Use the sliders to change the values of L1 (the Likelihood of the data given that H1 is true) and L0 (the Likelihood of the data given that H0 is true). Notice how BF10 (the Bayes Factor or Likelihood Ratio) changes. You can also drag the dot on the graph to change the values of L1 and L0.
| Likelihood Ratio (BF10) | = | L1 | ÷ | L0 |
| 1.0000 | = | 0.5000 | ÷ | 0.5000 |
A Likelihood Ratio (or Bayes Factor, BF) tells us how much the evidence favors one hypothesis over another hypothesis. It is how likely the observed data (the evidence) would be if one hypothesis were true compared to how likely the observed data would be if the other hypothesis were true. In our example the first hypothesis, H1 is that some research hypothesis is true. The second hypothesis, H0, is that it is false. So H1 is the research hypothesis, and H0 is the null hypothesis. And the Likelihood Ratio is the probability of observing our data if H1 were true (Likelihood1), divided by the probability of observing our data it H0 were true (Likelihood0). Bayes Factor (BF) is another term for the Likelihood Ratio, and BF10 means the Likelihood Ratio for L1 / L0. If you switch the subscripts, BF01 would be L0 / L1.
If the Likelihood Ratio (BF) = 1, then that means that our data would be just as likely to be observed when the hypothesis is false as when it is true. That means that the data tells us nothing about whether our hypothesis is true or not. So if BF10 = 1, we should ignore the data and not change our prior belief about the truth of the hypothesis at all.
If BF10 > 1, then that means the data is more likely if the hypothesis were true than if it were false. In that case we should update our prior belief to be more confident that the hypothesis is true than we were before seeing the data. The bigger the Bayes Factor BF10, the more our prior belief should change.
If BF10 < 1, then that means the data is less likely if the hypothesis were true than if it were false. In that case we should update our prior belief to be less confident that the hypothesis is true than we were before seeing the data. The smaller the Bayes Factor BF10, the more our prior belief should change towards disbelieving the hypothesis.
Here is something important to notice: The Likelihood Ratio (BF10) depends on two things: How strongly the hypothesis predicts the data, and how likely the data is in general. If our theory makes a strong prediction, one that has to be true if the theory is true, then L1 = 1, which is the maximum value for the numerator of the Likelihood Ratio, and will tend to make the Likelihood Ratio bigger. But if that prediction is likely to be true anyway even if our theory were false, then that would make L0 large too (maybe close to 1), and that big denominator would tend to make the Likelihood Ratio (BF10) smaller. So the math fits two of our intuitions about what makes good evidence for a theory: data that is strongly predicted by the theory, and data that would be surprising otherwise if the theory were not true.
Suppose we made a human saccrifice to the Sun God because we thought that would please him, and when he is pleased he makes the sun rise.
Our theory's prediction was confirmed! The sun rose today. How much more confident should we be in our theory now since we saw the sun rise as predicted?
It is 100% certain that if human saccrifice causes the sun to rise then the sun should rise after a human saccrifice is made. So L1 = 1.
But every day I've been alive I have seen the sun rise no matter what, so it is almost 100% certain that the sun will rise even if human saccrifice does not cause the sun to rise. So L0 = 1.
BF10 = 1 / 1 = 1
So seeing the sun rise today tells us nothing about the truth or falsity
of the theory that human saccrifice causes the sun to rise. The data will not change
our degree of belief in the theory at all.
Suppose we made a human saccrifice to the Sun God because we thought that would please him, and when he is pleased he makes a total solar eclipse.
Our theory's prediction was confirmed! There was a total solar eclipse today. How much more confident should we be in our theory now since we saw the eclipse as predicted?
It is 100% certain that if human saccrifice causes eclipses then there should be an eclipse after a human saccrifice is made. So L1 = 1.
Only about 1 out of 10,000 days when I've been alive I have seen a total eclipse, so the probability of an eclipse if human saccrifice does not cause eclipses is about 1 / 10,000. So L0 = 0.0001
BF10 = 1 / .0001 = 10,000
So seeing an eclipse today is very very strong evidence in favor of the theory.
So one thing these examples demonstrate is that evidence for a theory is much more impressive when that prediction would have been very unlikely unless the theory was true. Predicting an eclipse is more persuasive evidence than predicting a sunrise.
But something seems wrong about the second example, doesn't it? I'm still not ready to conclude that human saccrifice causes eclipses, are you? Why not?
I think this example demonstrates a second important point: Just because you calculate a large number for the Likelihood Ratio (BF), that does not guarantee the evidence is really strong. The BF provides an upper bound on how strong the evidence could be assuming that there is nothing wrong with the way the evidence was produced. If you suspect the research was badly designed or sloppily carried out, or that there was cheating involved, the data will not change your prior belief very much no matter how big the Bayes Factor is that was calculated. Was the day for the human saccrifice really chosen at random, or was it actually chosen because you knew from some other theory (like maybe astronomy and physics) that an eclipse would happen tomorrow? If cheating was involved, then the real likelihood for L0 was probably close to 1 rather than 0.0001, and the real BF10 was probably about 1 rather than 10,000. Just calculating is not enough. Judgment is also required.
Calculating a number does not substitute for critically evaluating the evidence, and that is true whether you are calculating a Bayes Factor, or a p value, or any other statistic.