This cover of Nate Silver's book neatly summarizes what's true for

many important decisions.

There's a small amount of signal in the world, as in the case of

the photoreceptive current, and an awful lot of noise relative to any particular

decision for the same reasons as we discussed in our last lecture.

A given choice establishes a certain set of relative stimulus aspects and

all other information, which may be very useful information for other purposes,

becomes noise.

In deciding whether to invest energy in reacting, you're not running away from

the tiger, calling in the bomb squad to detonate a shopping bag, asking a girl for

a date, the prior probability isn't the only factor.

One also might want to take into account the cost of acting or not acting.

So now let's assume there is a cost, or a penalty, for getting it wrong.

You get eaten, the shopping bag explodes.

And the cost for getting it wrong in the other direction, your photo gets spoiled,

you miss meeting the love of your life.

So how do we additionally take these costs into account in our decision?

Let's calculate the average cost for

a mistake, calling it plus when it is in fact minus.

We get a loss which we'll call L minus, penalty weight,

and for the opposite mistake, we get L plus.

So our goal is to cut our losses and make the plus choice when the average loss for

that choice is less than the other case.

So we can write this as a balance of those average losses.

The average or the expected loss from making the wrong decision, for

choosing minus when it's plus is this expression, the weight for

making the wrong decision multiplied by the probability that that occurs.

And now we can make the decision to answer plus when the loss for

making the plus choice is less than the loss for the minus choice.

That is, when the average loss for

that decision is less than the average loss in the other case.

So now, let's use base rule to write these out.

So now have L + P(r|-) P(r|-)

divided by P(r), all that to

be less than the opposite case,

P(r|+)P(r) divided by

the probability of response.

So now you can see that when we cancel out this common factor,

the probability of response, and rearrange this in terms of our likelihood ratio,

because now we have here the likelihood.

The probability of response given minus, on this side the likelihood for

the probability of response given plus, we can now pull those factors out as

the likelihood ratio and now we have a new criteria for our likelihood ratio test.

Now one that takes these loss factors into account.