Using historical data since 1930, I found 10 analog years in which the March through November average conditions in Barrow were both unusually warm and wet; specifically, I required both the mean temperature and total precipitation to be in the top 22 out of 84 years. Note that March through November 2013 was the fourth warmest and second wettest such period in Barrow's history since 1930. The analog years are 1950, 1951, 1989, 1997, 2003, 2004, 2009, 2010, 2011, and 2012; obviously we see the long-term trend at play here, with this year being the fifth consecutive year of relatively warm, wet conditions in comparison to the long-term average.
The charts below (click to enlarge) show the number of years (out of 10) in which the subsequent monthly mean temperature and total snowfall occurred in each of the three climatological terciles (below-normal, near-normal, above-normal), based on the 1981-2010 climatology, for Barrow, Fairbanks, and Anchorage. For example, for Barrow in January, 4 of the analog years produced below-normal temperatures (lower tercile), 2 years produced near-normal temperatures (middle tercile), and 4 years produced above-normal temperatures (upper tercile).
The most striking feature of the analogs is that 7 out of 10 years produced significantly colder than average conditions in Fairbanks in January, and only one analog year (2013) produced a notably warm January. The probabilities also appear to be shifted away from warm in February and March, though to a lesser extent.
For Barrow and Anchorage, the analog signals are less notable, but it is worth mentioning that warmth seems favored for March in Barrow, and above-normal snowfall seems somewhat likely for Anchorage in February. The lack of near-normal temperatures in Barrow appears to be mainly a result of the long-term trend; recent years have been unusually warm, but most of the earlier years were unusually cold, relative to the 1981-2010 normals.
It's interesting to look at the broader picture to see what large-scale patterns were characteristic of the analog years. The maps below show the percentage of years in which the monthly mean values were above or below the 1981-2010 mean for 500 mb height, sea-level pressure, and 2 m temperature. Note that these data are all from the NCEP/NCAR global reanalysis and the temperature values should be taken with a large pinch of salt.
First, January: the analog years tend to have upper-level ridging over the Bering Sea and high pressure over northern Alaska; this clearly favors colder conditions at the surface over interior Alaska and northwestern Canada.
The analogs show a nearly complete reversal in the pattern for February, with unusually low pressure aloft and at the surface over western Alaska. Above-normal temperatures are slightly favored in southern Alaska and western Canada, as the highest likelihood of unusual cold is in eastern Siberia.
Finally, for March: there isn't much of a signal for the broad circulation pattern.
To sum up, the history of similar past years based solely on the conditions in Barrow this year provides a hint as to what might transpire in January through March 2014. While the uncertainty is great, and we can't have high confidence in this kind of forecast, the analogs suggest a cold January may be in store for Fairbanks. In 9 out of the 10 analog years, the number of days in January with a daily minimum temperature of -20 °F or lower was at or above the 1981-2010 median of 13 days. However, the January and February weather patterns seem likely to be quite different from each other, with more unsettled conditions in February perhaps leading to a snowy month for Anchorage.