I've been thinking. Occam's Razor isn't just thing we use because we prefer simpler situations, it works mathematically.

Suppose there are two models that accurately describe a set of observations. Both work but are unproven. Model 1 explains the observations in terms of variables X and Y. Model 2 explains the observations of variables A, B and C.

Since Model 2, uses three variables compared with two in Model 1, Model 2 is clearly more complicated.

The variables A, B, C, X and Y are unproven so the probability that they are true is less than 1. Say, just for sake of argument that the probability of each of them being true is 0.9.

In order for the models to be true, the variable in terms of which they are described must be true. It follows that the probability of them being true is the product of the probabilities of the variables involved being true.

Therefore,
P(Model 1 is true) = P(X is true, Y is true)
=0.9 × 0.9 = 0.81
P(Model 2 is true) = P(A is true, B is true, C is true)
=0.9 × 0.9 × 0.9 = 0.729

Model 2 is the more complicated one and it has a smaller probability of being true than the simpler Model 1.

It is intuitive from basic arithmetic, that since probabilities cannot be greater than 1 by definition, the more variables involved, a the more probabilities involved in the product, in general, the smaller the probability of the model being true. Therefore, simpler models, ie ones with less variables, have higher probabilities of being true and are therefore the best ones to use without any other deciding factors. Just as specified in Occam's Razor.

Note that Model 1 hasn't been proven to be true while Model 2 is false, it simply states that based on the evidence available, which describes both models as being equally effective at explaining the observations, Model 1 is more likely to be true than Model 2.

You assume that A,B,C,X and Y all have equal probability. If X and Y have a 0.8 probability and A,B and C have a 0.9 probability, the #2 is more likely (0.729 vs 0.64). Beware of oversimplification!

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Any day you wake up on "the right side of the dirt" is a good day.

T. Anderson

You have to assume equal probability, given no other information. Once more information is available, a revised statistical estimate can be made. This remains consistent with Occam's Razor, which tells us to answer our question without having to invent anything new.

[Edited to add: "What does this have to do with Lunar Conspiracies?"]


<font size=-1>[ This Message was edited by: boron10 on 2002-12-13 20:18 ]</font>

There are fewer variables in 'we really landed on the Moon' than in 'it's all a big fake'? I'd take issue with the 'equal probability' thing though...

I was looking at the Moon on the way home tonight... gosh, it's neat being able to look up at this ball in the sky and think, "wow... people have been there."

Quote: I was looking at the Moon on the way home tonight... gosh, it's neat being able to look up at this ball in the sky and think, "wow... people have been there."

Ahh... but did you look at it when Armstong and Aldrin were actually there. I did.


<font size=-1>[ This Message was edited by: Kiwi on 2002-12-14 05:47 ]</font>

I remember looking at the moon when astronauts were there, but I don't remember which mission.

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Life is like a box of chocolates. All of your choices are bad for you.

The Christmas season is the time, I always look at the moon. It reminds me of the Apollo 8 mission and my start of my lifelong interest in astronomy, the space program, etc.

Especially lately I look at the moon with renewed fascination.

About Glom’s original post.

1) You are also assuming that ‘X’ and ‘Y’ are independent. Otherwise, it isn’t true that their joint probability is the product of individual probabilities.
2) What you call ‘variables’ would best be described as assumptions, I think. It makes no sense to attribute probability to a variable, only to a statement.

With the comparison Microsoft made to it's software and anything having to do with the moon, I look up at the moon and half expect it to crash any second now.

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"Some see the glass half full, some see it half empty, and some see it crawling with toxic alien parasites who want to devour your pancreas." - Sgt Aarhus, from the book Ascending by James Alan Gardner

The relevance to the issue is that on this forum, we discuss the pseudologic of the conspiracists. Occam's Razor is a very important part of that. So important that Jay dedicated an entire page to it.

The numerical example I gave was obviously an oversimplification, but I was trying to show how in general, the more variables you add to a model, and hence the more you complicate it, the lower the probability of it becoming true in the end. It doesn't always work, but then Occam's Razor does say the simplest solution is the best, but merely the best one to use in lack of any more evidence.

Possibly, I used the word 'variable' in the wrong context.

Luna is my favourite astronomical object to observe. I always think that first quarter is the best time to observe it when the shadows appear the longest. Montes Appeninus is stunning. Unfortunately, we've had nothing but perpetual cloud cover for the last month.

<font size=-1>[ This Message was edited by: Glom on 2002-12-14 08:21 ]</font>

<a name="2-12-14.Mh"> page= 2-12-14.Mh aka Math Hour
On 2002-12-14 07:53, Jetmech0417 wrote: To? 5:20 A.M. Pst
1 ever thinner
thats how I remember the story?
3 however I remember another 1
about the Area & volume of a )( conical section /
I dont recall the name of the shape now
its like a cornicopia wyhout a twist
really? glad its Saturday
now back to Other MATters

"Blue Moon of Death"?

__________________
Any day you wake up on "the right side of the dirt" is a good day.

T. Anderson

Isn't this using Occam's Razor to demonstrate that Occam's Razor works?

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Life is like a box of chocolates. All of your choices are bad for you.

[img]/phpBB/images/smiles/icon_lol.gif[/img]

It does seem like that, doesn't it? I was just saying that, in light of no other facts, Occam's Razor is consistent with a statisitcal estimate of independent variables. It would be tautological to show that Occam's Razor is consistant with Occam's Razor! [img]/phpBB/images/smiles/icon_biggrin.gif[/img]

Lucky! I'm not nearly old enough. I'm just hoping the Chinese make it.

Are you talking about the shape used to demonstrate the warping effect of a black hole? I don't recall if I've ever heard a name for it. Tell me, what about the area and volume of it?