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I don't think it is completely out of character, considering how desperate the situation is and how desperately he is willing to believe Melisandre. Though it is definitely a mistake, coupling danger to his legitimacy through his middle-aged heirlessness and the ugly image of burning a little girl to death in front of the men. 

I've heard that Shireen's demise is straight from GRRM, which has interesting implications considering how much of his army in the book is Northmen. Maybe in the show, Stannis is going to end up delivering an army to Sansa? One can hope. 
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