| ... | @@ -14,4 +14,18 @@ The evidential difference between 99 and 99.9% is now clearly huge. By performin |
... | @@ -14,4 +14,18 @@ The evidential difference between 99 and 99.9% is now clearly huge. By performin |
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This also explains why any opinion, except 0%, combined with 100% yields 100% in Bayes. The one exception, the 0% opinion, combined with 100% is not computable and yields no result. In the sense outlined here, a 100% opinion doesn't really exist because it implies that the scientist must perform an infinite number of experiments to attain it. The same is true for 0%. Anyone who performs, say 1000 experiments, and claims that all of them succeeded (thus yielding a "100%" success rate) can be challenged as simply not having performed enough experiments.
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This also explains why any opinion, except 0%, combined with 100% yields 100% in Bayes. The one exception, the 0% opinion, combined with 100% is not computable and yields no result. In the sense outlined here, a 100% opinion doesn't really exist because it implies that the scientist must perform an infinite number of experiments to attain it. The same is true for 0%. Anyone who performs, say 1000 experiments, and claims that all of them succeeded (thus yielding a "100%" success rate) can be challenged as simply not having performed enough experiments.
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Keep in mind that a probability is something that _can_ happen through random chance. We are not talking about logical or mathematical statements, for example. You cannot say that 2+2=4 100% of the time and call this a probabilistic statement. Such a statement is always true and there is no random event that can alter it. And, needless to say, statements of judgement are not probabilities either. If someone says they are 100% sure it will rain tomorrow, they are either rounding, estimating and exaggerating, or using 100% as an English synonym for "very certain". Whenever a probability is used, it must include the random chance that the opposite event will take place, however small it might be. |
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Keep in mind that a probability is something that _can_ happen, or not, through random chance. We are not talking about logical or mathematical statements, for example. You cannot say that 2+2=4 100% of the time and call this a probabilistic statement. Such a statement is always true and there is no random event that can alter it. And, needless to say, statements of judgement are not probabilities either. If someone says they are 100% sure it will rain tomorrow, they are either rounding, estimating and exaggerating, or using 100% as an English synonym for "very certain". Whenever a probability is used, it must include the random chance that the opposite event will take place, however small it might be.
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Let's relate this to Bayes using a concrete example. A test for a disease is 99% accurate. So, as we've just seen, the experiment leading to that 99% must have performed at least 100 tests to confirm. Let's suppose that a 2nd test is 10% accurate. So we have the following situation:
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100 healthy subjects ==\> Test 1 is 99% accurate ==\> 1 tests positive (false positive) ==\> Test 2 is 10% accurate ==\> .9 test positive again.
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100 sick subjects ==\> Test 1 is 99% accurate ==\> 99 test positive (true positive) ==\> Test 2 is 10% accurate ==\> 9.9 test positive again
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The number of people who tested positive twice = 0.9 + 9.9 = 10.8
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Of these, the number of people who are actually sick: 9.9
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P = Number of sick people who tested positive twice / Number of tested positive twice = 9.9/(9.9 + 0.9) = 0.917
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$P=a+b$ |
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