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I have tried it in the past. They all predicted a page rank of 6 for one of my pages last time and it got pr3.
It's entertainment value only, much as pr itself is.
I really like Rustybrick PR Prediction tool. It often predict one of my sites to have PR 10 in next update. Very cheering up just to look at such prediction.
It's possible to predict PR if you know all your backlinks and can approximate their influence. I usually have no problem with predicting the PR of most of my sites, because I know what backlinks they have gained in last few months. Sometimes I mistake one point of the scale - for example the site gets strong PR 3 instead of weak 4 - but generally I can say which sites are likely to go one point up and which will stay with the same PR.
It's more difficult to do it automatically - Google doesn't show backlinks, so you'd have to check backlinks with MSN, then check PR of linking pages, apply some calculation, and it might work, but wouldn't be too reliable.
I noticed that the Google directory ranks sites by PR so thats a round about way to do it.
Yahoo usually lists highest PR sites first in it's link search so that's another way to gague how a site is doing.
Be nice to have a more accurate estimate.
To estimate the PR, you just have toolbar and directory data. Talking all data (i.e. toolbar is showing x/10, the directory is showing y/7 plus the PR of pages which are linked / linking to this page) you can estimate the PR - not more. (There are few cases when these data are enough to get precise values like 4.6)
Yahoo doesn't use PR. Of course, there might be a correlation between PR and the position for a link search, but this would be because both values are correlated to a third parameter (number of backlinks).
However, I see the way to estimate real PR with good accuracy, it's just difficult and resource-consuming task. You have to write a crawler, crawl pages you're interested in and as many related pages as possible, get toolbar PR for all of them, and use data-mining methods to develop an algo that predicts real PR for these pages, fitting the algo variables so toolbar PR estimated by your algo will be closest to toolbar PR you got from Google. For better results, use outdated crawl data (about 3-4 month old perhaps) to match toolbar PR better.
Of course, to do this you need impressive bandwidth and computing power, smaller than Google, but still impressive. I wonder how many pages it would be needed to crawl to get sensible results (millions for sure), but I find it doable anyway :)