Stuck on Problem 1.7

[i_need_some_help]

On part (a), I tried a few different things. Most recently, 1 - binomcdf(1 to 10 | N, mu) ^ (number of coins), but this doesn't seem to be correct.

In the case where mu is 0.05 and we try with one coin, the probability should be 1 - 0.05**10. I can't figure out how to generalize that to multiple coins.

WON'T YOU HELP??

[magdon]

First you need to compute the probability that any one specific coin has nu=0. Call this probability P. Now the number of coins that have nu=0 is itself a binomial distribution with probability P.

You can use the above observation, or you can use a trick: the probability that at least one coin has nu=0 is related in a simple way to the probability that all coins have

Originally Posted by i_need_some_help

On part (a), I tried a few different things. Most recently, 1 - binomcdf(1 to 10 | N, mu) ^ (number of coins), but this doesn't seem to be correct.

In the case where mu is 0.05 and we try with one coin, the probability should be 1 - 0.05**10. I can't figure out how to generalize that to multiple coins.

WON'T YOU HELP??

[foenix]

Greetings,

This is in reference to part (b) of Problem 1.7
I am having some difficulty understanding what is meant by using the max over two coins.

My understanding is that the two coins both have binomial distributions for nu with 6 trials and a mu of .5. As such, wouldn't the probability of a given nu be the same for both coins? What am I missing here?

Appreciate the help.

[magdon]

Your understanding is correct that each has the same binomial distribution. What you are missing is that each coin is tossed independently. An example might help.

Toss the two coins 6 times and suppose the first has 3 heads and the second gas 2 heads.

.

Define the deviations:

.

Define the worst deviation:

.

In this particular instance the deviation is 0.16666. You are asked to compute the probability distribution for . In particular,

Originally Posted by foenix

Greetings,

This is in reference to part (b) of Problem 1.7
I am having some difficulty understanding what is meant by using the max over two coins.

My understanding is that the two coins both have binomial distributions for nu with 6 trials and a mu of .5. As such, wouldn't the probability of a given nu be the same for both coins? What am I missing here?

Appreciate the help.

[MaciekLeks]

Hello Professor Malik Magdon-Ismail,

I did Exercise 1.10/c and Problem 1.7. I've done it but I still do not know how to interpret the results correctly.

Questions:
1. How the worst deviation method you mentioned is related to the hint in the book (the sum rule)?
2. If I use Hoeffding Inequality with your method, should I multiply RHS of Hoeffding Inequality by a number of coins (in this case M=2)? IMHO, I should not.
3. Why cmin in (Exercise 1.10) does not hold the hoeffding bound while max deriation in Problem 1.7 holds the bound (see the plot and the linked post)?
4. While increasing the sample size to N=10 the Hoeffding bound is not longer held for Problem 1.7. Why?

The plots:

N=6, RHS=2.0*exp(-2.0*(ε^2.0)*N)


N=10, RHS=2.0*exp(-2.0*(ε^2.0)*N)


P.S. Please see my unanswered question for Exercise 1.10 (c) here: http://book.caltech.edu/bookforum/showthread.php?t=4616

[henry2015]

1. I think here is how P[max(..)>] relates to the Rule of Addition:

Let's say A=|v0-u0|>, B=|v1-u1|>
P[max(..)>]
<= P[A or B]
= P[A] + P[B] - P[A and B]

Since P[A] and P[B] are both bound by Hoeffding Inequality (HI), hence,
P[A] + P[B] - P[A and B]
<= 2 * (HI's bound) - P[A and B]
<= 2 * (HI's bound) because P[A and B] >= 0

In this case, we can see that M is 2 (as expected because we are using 2 coins).

2. See above.

3. I believe that if you rerun your script several times, you will see that the vanilla HI doesn't apply but M*HI's bound does. The reason your plot showed that HI applied because you were just lucky, and HI is talking about the upper bound, which indicates "always true" if the condition meets; that's why Exercise 1.10 asks the reader to run 100000 trials and then pick the min to make sure the reader won't be "lucky" to have vmin bound by the vanilla HI but M * HI's bound.

4. See #3.

I am not an expert so I could be wrong.

If any prof can confirm, it would be benefit for everyone reading here

Thanks!

Originally Posted by MaciekLeks

Hello Professor Malik Magdon-Ismail,

I did Exercise 1.10/c and Problem 1.7. I've done it but I still do not know how to interpret the results correctly.

Questions:
1. How the worst deviation method you mentioned is related to the hint in the book (the sum rule)?
2. If I use Hoeffding Inequality with your method, should I multiply RHS of Hoeffding Inequality by a number of coins (in this case M=2)? IMHO, I should not.
3. Why cmin in (Exercise 1.10) does not hold the hoeffding bound while max deriation in Problem 1.7 holds the bound (see the plot and the linked post)?
4. While increasing the sample size to N=10 the Hoeffding bound is not longer held for Problem 1.7. Why?

The plots:

N=6, RHS=2.0*exp(-2.0*(ε^2.0)*N)


N=10, RHS=2.0*exp(-2.0*(ε^2.0)*N)


P.S. Please see my unanswered question for Exercise 1.10 (c) here: http://book.caltech.edu/bookforum/showthread.php?t=4616