Bayesian Belief Networks
One potential problem with event trees is that they assume cause and effect. For example, part of a binary event tree for volcanic hazards might show that unrest may be related to magma ascent, which may result in eruption (see binary tree link). A problem is that magma ascent causes volcano unrest, not the other way around. For example, magma ascent might result in high RSAM. High RSAM does not cause magma ascent. Therefore, an event tree in which unrest leads to magma ascent leads to eruption seems to violate one of its own assumptions, that events follow from one another in an orderly direction with time. Nevertheless, it is quite important to capture the notion that high RSAM is an indicator of increased likelihood of magma ascent. An alternative structure to the event tree is called a Bayesian Belief Network (BBN), first proposed for use in volcano eruption forecasting, to my knowledge, by Thea Hincks, Willy Aspinall and colleagues.
In this simplest BBN, magma ascent is a hidden variable (gray circle), meaning we have no direct information about whether it is occurring or not. Magma ascent causes high RSAM and magma ascent leads to eruption. High RSAM and eruption are both observed variables (red circles), which are both related to the hidden variable. So we are interested in the Bayesian "inverse" probability, given high RSAM, what is the probability of magma ascent? Given high RSAM, what is the probability of eruption?
The formulas:
As before (see Bayes' rule link), \(P[A]\) is the probability magma is ascending beneath a volcano, so \(P[\neg A] = 1 - P[A]\) is the probability magma is not ascending. We want to know the probability magma is ascending if we observe high RSAM values. This is the probability \(P[A|B]\), read: the probability that magma is ascending given that we observe high RSAM. If we assume values for \(P[A]\), the probability of high RSAM given magma ascent (\(P[B|A]\)) and the probability of not high RSAM given no magma ascent (\(P[\neg B | \neg A]\))then we can calculate the probability:
\(P[A]\) is the probability of magma ascent (we guess!), \(P [\neg A] = 1 - P[A]\) is the probability magma is not ascending.
\(P[B|A]\) is the probability RSAM is high, given that magma is ascending; \(P[\neg B|A] = 1 - P[B|A]\)
\(P[\neg B|\neg A]\) is the probability RSAM is not high, given that magma is not ascending; \(P[B|\neg A] = 1 - P[\neg B|\neg A]\)
\(P[B]\) is the probability RSAM is high: $$P[B] = P[B|A] P[A] + P[B| \neg A] P[\neg A]$$
\(P[ \neg B]\) is the probability RSAM is not high: $$P[\neg B] = 1 - P[B] = P[\neg B|A] P[A] + P[\neg B| \neg A] P[\neg A]$$
\(P[C]\) is the probability of eruption in some timeframe, given magma is ascending
Now we apply Bayes' theorem. \(P[A|B]\) is the probability magma is ascending if RSAM is high: $$P[A|B] = \frac{P[B|A] P[A] }{P[B]}$$
\(P[\neg A|B]\) is the probability magma is not ascending if RSAM is high: $$P[\neg A|B] = 1 - P[A|B]$$
\(P[A|\neg B]\) is the probability magma is ascending if RSAM is not high: $$P[A| \neg B] = \frac{P[\neg B|A] P[A] }{P[\neg B]}$$
Then the probability of eruption in the timeframe, given RSAM is high is: $$P[C| A|B] = P[C] P[A|B]$$
Some References
Hincks et al., 2014 discuss Bayesian networks applied to volcano eruption forecasting.
There is lots of information on application of Bayesian networks from fields like machine learning and biology. Nagarajan et al. is a good introduction using R.