Absence as a "Complementary Hypothesis"

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Absence as a "Complementary Hypothesis"

Postby whomever on November 16th, 2013, 4:51 pm 

I've come up with this idea of the "complementary hypothesis". I don't know wheter it's truly original.
The idea: In a pair of complementary hypotheses, one is the explanatory hypothesis (as in the scientific method), and the other is the negation of the explanatory hypothesis. Because they are absolute complements, their respective probabilities must always add up to 1, and falsifying one proves the other.

When you take this view, hypotheses seem more testable. The most basic way to judge an hypothesis would be to compare it to its complement, and ask which does the data fit better? For this comparison, I think the bottom-line is the likeliness of the actual observations. Certain observations or sets of observations may be more probable in one condition than the other. For example:

"Somebody is sneaking into my room" (A) and
"Nobody is sneaking into my room" (AC) are complements.
Somebody ≥ 1 ... Nobody = 0 ... discrete counting using positive integers.
A is unfalsifiable because a really good sneaker will not leave any evidence of their sneaking; no observation would be inconsistent with A. However, there are some observations that would be inconsistent with AC, such as written messages appearing on the room's walls before I return to it. It's extremely unlikely that written messages would appear on my walls if nobody was sneaking in. In the absence of such evidence, I can at least say that this absence of evidence (B) is more likely given AC.
Condition A offers the absence of evidence as a possibility (P(B|A) ≤ 1.0), but only AC assures that it will be the case (P(B|AC) = 1.0).
https://en.wikipedia.org/wiki/Conditional_probability

Of course, scientists like to collect more information before reaching conclusions.
Any observation that supports A is, in other words, an observation that can be explained by A. But A may not be the only way of explaning the observation. We can't test A, but we may be able to test alternative explanations.
By supporting an alternative explanation, I discredit some of the evidence for A. And remember that the absence of evidence is the advantage that AC has over A.

Suppose I have no evidence that somebody is sneaking in except that Delilah, who sits in my room 24/7, keeps telling me so. This is unsettling. The fact that Delilah makes this claim is evidence for A, but I still have no way of refuting A, nor does there appear to be any evidence that confirms A. However, I could also explain Delilah's claims as hallucinations, and I can test whether she is delusional. Suppose I test her for delusions and rest on a 0.75 chance that her observations are hallucinations. The possibility of hallucination (D) is represented by the circle labeled "Delilah Delusional", whereas the possibility that her observations are genuine sensory experiences (G) is the rest of what is inside the circle labled "Delilah Claims". To adjust the probability of A with this incoming information:
P(A|D) ⋅ P(D) + P(A|G) ⋅ P(G)
P(A|D) ⋅ 0.75 + P(A|G) ⋅ 0.25

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Last edited by whomever on November 16th, 2013, 5:00 pm, edited 2 times in total.
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Re: Absence as a "Complementary Hypothesis"

Postby Venus on November 16th, 2013, 4:59 pm 

whomever wrote:The fact that Delilah makes this claim is evidence for A

How so?

Sounds to me this is a weak argumentum ad numerum fallacy.
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Re: Absence as a "Complementary Hypothesis"

Postby whomever on November 16th, 2013, 5:07 pm 

Venus wrote:
whomever wrote:The fact that Delilah makes this claim is evidence for A

How so?

Sounds to me this is a weak argumentum ad numerum fallacy.

She would be more likely to claim A if A was true. It's a simple matter of whether she is telling the truth and really has observed someone sneaking into the room.
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Re: Absence as a "Complementary Hypothesis"

Postby neuro on November 18th, 2013, 5:43 pm 

Actually, you don't need hypoteses to be absolute complements of each other: a pretty commonly used approach is to estimate the likelihood of an hypthesis (probability of observing exactly the set of data you did observe if the hypothesis is true). Such likelihood is a quite low number in most experimental conditions (e.g. the probability of measuring exactly the values you measure for the weight of ten people is bound to be very low, whatever is your hypothesis). However, the likelihoods of several hypotheses can be mutually compared to pick the most likely.
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Re: Absence as a "Complementary Hypothesis"

Postby whomever on November 18th, 2013, 9:41 pm 

That's what I initially thought, that each mathematical hypothesis (if/then statement) is viewed on its own. I began to wonder why people never state the complementary hypothesis and test that instead. For example, if I want to know whether somebody is sneaking into my room (A), I could test the complementary hypothesis (AC) instead.
After some thought, I decided that they should be taken together. After all, everything could be consistent with AC even if A really is true.

One problem with that position is that the probability of an event could be high for both conditions.
Suppose I am doing a double-slit experiment, but I only have one electron to shoot, and I need to know whether it travels as a particle or a wave. My electron gun is a tad wobbly, and the sensor is quite close to the gun.
If the electron is a particle or a wave, it has a 0.9 or 0.25 chance of landing in a particular region on the sensor, respectively.

In addition, a person could make unrelated propositions that have the same high probability whether the hypothesis is true or not. Given that we already know the sky to be blue, 1 and 2 are both true.
(1) "If carbon monoxide is dangerous, the sky should be blue."
(2) "If carbon monoxide is not dangerous, the sky should be blue."

What if I presented this to someone who had never seen the sky before:
  • If carbon monoxide is not dangerous, the sky should be blue.
  • The sky is blue.
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Re: Absence as a "Complementary Hypothesis"

Postby Gregorygregg1 on November 18th, 2013, 10:57 pm 

whomever wrote:I've come up with this idea of the "complementary hypothesis". I don't know wheter it's truly original.
The idea: In a pair of complementary hypotheses, one is the explanatory hypothesis (as in the scientific method), and the other is the negation of the explanatory hypothesis. Because they are absolute complements, their respective probabilities must always add up to 1, and falsifying one proves the other.


Thesis, antithesis, synthesis.
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Re: Absence as a "Complementary Hypothesis"

Postby owleye on November 19th, 2013, 11:46 am 

Interesting analysis, that is until you began to discuss electron guns. However it prompted me to think of the classes I took while a philosophy major that dealt with Jerrold Katz's "Realistic Rationalism", a book dedicated to the philosophy of mathematics. I think I still have that book in my library.

It's been awhile now, but one of the things that struck me was how he likened it to Sherlock Holme's rationalism, one in which goes something like this: If it isn't A, or B, or C, then it must be D (presumably because that's the total number of realistic ways that X can occur) in which it becomes elementary, my dear Watson.

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Re: Absence as a "Complementary Hypothesis"

Postby whomever on November 19th, 2013, 2:31 pm 

Physics 101 is just on the horizon. Until I arrive there, I will imagine that the electrons are being "shot" from a "gun".
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