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.