Let's start with the Wikipedia article. It says "The polar bear is classed as a vulnerable species, with 5 of the polar bear subpopulations in decline. For decades, unrestricted hunting raised international concern for the future of the species; populations have rebounded after controls and quotas began to take effect." Further down, in the section on population, it says "It is difficult to estimate a global population...however biologists use a working estimate of about 20,000-25,000."
The most accurate counts involve flying a helicopter, shooting a tranquilizer dart at a bear, and landing to tag it, according to Wikipedia. Exactly how that is used to count the bears is not explained, but below on this page you will see a reference to "capture / recapture data" that gives a hint. Another technique is to count (from an airplane) the number of polar bear dens.
Polar bears may be threatened by hunting, pollution, increased human activity in their habitat, and by changes in temperature that cause ice floes to break up earlier in the year, "driving the bears to shore before they have built sufficient fat reserves to survive the period of scarce food in the late summer and early fall." In addition to this threat, "reduction in sea-ice cover also forces bears to swim longer distances, which further depletes their energy stores and occasionally leads to drowning."
The factual question as to what is currently happening to polar bear populations is thus hard to answer. One finds the following repeated in numerous places, including the Wikipedia article: "Of the 19 recognized polar bear subpopulations, 8 are declining, 3 are stable, 1 is increasing, and 7 have insufficient data." This seems to trace back to the Proceedings of the 14th Working Meeting of the IUCN/SSC Polar Bear Specialist Group, which contains a report Status of the Polar Bear. The link from the Wikipedia article is no longer valid, and the date is 2005.
But the IUCN/SSC Polar Bear Specialist Group has its own website, where with a couple of clicks you can come to the status table. There you can see a row for each of the 19 populations, and the estimates of population, date of estimate, and whether the authors think the population is declining or increasing or stable. For example, the first row with an estimate is the Southern Beaufort Sea population, with a 2006 estimate of 1500. The comments reveal that this number is based on a "preliminary analysis of capture-recapture data collected jointly by the U.S. and Canada, from 2001-2006." A final analysis is promised for 2007, but isn't there. Their estimate of the accuracy of this count is "1000-2000". No justification is given for concluding that this population is "declining". Only three of the 19 populations have two counts given (for different years) in the table--one each of declining, increasing, and stable.
The table also has a column for the 5-year mean of "annual removals" (presumably that means bears killed by hunters). The total is about 700 per year. That's a lot of polar bears out of a population of 20,000-25,000. Clearly it will be hard to see the effect of earlier ice melting against that background, if to count the bears you have to rely on the techniques described above!
Ian Plimer's book Heaven and Earth refers to a report of the US Fisheries and Wildlife, dated April, 2006. The conclusions he states from this report are:
"There are some 22,000 polar bears in about 20 distinct populations worldwide. Only two bear populations, accounting for 16.4% of the total, are decreasing, and they are in areas where air temperatures have actually fallen, such as the Baffin Bay region. By contrast, another two populations, about 13.6% of the total number, are growing, and they live in areas where air temperatures have risen, near the Bering Strait and the Chukchi Sea." In the IUCN's table, the Chukchi sea population is listed as having 2000 bears in 1993, but in the status column, it says "data deficient", so they apparently neither agree nor disagree with the US Fisheries and Wildlife that the population is increasing there. The comments explain that the Chukchi sea data were based on "extrapolation of multiple years of spring den numbers collected on Wrangel Island". The original paper (1998) estimated 2000-5000 and in 2002 that was "revised to 2000 animals with low confidence". Apparently the decrease from 5000 to 2000 is the reason why "the subpopulation trend is believed to be declining." But they didn't believe it enough to put it in the table--that's only in the comments.
OK, so it's hard to count polar bears. The other facts relevant to this issue are the dates each year when the ice floes break up. Do we know the facts about that? Let's turn again to the IUCN/SSC website, where there is a section on "Population Information". It says, "This section is under development...a final version will probably not be published before 2010. In this section you will eventually find extensive information about different populations...Three populations will serve as "pilot" populations. There will be information about sea ice and snow cover, trend information (where available),..." However, we don't have to wait until 2010 for data. We find some interesting historical data:
April sea ice extent from historic records
Sea ice extent in the North Atlantic 1864-1998 (from Vinje 2001), who constructed maps of sea ice extent from old ship logs and records from e.g. seal hunters from the period 1864-1998. In the figure, the term Nordic Seas refer to the Greenland, Iceland, Norwegian, Barents and western Kara Seas.
Too bad this graph stops 11 years ago in 1998.
So is this trend in the April ice extents dangerous to polar bears? Does it threaten their survival? How might we approach this problem? Polar bears, as a species, diverged from the brown bears about 200,000 years ago, according to the Wikipedia article. Therefore, they have survived all the climate changes that have occurred in the past 200,000 years. Thus, we should ask what the above graph looks like if extended backwards in time for 200,000 years. That, I don't think we know.
But we do know that 10000 years ago, and 8500 years ago, the Baffin Island lakes in northern Canada were 5 degrees C warmer than now. [There are three footnotes on p. 258 of Plimer to support this claim. One study was "multi-proxy", one studied pollen grains, one studied diatom fossils.] Another line of evidence involves the northern boundary of the forests of Siberia, which advanced northwards from 10,000 years ago until 7000 years ago, then retreated to their present position, so 7000 years ago, it was warmer than now. Plimer says, without a reference, 2.5 to 7 degrees warmer. And in recent times, 1920-1940, the Arctic was warmer than now. In 1920-1939, summer sea ice in the Barents Sea was 12% less in area than in 1898-1920. Page 259 of Plimer has many other historical citations to prove that the Arctic was warmer then; and indeed, the above graph shows less sea ice in 1920-1940 than in the decades before and after that period.
When their habitat was 5 degrees C warmer than now, the polar bears did not die out, because they are still here. The IPCC's projected temperature rise in the 21st century is a couple of degrees. So it seems the polar bears have a good chance of surviving that temperature rise, or even a larger one. That is, if they survive being hunted, and having their habitat encroached upon for oil exploration, tourism, etc. But those are not global warming issues.