[EM] Richard's frontrunners example

[MIKE OSSIPOFF]

>You see, the problem with this argument is as follows:
>
>b is a conditional statement of the form, if X then Y. Now b is true 
>whenever
>X and Y are both true. But b is also true if X is false and Y is true. 
>What's
>more, it's true if X and Y are both false. The only way statement b can
>be false is if X is true and Y is false.
>The consequence of this is that b can be true at times when a is not.

>b will
>be true whenever a is true; it will also be true whenever X (there is a 
>tie)
>is false. If the probability that there is no tie is 99%, then b will be 
>true
>at least 99% of the time.

So you're saying that b is true when it isn't making any prediction
about Y. When X is false, b says nothing true about whether Y is
true, and says nothing false about whether Y is true. And you claim
that b is true when it isn't taking a position about Y.

You know there are statements that aren't true or false, and that's
one of them, in that situation.

Of course then you can say "Then when a is true, and X is false,
then b makes no claim, true or false, about Y. So b isn't always
true when a is true."

That would be a more convincing criticism. It just occurred to me
now. I'll comment more on a subsequent day.

Just for now, though, what convinced me that b is true when a is,
was the notion of b being a statement whose truth or falsity isn't
determined by whether it can be answered in any particular situation.
A statement only about when X is true. So we don't even score b's
truthfulness when X isn't true. We ignore that situation.

What I mean is that since b's accuracy can only be tested when X is
true, then we must judge its truthfulness by how accurate it is when
X is true. When b is inapplicable because X is false, then I say
"Ignore that. Tideman's Pij definition isn't intended to consider what
happens when there's no tie." You'll say that's a weak position. I
don't know. Strictly speaking, a is true and b isn't true, in that
situation. But what if we take b to mean "If there is or were a
2-way tie for 1st place, then it is or would be between i & j"?

Then, when there isn't such a tie, be is about if there were a
2-way tie for 1st place. And if a is true, and if there were a 2-way
tie for 1st place, then it would be between i & j. So when a is true,
b is true.

Now, it seems to me that there's a good case for saying that that
is what Tideman actually meant. That's how I interpreted it, and
maybe that's how he meant it.

If it's interpreted in that way, then both Pij definitions are the
same. I suggest that you've pointed out that your interpretation of
what Tideman meant can act unexpectedly. If anything, that suggests
that my Pij definition is better. Or so it seems to me right now.
The objection that b might not be true when a is is new to me today,
and so right now I'm not saying anything for sure. Today you're speaking
with more assurance than I am.

I haven't checked whether the those 2 interpretations of Tideman's
Pij definition actually result in different Approval strategies.
Most likely it's just a logic debate without material consequences.
But, again, it's too soon for me to be sure, about an objection that
only occurred to me while writing this letter.



>If the equivalence is not in evidence to begin with, you can't use that
>equivalence to prove the probabilities are equal.

Of course not. I've been saying all this time that either a & b are
both true or both false.

> > But please note that neither Bart nor I said anything about the
> > probability that i & j are in a 2-way tie for 1st place. And neither
> > Bart nor I said anything about the probability that there will be
> > a tie for 1st place. So this just isn't a Bayes problem.
>
>How can you deduce whether or not Bayes applies from what
>you or Bart never said? Let's look instead at what was said. Here's
>what Bart said:
>
>Slight correction/clarification:  The precise meaning of Pij is usually
>taken to mean the probability, given a tie exists for first place, that
>i and j will be involved in that tie.
>
>This *is* a conditional probability.

Sure. But I've been saying that it doesn't matter what's inside
statement b.


Mike Ossipoff

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