Question regarding online bagging (bootstrap)

[cc2709]

Hello guys,

I'm trying to understand the proof of convergence for online bagging/bootstrap, in the Oza's paper an expression says:

θ ~ $\sum$$^{N}_{t=0}$P(Poisson(N)=t)Multinomial(t,1/N)

θ may represents a vector with each element being a real value;
Poisson() is the Poisson distribution;
t is certain real value;
Multinomial() is the multinomial distribution...

But I am totally lost when put all these stuff together
what's that mean by adding up the multiplication of a probability with a distribution?

I don't have a solid background on probability and I've been tortured by these for a while...hopefully I can get some suggestion...

Thanks a lot in advance!

 

[mmwave]

Thank you for your question. The expression you mentioned is used to describe the convergence of online bagging/bootstrap. Let me break it down for you to better understand it.

First, the expression is using the symbol θ to represent a vector with each element being a real value. This vector represents the parameters or weights of a model.

Next, the Poisson distribution is used to model the number of events that occur in a fixed interval of time or space. In this case, it is used to model the number of times a data point is sampled in the online bagging/bootstrap process. The parameter N in the Poisson distribution represents the average number of events in the interval.

Moving on to the term P(Poisson(N)=t), this represents the probability of the Poisson distribution where t is a certain real value. This can be interpreted as the probability of a data point being sampled t times in the online bagging/bootstrap process.

The Multinomial distribution is used to model the probability of multiple outcomes occurring simultaneously. In this context, it is used to model the probability of a data point being selected for a specific model in the online bagging/bootstrap process. The parameter t in the Multinomial distribution represents the number of trials, and 1/N represents the probability of selecting a specific model.

Finally, the expression is adding up the multiplication of the probability of a data point being selected t times with the probability of selecting a specific model. This is essentially calculating the overall probability of a data point being selected for a specific model in the online bagging/bootstrap process.

I hope this explanation helps you better understand the expression and the concept of convergence in online bagging/bootstrap. Let me know if you have any further questions. Good luck with your studies!