Hi all,
In the second problem of our homework, we have to calculate a maximum likelihood estimate,
but y (the number of "successes") can vary in the interval between 5 and 15.
Do we just use a number of successes as a random variable on {5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15}, in which case we work with a set of binomial pdf-s, and calculate an MLE
interval instead of maximum likelihood estimate? In that case our pdf would look like
p?(y??)=(100!/y!(100-y)!)* ?^y*(1-?)^(100-y), y ? {5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15}.
If anyone has a different idea, please share with the rest of us.
Thanks,
Nino Malekovic
MPA Candidate, Class 2011
Harvard Kennedy School
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Today's Topics:
1. Re: Bayes Rule (Maya Sen)
2. looking for a co-author? (Maya Sen)
3. Model buildilng, and "the rules" (Lin, Eric)
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Message: 1
Date: Fri, 5 Feb 2010 14:40:07 -0500
From: Maya Sen <msen at fas.harvard.edu>
Subject: Re: [gov2001] Bayes Rule
To: "Class List for Gov 2001/E-2001" <gov2001-l at lists.fas.harvard.edu>
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<16e0be401002051140h22f5ebe9l4526652c154596ff at mail.gmail.com>
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Hi Bob,
So, you've got two questions here. The first one is if there are conditions
that must be in place prior to using Bayes Rule. I'm not exactly sure of
what conditions you're looking for here, but you do have to have all of the
constituent parts of the equation. Bayes Rule is
P(A|B) = P(B|A)*P(A)/P(B)
where
P(B|A) = the conditional probability of B given A
P(A) = the marginal probability of A
P(B) = the marginal probability of B
A lot of times you don't have the marginal probabilities of A and B, in
which case you can use the law of total probability (which we talked about
in class):
P(B) = P(B|A_1)P(A_1) + ... + P(B|A_n)P(A_n)
In terms of what's the simplest way to identify and label the correct
events, I think it's useful to see what in the problem is giving you clues
about P(A), P(B), etc. You should also look for clues that you are being
provided with a conditional probability by identifying words such as "given
that" etc.
It also helps to work through examples, so I've taken the following example
directly from the Wikipedia page on Bayes Rule:
Suppose there is a school with 60% boys and 40% girls as students. The
female students wear trousers or skirts in equal numbers; the boys all wear
trousers. An observer sees a (random) student from a distance; all the
observer can see is that this student is wearing trousers. What is the
probability this student is a girl? The correct answer can be computed using
Bayes' theorem.
The event *A* is that the student observed is a girl, and the event *B* is
that the student observed is wearing trousers. To compute P(*A*|*B*), we
first need to know:
- P(*A*), or the probability that the student is a girl regardless of any
other information. Since the observers sees a random student, meaning that
all students have the same probability of being observed, and the fraction
of girls among the students is 40%, this probability equals 0.4.
- P(*B*|*A*), or the probability of the student wearing trousers given
that the student is a girl. As they are as likely to wear skirts as
trousers, this is 0.5.
- P(*B*), or the probability of a (randomly selected) student wearing
trousers regardless of any other information. Since P(*B*) = P(*B*|*A*)P(
*A*) + P(*B*|*A*')P(*A*'), this is 0.5?0.4 + 1?0.6 = 0.8.
Given all this information, the probability of the observer having spotted a
girl given that the observed student is wearing trousers can be computed by
substituting these values in the formula:
P(A|B) = P(B|A)P(A)/P(B) = .5*.4/.8 = .25
hope that helps --
Maya
On Thu, Feb 4, 2010 at 5:43 PM, Bobby L. Woods <blwoods at fas.harvard.edu>
wrote:
Are there conditions that must met prior to using
Bayes Rule, or can it be
used
for any probability question? Also, what is the
simplest way to identify
and
label the correct events?
Thanks,
Bob
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