Essay:A new approach to probability

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These are some ideas floating around in my head... please let me know if I've screwed up the maths, I probably have...

In this essay I want to introduce a new approach to probability. I don't mean this approach as a replacement for traditional approaches, just as another tool which can be useful in certain circumstances.

Normally we express probabilities as a real number between 0.0-1.0. However, sometimes we deal with scenarios where the probabilities are very close to 0 or 1, and in those circumstances a more logarithmic scale may be more useful. Also, attempts to assign precise probabilities suffers from the metaprobability problem — how do we know the probabilities we have assigned are the right ones?[1]We may often feel very uncertain about assigning an exact probability, but less uncertain in indicating a rough order of magnitude. So, using a logarithmic scale for probability has its advantages.

So, let me introduce four new measures of probability:

• positive certainty: this is a measure of closeness to certainty (1.0). It can be defined as $-\log{(1 - p)}\,\!$. For example, using base 10 logarithms, if p=0.9, then we have a positive certainty of 1, if p=0.99, a positive certainty of 2, if p=0.999, a positive certainty of 3, etc. If p=1.0 exactly, that is infinite positive certainty; p=0.5 is a positive certainty of about 0.30, p=0.0 is a positive certainty of 0.
• negative certainty: this is a measure of closeness to impossibility (0.0). It can be defined as $-\log{p}\,\!$. For example, using base 10 logarithms, if p=0.1, then we have a negative certainty of 1, if p=0.01, a negative certainty of 2, if p=0.001, a negative certainty of 3, etc. If p=0.0 exactly, that is infinite negative certainty; p=0.5 is a negative certainty of about 0.30, p=1.0 is a negative certainty of 0.
• positive uncertainty: this is a measure for probabilities near to but greater than 0.5. It can be defined as $-\log{(p - 0.5)}\,\!$. For example, using base 10 logarithms, if p=0.51, then we have a positive uncertainty of 1, if p=0.501, a positive uncertainty of 2, etc. If p=0.5 exactly, that is infinite positive uncertainty; if p=1.0 exactly, that would be a positive uncertainty of about 0.30. Since 0.30 is the minimum, that suggests redefining it as $\log{0.5} - \log{(p - 0.5)}\,\!$, to bring the minimum of the scale back down to zero. If p is less than 0.5, the positive uncertainty is undefined.
• negative uncertainty: this is a measure for probabilities near to but less than 0.5. It can be defined as $-\log{(0.5 - p)}\,\!$. For example, using base 10 logarithms, if p=0.49, then we have a negative uncertainty of 1, if p=0.499, a negative uncertainty of 2, etc. If p=0.5 exactly, that is infinite negative uncertainty; if p=0.0 exactly, that would be a negative uncertainty of about 0.30. Since 0.30 is the minimum, that suggests redefining it as $\log{0.5} - \log{(0.5 - p)}\,\!$, to bring the minimum of the scale back down to zero.

Other approaches to rescaling the uncertainties are possible; one could instead use something like: ${-\log{(p - 0.5)} \over \log{0.5} } - 1\,\!$. This measure has the interesting property of being independent of the choice of logarithmic base; it is essentially choosing a logarithmic base of 0.5.

 Which logarithm base to use?

The obvious choices are 2, 10, and e. I am wondering, could these measures be given an information-theoretic interpretation? If we used a logarithm base of 2, might it be reasonable to use bits as units for some of these measures??

 Footnotes

1. We can interpret probability either objectively (probability as some real aspect of external reality) or subjectively (probability as a measure of our degree of confidence in our own beliefs). Either way, the problem remains, how do we know we have chosen the right probability? If probabilities are objective, we can ask what is the probability that the probability we have chosen is the objectively correct one? If probabilities are subjective, we can ask whether the probability we have chosen is an accurate measure of our own confidence.