Monday, October 20, 2008

The Bayesian probability of an Obama majority is 53%

Today's Gallup Poll states:
Gallup Poll Daily tracking from Friday through Sunday gives Barack Obama an 11 percentage point lead over John McCain in the presidential vote preferences of all registered voters, 52% to 41%.
Sounds like Obama will win by something like 11%, but here's an alternative Bayesian interpretation, just for fun:

The survey has a +/- 2 % sampling error, implying that the true Obama vote is between 54 and 50 percent (for what I suspect is a 95% confidence interval), and between 43 and 39 for McCain. The relevant question is actually: What is the probability that more registered voters will vote for Obama (BO) than McCain, based on the polling data we have available? In other words, we want to know p(BO|poll data). Bayes' Theorem tells us that this is proportional to p(BO) multiplied by p(poll data|BO).

If we ignore the other candidates, a binomial distribution can give us the likelihood for p(poll data|BO) by defining a "success" as a vote for Obama, and a "failure" a vote for McCain. From the poll cited above and the 3 previous polls prior to it, we can adjusting to eliminate the third parties, we can create the following Table:

Oct. 19-16 Oct. 15-13 Oct. 13-11 Oct. 12-10 Totals
Obama 1455 1365 1424 1419 5664
McCain 1175 1198 1173 1141 4687

Total 2630 2563 2597 2560 10351

So, using these we get p(poll data|BO) = BO^{1455} (1-BO)^{1175}, where BO is the proportion of the population voting for Obama. Now we need to define the prior p(BO), which would probably be best served by defining it as a beta distribution, using the sums of the 3 prior polls as alpha and beta:
p(BO)=BO^{4209-1} (1-BO)^{3512-1}
So, multiplying these functions will yield Bayes' Theorem as:
p(BO|poll data) "propto" BO^{5663}(1-BO)^{4686}
Graphically, it takes the following shape:
The horizontal axis is the proportion of the population voting for Obama, and the vertical axis demonstrates the result of this function after normalization. To answer to the question "what is the probability Obama will receive more votes than McCain?" is the result of summing the area under curve to the right of 0.50 on the x-axis, which is approximately 53.12%.

The interpretation is a bit clearer than the confidence intervals generated by Gallup and it incorporates prior survey results as well. The Gallup poll simply told us that the true share of the voters for Obama based on the single survey data will be somewhere between 50 and 54% and that McCain would get between 39 and 43%.

Here is my Excel file, if you care to play with it or change the numbers to try different assumptions. Enjoy!

Note: The poll data is different from the CNN Opinion poll currently occupying their homepage, even though they are frequently citing Gallop polls.
Update: This CNN story could use a refresher course in Bayesian Theory.

1 comment:

Anonymous said...

Forward Thinking 101
using simple deduction...

looks like Obama will win the election...

the people have spoken, they want change...finally someone that will change things...hmmm...

1. The overall media controlled by you know who was totally behind Obama...from the start...

2. 28% of Obama's money was donated by republicans...and More than half of the whopping $426.9

million Barack Obama has raised has come from small donors whose names the Obama campaign won't

disclose, plus secret donors from overseas. The internet service providers (ISP) they were able to

trace were from Saudi Arabia , Iran , and other Middle Eastern countries. One of the banks used for

fund transfers was also located in Saudi Arabia. Another concentrated group of donations was traced

to a Chinese ISP with a similar pattern of limited credit card charges. 13,000,000 clustered

donations around $15 each...hmmm when he wins maybe the extra half a billion money donated can be

used for the economy...

3. it was discovered that Obama is a distant cousin of Lynn Cheney...

4. the republicans have never had the courage or concept of picking a woman as a vice presidential

candidate, its not their style... Obama refused several woman and McCain chose a woman...kind of
reversed roles...

5. All Things Considered, August 29, 2008 · The Country song Only in America filled the stadium

Thursday night after Barack Obama's acceptance speech. The song by Brooks and Dunn was also featured
in the last GOP convention and the campaign of George W. Bush.

6. WASHINGTON – With or without a formal title, Colin Powell will have Barack Obama's ear if the

Democratic presidential candidate wins the White House in the Nov. 4 election, the candidate said

Monday Oct. 20th 2008,
"He will have a role as one of my advisers," Barack Obama said on NBC's "Today" in an interview

aired Monday, a day after Powell, a republican, a four-star general and President Bush's former

secretary of state, endorsed him.


wonder why all the republicans have picked Obama...
maybe they want change...I guess either way they win?

they did it better this time...
very clever concept, and they get more clever every time...

we the people feel great we won...
now be happy and get back to work...

any questions....