[gov2001-l] Independently and identically distributed draws - Gov2001

kimai＠fas.harvard.edu

22 Feb 22 Feb

6:28 p.m.

For now, let's stick with i.i.d. case... Kosuke On Sat, 22 Feb 2003, Andrew Reeves wrote:

Problem 2c asks to write a function which generates iid draws from the beta-binomial distribution. Slide 47 of the lectures notes says to ``Begin with N Bernoulli trials with parameter pi_j,j=1,...,N(not necessarily iid).'' Are Bernoulli trials not iid? Could someone perhaps clarify this concept? Thanks, Andrew -- Andrew Reeves reeves(a)fas.harvard.edu 617.493.3485 tel. 301.639.8369 cell. http://people.fas.harvard.edu/~reeves/ _______________________________________________ gov2001-l mailing list gov2001-l(a)fas.harvard.edu http://www.fas.harvard.edu/mailman/listinfo/gov2001-l

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brightjo＠gse.harvard.edu

3:07 p.m.

New subject: [gov2001-l] 2 c) Beta-binomial & iid

Hi Stan, I think we are drawing pi from the beta distribution. This makes pi a random variable in the "beta-binomial distribution", while it was a fixed parameter in the binomial distribution. If I understand correctly then, a draw from the beta-binomial distribution is just a draw from the binomial distribution where pi is randomly sampled from the beta distribution for every observation (for the N total observations). Or, maybe a way to frame it more consistent with the lecture notes is that we make N Bernoulli draws with an "evolving" pi as drawn from the beta distribution. I believe then our parameters are just mu and gamma (not the gamma function), the parameters for the beta distribution (or I guess you could use the standard parameterization with alpha and beta), and our arguments for drawing from the beta-binomial distribution, given a specific mu and gamma, is just N, the number of total draws. Again, y will be the number of successes like in the normal binomial distribution ... I think. Hopefully, this helps. John. On 2/23/03 1:51 PM, "Stanislav Markus" <smarkus(a)fas.harvard.edu> wrote:

...

If a beta-binomial distribution is iid, isn't it then the same as binomial distribution? I thought that the whole point of a *beta-binomial* distribution is that you drop the iid assumption and let pi vary. So, for question 2c, can we draw from a binomial distribution - or am I confusing the issues here? **************************** Stanislav Markus Ph.D. Candidate Harvard University Department of Government e: smarkus(a)fas.harvard.edu t: 617.513.5407 -----Original Message----- From: gov2001-l-admin(a)fas.harvard.edu [mailto:gov2001-l-admin@fas.harvard.edu] On Behalf Of Kosuke Imai Sent: Saturday, February 22, 2003 6:28 PM To: Andrew Reeves Cc: gov2001 Subject: Re: [gov2001-l] Independently and identically distributed draws For now, let's stick with i.i.d. case... Kosuke On Sat, 22 Feb 2003, Andrew Reeves wrote:

Problem 2c asks to write a function which generates iid draws from the beta-binomial distribution. Slide 47 of the lectures notes says to ``Begin with N Bernoulli trials with parameter pi_j,j=1,...,N(not necessarily iid).'' Are Bernoulli trials not iid? Could someone perhaps clarify this

concept?

Thanks, Andrew -- Andrew Reeves reeves(a)fas.harvard.edu 617.493.3485 tel. 301.639.8369 cell. http://people.fas.harvard.edu/~reeves/ _______________________________________________ gov2001-l mailing list gov2001-l(a)fas.harvard.edu http://www.fas.harvard.edu/mailman/listinfo/gov2001-l

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brightjo＠gse.harvard.edu

3:30 p.m.

New subject: [gov2001-l] 2 c) Beta-binomial & iid

I see your point now. Two thoughts come to mind. 1. It means mu and gamma should remain constant, and thus the iid refers to those parameters (and not pi) ,or 2. we should choose values for parameters of mu and gamma that give us mean mu (E[pi]=mu) and the probability of drawing a pi from the beta distribution *not* equal to mu to be essentially 0. I am not sure which option is the correct one for this problem. John. On 2/23/03 3:15 PM, "Stanislav Markus" <smarkus(a)fas.harvard.edu> wrote:

...

John, thanks for your comments, that's exactly how I conceive beta-binomial distribution, too. However, sampling pi on each draw *is not iid*, since by definition "identically distributed" means same pi across draws. Hence, we for question 2c) we should keep pi fixed - which was the reason for my question: is a beta-binomial distribution in which pi is fixed identical to a binomial distribution? I believe so, since point 3 on page 47 of the slides says that if we do not sample pi, we get binomial draws. Please correct me if I'm wrong! Thanks, Stan **************************** Stanislav Markus Ph.D. Candidate Harvard University Department of Government e: smarkus(a)fas.harvard.edu t: 617.513.5407 -----Original Message----- From: John Bright [mailto:brightjo@gse.harvard.edu] Sent: Sunday, February 23, 2003 3:08 PM To: Stanislav Markus; 'gov2001' Subject: Re: [gov2001-l] 2 c) Beta-binomial & iid Hi Stan, I think we are drawing pi from the beta distribution. This makes pi a random variable in the "beta-binomial distribution", while it was a fixed parameter in the binomial distribution. If I understand correctly then, a draw from the beta-binomial distribution is just a draw from the binomial distribution where pi is randomly sampled from the beta distribution for every observation (for the N total observations). Or, maybe a way to frame it more consistent with the lecture notes is that we make N Bernoulli draws with an "evolving" pi as drawn from the beta distribution. I believe then our parameters are just mu and gamma (not the gamma function), the parameters for the beta distribution (or I guess you could use the standard parameterization with alpha and beta), and our arguments for drawing from the beta-binomial distribution, given a specific mu and gamma, is just N, the number of total draws. Again, y will be the number of successes like in the normal binomial distribution ... I think. Hopefully, this helps. John. On 2/23/03 1:51 PM, "Stanislav Markus" <smarkus(a)fas.harvard.edu> wrote:

If a beta-binomial distribution is iid, isn't it then the same as binomial distribution? I thought that the whole point of a *beta-binomial* distribution is that you drop the iid assumption and

let

pi vary. So, for question 2c, can we draw from a binomial distribution

-

or am I confusing the issues here? **************************** Stanislav Markus Ph.D. Candidate Harvard University Department of Government e: smarkus(a)fas.harvard.edu t: 617.513.5407 -----Original Message----- From: gov2001-l-admin(a)fas.harvard.edu [mailto:gov2001-l-admin@fas.harvard.edu] On Behalf Of Kosuke Imai Sent: Saturday, February 22, 2003 6:28 PM To: Andrew Reeves Cc: gov2001 Subject: Re: [gov2001-l] Independently and identically distributed

draws

For now, let's stick with i.i.d. case... Kosuke On Sat, 22 Feb 2003, Andrew Reeves wrote: > Problem 2c asks to write a function which generates iid draws from

the

> beta-binomial distribution. > > Slide 47 of the lectures notes says to ``Begin with N Bernoulli

trials

with parameter pi_j,j=1,...,N(not necessarily iid).'' Are Bernoulli trials not iid? Could someone perhaps clarify this

concept?

Thanks, Andrew -- Andrew Reeves reeves(a)fas.harvard.edu 617.493.3485 tel. 301.639.8369 cell. http://people.fas.harvard.edu/~reeves/ _______________________________________________ gov2001-l mailing list gov2001-l(a)fas.harvard.edu http://www.fas.harvard.edu/mailman/listinfo/gov2001-l

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brightjo＠gse.harvard.edu

4:08 p.m.

New subject: [gov2001-l] 2 c) Beta-binomial & iid

I think the parameters mu and gamma provide a more intuitive feeling for what the corresponding distribution will look like. That is, mu is just the mean of the beta distribution and mu*gamma is the variance of the beta distribution for this parameterization. Alpha and Beta correspond to the "weights" of the probability of getting "0" or "1". I am not sure if the values for alpha and beta (or mu and gamma) are just any old, garden variety values. I think we might be looking for the parameter values that satisfy certain conditions, namely the iid business. I dunno. I am drawing from my beta-binomial the same way as you describe below (your "10 numbers" refers to your N draws?). John. On 2/23/03 3:54 PM, "Nirmala Ravishankar" <ravishan(a)fas.harvard.edu> wrote:

...

So, here are some more questions about the beta-binomial: 1) Is there only one beta distribution for which we randomly assign a alpha and beta? 2) I am still confused about how mu and gamma work into this. Are mu and gamma in this case alpha and beta respectively. King's notes say that mu = E(pi). How does that work into this whole thing. Currently, I am generating a beta distbution with alpha = 0.5 and beta = 1. Then I randomly sample 10 numbers from this distribution and use them as my probabilities for the bernoulli. Does this make sense?? - Nirmala On Sun, 23 Feb 2003, John Bright wrote:

Hi Stan, I think we are drawing pi from the beta distribution. This makes pi a random variable in the "beta-binomial distribution", while it was a fixed parameter in the binomial distribution. If I understand correctly then, a draw from the beta-binomial distribution is just a draw from the binomial distribution where pi is randomly sampled from the beta distribution for every observation (for the N total observations). Or, maybe a way to frame it more consistent with the lecture notes is that we make N Bernoulli draws with an "evolving" pi as drawn from the beta distribution. I believe then our parameters are just mu and gamma (not the gamma function), the parameters for the beta distribution (or I guess you could use the standard parameterization with alpha and beta), and our arguments for drawing from the beta-binomial distribution, given a specific mu and gamma, is just N, the number of total draws. Again, y will be the number of successes like in the normal binomial distribution ... I think. Hopefully, this helps. John. On 2/23/03 1:51 PM, "Stanislav Markus" <smarkus(a)fas.harvard.edu> wrote:

If a beta-binomial distribution is iid, isn't it then the same as binomial distribution? I thought that the whole point of a *beta-binomial* distribution is that you drop the iid assumption and let pi vary. So, for question 2c, can we draw from a binomial distribution - or am I confusing the issues here? **************************** Stanislav Markus Ph.D. Candidate Harvard University Department of Government e: smarkus(a)fas.harvard.edu t: 617.513.5407 -----Original Message----- From: gov2001-l-admin(a)fas.harvard.edu [mailto:gov2001-l-admin@fas.harvard.edu] On Behalf Of Kosuke Imai Sent: Saturday, February 22, 2003 6:28 PM To: Andrew Reeves Cc: gov2001 Subject: Re: [gov2001-l] Independently and identically distributed draws For now, let's stick with i.i.d. case... Kosuke On Sat, 22 Feb 2003, Andrew Reeves wrote:

Problem 2c asks to write a function which generates iid draws from the beta-binomial distribution. Slide 47 of the lectures notes says to ``Begin with N Bernoulli trials with parameter pi_j,j=1,...,N(not necessarily iid).'' Are Bernoulli trials not iid? Could someone perhaps clarify this

concept?

Thanks, Andrew -- Andrew Reeves reeves(a)fas.harvard.edu 617.493.3485 tel. 301.639.8369 cell. http://people.fas.harvard.edu/~reeves/ _______________________________________________ gov2001-l mailing list gov2001-l(a)fas.harvard.edu http://www.fas.harvard.edu/mailman/listinfo/gov2001-l

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