Someone explain the appeal of Old Man Henderson?

[Kaelik]

Third:
Okay, we have someone who eats pizza 1 out of 50 nights. Prior 0.02.
This person throws the box away the same night 50% of the time and throws it away the next night 50% of the time. Other factors are considered negligible.
If we see a pizza box, the probability that he had pizza tonight goes up to 0.5, as does the probability that he had pizza last night.
If we don't see a pizza box, the probability that he had pizza tonight must now be the probability that he had pizza tonight and did not throw the box away tonight - 0.02*0.5=0.01. Not looking the the trash can leaves us at 0.02, but looking and seeing no box makes us slightly more certain that he did not (98%->99%).

Yes, for that number. But it is trivially easy (for people who do math) to move the numbers around so that you hit the prior.

[virgil]

Are we trying to apply Bayesian Philosophy toward the subjective valuation of Zak's diplomacy system or something?

[Kaelik]

Are we trying to apply Bayesian Philosophy toward the subjective valuation of Zak's diplomacy system or something?

No we just diverted the topic to something actually kind of worth talking about.

[momothefiddler]

Third:
Okay, we have someone who eats pizza 1 out of 50 nights. Prior 0.02.
This person throws the box away the same night 50% of the time and throws it away the next night 50% of the time. Other factors are considered negligible.
If we see a pizza box, the probability that he had pizza tonight goes up to 0.5, as does the probability that he had pizza last night.
If we don't see a pizza box, the probability that he had pizza tonight must now be the probability that he had pizza tonight and did not throw the box away tonight - 0.02*0.5=0.01. Not looking the the trash can leaves us at 0.02, but looking and seeing no box makes us slightly more certain that he did not (98%->99%).

Yes, for that number. But it is trivially easy (for people who do math) to move the numbers around so that you hit the prior.

Yes, but at that point you're saying that P(pizza|~box)=P(pizza), meaning pizza and ~box are independent, so pizza and box must necessarily be independent too.

For instance, if the man eats pizza 1 out of 50 nights and throws the box away sometime in the next 50 days, finding the trash can empty doesn't give you any further information but neither does finding a box.

Are we trying to apply Bayesian Philosophy toward the subjective valuation of Zak's diplomacy system or something?

No we just diverted the topic to something actually kind of worth talking about.

This

[RadiantPhoenix]

Well, let's see...

We observe "box" or "~box"
We do not directly observe "pizza" or "~pizza"

	pizza	~pizza
box	P(pizza|box)	P(~pizza|box)
~box	P(pizza|~box)	P(~pizza|~box)

For obvious reasons, each row adds up to 1. Each column, however, does not necessarily add up to 1.

If we look at this as an arbitrarily large sample, so we can ignore noise

	pizza	~pizza	n
box	n(pizza & box)	n(~pizza & box)	n(box)
~box	n(pizza & ~box)	n(~pizza & ~box)	n(~box)
n	n(pizza)	n(box)	n(1)

It looks like:
P(pizza|box) = n(pizza & box) / n(box)
P(~pizza|box) = n(~pizza & box) / n(box)
P(pizza|~box) = n(pizza & ~box) / n(~box)
P(~pizza|~box) = n(~pizza & ~box) / n(~box)


But we don't actually have the n values, we have:

	pizza	~pizza
box	P(box|pizza)	P(box|~pizza)
~box	P(~box|pizza)	P(~box|~pizza)

And now the columns need to add up to 1, but the rows don't. Let's simplify:

	pizza	~pizza
box	P(box|pizza)	P(box|~pizza)
~box	{1 - P(box|pizza)}	{1 - P(box|~pizza)}

If we also have P(pizza), then we get a table that looks like:

	pizza	~pizza
box	P(box|pizza) * P(pizza)	P(box|~pizza) * P(~pizza)
~box	P(~box|pizza) * P(pizza)	P(~box|~pizza) * P(~pizza)

Now the whole table adds up to 1.

Simplified, that's:

	pizza	~pizza
box	P(box|pizza) * P(pizza)	P(box|~pizza) * {1 - P(pizza)}
~box	{1 - P(box|pizza)} * P(pizza)	{1 - P(box|~pizza)} * {1 - P(pizza)}

Assuming I didn't screw up, this is our table of "n" values. (or rather, a proportional scaling of it, which is good enough)

So, if we add the components up...

	pizza	~pizza
box	P(box|pizza) * P(pizza)	P(box|~pizza) * {1 - P(pizza)}
~box	{1 - P(box|pizza)} * P(pizza)	{1 - P(box|~pizza)} * {1 - P(pizza)}

If we want to make P(pizza | ~box) = P(pizza) we go:

• [{1 - P(box|pizza)} * P(pizza)]/[{1 - P(box|pizza)} * P(pizza) + {1 - P(box|~pizza)} * {1 - P(pizza)}] = P(pizza)
• {1 - P(box|pizza)}/[{1 - P(box|pizza)} * P(pizza) + {1 - P(box|~pizza)} * {1 - P(pizza)}] = 1
• {1 - P(box|pizza)} = {1 - P(box|pizza)} * P(pizza) + {1 - P(box|~pizza)} * {1 - P(pizza)}
• {1 - P(box|pizza)} * {1 - P(pizza)} = {1 - P(box|~pizza)} * {1 - P(pizza)}

Now, if P(pizza) is not 1...

• {1 - P(box|pizza)} = {1 - P(box|~pizza)}
• P(box|pizza) = P(box|~pizza)

So, if P(pizza | ~box) = P(pizza), then P(box|pizza) = P(box|~pizza)

... and now I've forgotten what we were trying to argue.

[Korwin]

... and now I've forgotten what we were trying to argue.

I never knew what you tried.

[momothefiddler]

So, if P(pizza | ~box) = P(pizza), then P(box|pizza) = P(box|~pizza)

... and now I've forgotten what we were trying to argue.

Well I'll admit I got lost pretty quickly in all that, but it looks like your final result is "If pizza and box are independent, then box and pizza are independent", and I agree with that, at least.

[Mistborn]

You're terrible at reading. What am I accomplishing? The lie on the page is next to the correction of the lie. That's what's accomplished. This is at least the 3rd time I've repeated this simple idea to you, I guess I might as well repeat: "You're terrible at reading."

No you're just terrible a arguing. We do read your posts (and then immediately regret doing so) it's just that your arguments are bad arguments, ones we've heard and refuted many times before. You are not the first person to come charging into the Den to have this argument with us. we've had this argument many times already and we have become we have become exceedingly efficient at it.

[RadiantPhoenix]

... and now I've forgotten what we were trying to argue.

I never knew what you tried.

Well, now we're in the same boat.

[Zak S]

You're terrible at reading. What am I accomplishing? The lie on the page is next to the correction of the lie. That's what's accomplished. This is at least the 3rd time I've repeated this simple idea to you, I guess I might as well repeat: "You're terrible at reading."

No you're just terrible a arguing. We do read your posts (and then immediately regret doing so) it's just that your arguments are bad arguments, ones we've heard and refuted many times before. You are not the first person to come charging into the Den to have this argument with us. we've had this argument many times already and we have become we have become exceedingly efficient at it.

Everybody was having a nice conversation about formal logic and you went and did that. Just move on, small troll. Think about other things.

[Mistborn]

Everybody was having a nice conversation about formal logic and you went and did that. Just move on, small troll. Think about other things.

[RedstoneOrc]

we have become we have become

Oh no Lord Mistborn made a mistake! Zak S was correct all is lost.
Zak is right Zak is right Zak is right /sarcasm.
Back to popcorn.

[codeGlaze]

Back to popcorn.

(Thank you Surgo)