes is a sequence of individual outcomes all from the first category.
The most probable type (i.e. empirical distribution) is one to the possible empirical distributions closest to the theoretical distribution (p1,p2,⋯,pM) .
.
rephrasing (but maybe i misunderstood) :
how large should be a sample from a multinomial to have a guarantee that the empirical mode (most frequent class in the sample) is among the m0 classes of true highest probabilities ?
or more accurately
what is the relationship between the probability of success (the empirical mode in the sample is indeed among the m0 classes with highest probabilities) and the size of the sample ?
.
correct ?
.
Chao-Hua Yu
One solution is to use simulation with R or Mathematica.
Chao-Hua Yu >: So my question is what size of N we should take to make the most frequently occuring outcome is less than some predetermined number m0 (1 p2=>...=>pm >0 ], not talking about non-uniqueness of J (!). Thus the closest natural replacements for this question should possibly sound as follows: For given N and probabilities of the classes p1, p2,..., pm ....
Q1: .... find the probability distribution (p.d.) of J, given N and p1,2,...,pm.
Q2: .... find the probability of the event {J Pr{J=m}
EX4: For m=2, the p.d. of N_i is binomial B(N,pi}, for i=1,2. Therefore Pr{J=1} = Pr{ N_1 => N2 } = Pr{ N_1 => N/2} = \sumj=N/2N b_j ,
where b_j = N! p1j (1-p1)N-j/ [ j! (N-j)! ], j=0,1,..., N.
A particular related problem is to find the limit p.d. of J as N approaches infinity. Indeed, we know from the CLT that then the joint p.d. of the random vector
[(N_1, N_2, ..., N_m) - N (p1,p2,...,pm)] / \sqrt{N}
approach a normal p.d with mean vector (0,0,...0) and covariance matrix
C=[cj,k : j,k =1,2,...,m]
where cj,k= p_j (1-p_j) for j=k, and cj,k= - p_j p_k when j is not equal k. Possibly, this will allow to find the asymptotic p.d. of J.
Good luck in further steps!
Joachim
- Badges
- Science topic
- Similar topics
- Mathematics
- Statistics
- Probability