One measure of probability is log-odds (or logit), which is the logarithm of the odds-ratio (the base of the logarithm doesn’t matter at the moment). That is, for an event with probability p, the log-odds is
And to convert back, for an event with log-odds (base b) of q the probability is frac{b{q}}{1b{q}}
Log-odds are a measure of the amount of information or evidence in support of that event. With a probability of 0.5, the log-odds is 0 (i.e. no net evidence either way), with a probability less than 0.5, log-odds is less than 0 (net evidence against the event), and with p > 0.5, the log-odds is greater than 0 (net evidence for the event).
In the odds world, a bel of evidence means multiplying the odds ratio by 10, so after observing a bel of evidence for it, an event goes from 1:1 to 10:1, or 1:30 to 1:3. In the base-10 log-odds world, this is equivalent to adding 1 to the log-odds, so, those examples become 0 → 1, and −1.48 → −0.48. A decibel is adding 0.1 to the log-odds (i.e. a tenth of a bel).
For the example given, 3 decibels turns 0 (log-odds for 50%) into 0.3, and the conversion back gives p = 66%. For 5% the log-odds are −1.279, so 3 decibels turns that into −0.979, which corresponds to p = 9.5%. Similarly, 1% becomes 1.9%, and 0.1% becomes 0.2%.
One measure of probability is log-odds (or logit), which is the logarithm of the odds-ratio (the base of the logarithm doesn’t matter at the moment). That is, for an event with probability p, the log-odds is
And to convert back, for an event with log-odds (base b) of q the probability is frac{b{q}}{1 b{q}}
Log-odds are a measure of the amount of information or evidence in support of that event. With a probability of 0.5, the log-odds is 0 (i.e. no net evidence either way), with a probability less than 0.5, log-odds is less than 0 (net evidence against the event), and with p > 0.5, the log-odds is greater than 0 (net evidence for the event).
In the odds world, a bel of evidence means multiplying the odds ratio by 10, so after observing a bel of evidence for it, an event goes from 1:1 to 10:1, or 1:30 to 1:3. In the base-10 log-odds world, this is equivalent to adding 1 to the log-odds, so, those examples become 0 → 1, and −1.48 → −0.48. A decibel is adding 0.1 to the log-odds (i.e. a tenth of a bel).
For the example given, 3 decibels turns 0 (log-odds for 50%) into 0.3, and the conversion back gives p = 66%. For 5% the log-odds are −1.279, so 3 decibels turns that into −0.979, which corresponds to p = 9.5%. Similarly, 1% becomes 1.9%, and 0.1% becomes 0.2%.
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