Re: Bayesian boxes and expectation value

From: Jacques M Mallah <jqm1584.domain.name.hidden>
Date: Wed, 19 May 1999 18:24:27 -0400

On Wed, 19 May 1999 GSLevy.domain.name.hidden wrote:
> In a message dated 99-05-18 17:15:37 EDT, Jacques Mallah writes:
> << You don't seem to understand: that's NOT how to take an
> expectation value. It bears little resemblance to the formula for an
> expectation value, regardless of what "the distribution of m" is. >>
>
> The concept of scales and distributions are kind of related: having a uniform
> distribution over a logarithmic scale is equivalent to having an exponential
> distribution over a linear scale.
>
> The result if a calculation of an expectation value is predicated on the
> scale used. For example if the expectation value of the sound intensity is
> desired it could be calculated either using watts or using decibels with
> vastly different results since the sound intensity in decibels has a
> logarithmic relationship with its intensity in energy units.

That is not an example of two different ways to take an
expectation value, it's an example of taking two different expectation
values of different functions.
In the boxes case, the function we were trying to take the
expectation value of is the value, not the log of the value.

> Who is to say that one particular scale is more "natural"
> than the other?

Irrelevant. The only correct function to use is the one you're
trying to take the expectation value of.
Because our assumed utility function was proportional to the
value in the box, the expectation value of that is of interest in decision
making. If we had some other utility function, we would need the value of
that for decision making.
Regardless, once we fix some quantity as the one we are
considering, we have to use the usual formulas.

> I still maintain that the scale issue in the Bayesian problem comes in the
> back door and must be dealt with. In the Bayesian case, the problem itself
> defines the scale to be used. And selecting a logarithmic scale (for example)
> guarantees that the expectation value of the box not chosen is exactly equal
> to m, which agrees with common sense.

I hope you now see the error in your above paragraph. The
log of the expectation value of the amount in the box is not equal to the
expectation value of the log of the amount.

- - - - - - -
Jacques Mallah (jqm1584.domain.name.hidden)