Risk functional vs Loss function

[sgomezr]

What is exactly the difference between a risk functional and a loss function? Last class, we discussed MSE (minimum squared error) and ARI (Arbitrary rand index) as good examples for the loss function, and defined the risk functional as l:ARI. Does this means that the risk functional is dependent on the loss function?

[kristinmg]

This is confusing me as well so I'd appreciate if someone could explain

[maxcollard]

(Potential duplicate of https://github.com/Statistical-Connectomics-Sp15/intro/issues/97)

[sgomezr]

Not really, because I want more of a comparison. Especially because I have noticed that some functions (like the MSE) can be sometimes used as decision rule (if using Bayes), loss function and risk functional...

[maxcollard]

I think there's some wiggle room in the particulars of the formalism, but generally, loss is measuring how bad your protocol is at assigning actions to a particular sample, where as risk is measuring how bad your protocol is at assigning actions overall to samples that are realizations of a specific random variable in your model class. So, your decision protocol might be really good at figuring out the "truth" for a particular graph g---i.e., Loss(g) ~ 0---but terrible at figuring out the "truth" for graphs drawn from a particular distribution---e.g., Risk( ErdősRényi(6, 0.01) ) >> 0.

[akim1]

This is my understanding of loss and risk. if you were to pull on a slot machine once, that would give you a loss (e.g., the amount of money you won/lost). If you were to pull on a slot machine for a sufficiently large number of time, you will get risk which tells you, on average, how much money you will lose/win.

The easiest way to think of this is that whatever decision framework you will have will sometimes give you wrong answers and sometimes give you right answers (i.e., loss). The loss function is just a formalism to determine how correct our answer was. Since you don't really know what the right answer is in most real problems, you can construct a quantity that gives a reasonable measure of how correct you are based on your problem (e.g., adjusted rand index).

Risk is just a measure of how correct you will be, on average, with your framework. The usefulness of a given framework essentially depends on your being more correct than wrong (both in probability and magnitude).

[adjordan]

I believe that is the correct answer. Your risk is the probability of a loss.

[sgomezr]

Thanks guys! ^^

[jovo]

yup, more generally, risk can be defined as any functional of the distribution of loss, with respect to the distribution of the data.

so, \int f(loss(x,y)) * prob(x) dx, where y is the truth, and f(.) is some function, such as expectation or median.

On Fri, Feb 20, 2015 at 2:05 PM, Sandra Gomez notifications@github.com wrote:

Thanks guys! ^^

— Reply to this email directly or view it on GitHub https://github.com/Statistical-Connectomics-Sp15/intro/issues/99#issuecomment-75299434 .

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