The causes in the BBC program were said to be a combination of (1) economic: the younger the girl is married off, the less dowry has to be paid. As she approaches even the tender age of 20, she is so costly to marry off that many, especially in the countryside, cannot afford it. And (2), so called "Eve Teasing", where young girls are molested by boys and men, every time they leave home. This is not the common-garden "teasing", wolf whistles and so on, but aggressive bullying, including the throwing of acid on girls. They have to be escorted to school -- or they are married off early to get the "protection" of their young husbands....
Nowhere in the program, however, is the possibility of a religious aspect -- Islam, to be precise -- mentioned.
So I thought I'd to some research and the results are here. I've compared the figures for the percentage of girls married while still minors -- which is taken as below 18 years-old in the UNESCO study I refer to. And for the percentage of Muslim population, I've taken the CIA World Fact Book figures, which are widely referred to as being reliable.
The correlations are: for the top ten countries: 70% and for the top twenty countries: 60%. These are statistically significant figures, especially for the top ten.
Apologists of Islam will be quick to say: "correlation does not imply causation". Indeed it does not. But the opposite is equally true, probably more so: "correlation can strongly suggest causation", in which case it's incumbent on the analyst to suggest what it is that may cause the correlation.
In the case of Islam, it's not hard to find reasons why its doctrines may lead to the phenomenon of Child Brides.
Mohammed is the first and strongest witness for the prosecution: he was married to Aisha when she was 9 and consummated that marriage when she was 12 (later: sorry, commenters have noted the figures are 6 and 9 respectively; I was just going on memory. So, my point holds a fortiori). Since Mohammed is the "best of examples", held in esteem almost equal to Allah, his actions are normative for Muslims. It's for that reason that Iran, as just one example, allows under-age marriage.
Second item for the prosecution are the Koran and Hadith, in which the role of women is to submit to men; they are clearly considered second class items of property in the texts of Islam. This may account for the horrid practices of "Eve Teasing" in Bangladesh. At the very least they do not discourage them.
I don't suggest that Islam is the only reason for Child Marriage in the countries I've looked at. Clearly there are other longer-term cultural and social issues: vide, for example, the number that are in Africa, where no doubt ancient tribal and other social factors are at work.
What I do suggest is that Islam as the very least allows, enables and even encourages the practice of child brides: it's there in the "perfect example" of Mohammed and in the core texts of Islam which dehumanize and demean women and young girls.
LATER: linked by BCF.
LATER (9 NOV 12): "Saudi Arabia: 5,000 child brides under age 14".
"Early Marriage: a harmful traditional practice", UNICEF, 2005