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I'm trying to get a grasp on utility and loss functions, and at first I thought that a utility function was the flipside of a loss function and vice versa. Kind of like how if you know the probability of getting heads is p, you know the probability of getting tails (1-p). However, I can't find anything on the internet to that effect. I might be thinking of the regret function, but I'm not sure. Am I missing something?

kjetil b halvorsen
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Katherine Savoie
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    You are correct, the loss function is the negative of the utility function. In optimization we either maximize the utility or minimize the loss, but that's just a choice of convention, the optimization is mathematically equivalent. – Hugh Feb 22 '17 at 17:53
  • I don't think it's universally true. They're certainly related though – Aksakal Feb 22 '17 at 21:01
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    I don't think you're missing anything; to the extent that you can specify utility, the loss function should be chosen so that minimizing the loss maximizes the utility. – Glen_b Feb 23 '17 at 00:27

2 Answers2

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Loss is a negative utility. If you need an authoritative source for this, check the Statistical Decision Theory book by James O. Berger (p. 53):

Once $U(\theta, a)$ has been obtained, the loss function can simply be defined as

$$ L(\theta, a) = -U(\theta, a). \tag{2.3} $$

The same is stated by Christian P. Robert in his book The Bayesian Choice, who introduces utility with the following notion (p. 54):

The notion of utility (defined as the opposite of loss) is used not only in Statistics, but also in Economics and in other fields like Game Theory where it is necessary to order consequences of actions or decisions. Consequences (or rewards) are generic notions which summarize the set of outcomes resulting from the decision-maker’s action. In the simplest cases, it may be the monetary profit or loss resulting from the decision. (...)

Tim
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$U(x)=-\mathcal{L}(x)$. I cannot imagine you will find anything on the internet. I believe you can find a formal treatment of these functions in Geweke's "Contemporary Bayesian Econometrics and Statistics."

Dave Harris
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  • We could perhaps be more general than that. I think some other choice of $\mathcal L$ such that minimizing it will maximize $U$ should suffice. That includes $U = -\mathcal{L}$ but really, $U=m(-\mathcal{L})$ for some sufficiently nice monotonic function $m$ should work just as well. (would continuity of $m$ suffice? Is it more than required? It would at least be desirable to have it, for several reasons). It's a bit like maximizing likelihood vs log-likelihood vs 2 log likelihood -- the negative of any of them could be taken as a loss function (any many other choices beside) – Glen_b Feb 23 '17 at 00:29