Bayesian belief networks with two monitored parameters
Consider a Bayesian belief network (BBN) with two eruption monitoring parameters. For example, a BNN might include a hidden node that describes the probability that magma is ascending beneath the volcano. There are two monitoring observations, whether gas flux is high (or not) and whether RSAM is high (or not). How do these two variables inform the eruption probability?
The simplest procedure is to calculate the conditional probabilities P[high RSAM | magma is ascending] and P[high gas flux | magma is ascending] as before using Bayes' rule. We then calculate the probability that both RSAM and gax flux are high, given that magma is ascending and use these values to calculate the "inverse" probability, the probability that magma is ascending given that both gas flux and RSAM are high.
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 (1) the probability magma is ascending if we observe high RSAM values. This is the probability \(P[A|B1]\), 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 (\(P[B1|A]\)) and the probability of not high RSAM given no magma ascent (\(P[\neg B1 | \neg A]\)). Similarly, \(P[A|B2]\) is the probability that magma is ascending given that we observe high gas flux.
\(P[A]\) is the probability of magma ascent (we guess!), \(P [\neg A] = 1 - P[A]\) is the probability magma is not ascending.
\(P[B1|A]\) is the probability RSAM is high, given that magma is ascending; \(P[\neg B1|A] = 1 - P[B1|A]\)
\(P[\neg B1|\neg A]\) is the probability RSAM is not high, given that magma is not ascending; \(P[B|\neg A] = 1 - P[\neg B1|\neg A]\)
\(P[B1]\) is the probability RSAM is high: $$P[B1] = P[B1|A] P[A] + P[B1| \neg A] P[\neg A]$$
\(P[ \neg B1]\) is the probability RSAM is not high: $$P[\neg B1] = 1 - P[B1] = P[\neg B1|A] P[A] + P[\neg B1| \neg A] P[\neg A]$$
\(P[C]\) is the probability of eruption in some timeframe, given magma is ascending
Now we apply Bayes' theorem for RSAM: \(P[A|B1]\) is the probability magma is ascending if RSAM is high: $$P[A|B1] = \frac{P[B1|A] P[A] }{P[B1]}$$
\(P[\neg A|B1]\) is the probability magma is not ascending if RSAM is high: $$P[\neg A|B1] = 1 - P[A|B1]$$
\(P[A|\neg B1]\) is the probability magma is ascending if RSAM is not high: $$P[A| \neg B1] = \frac{P[\neg B1|A] P[A] }{P[\neg B1]}$$
And for gas flux in exactly the same way: \(P[A|B2]\) is the probability magma is ascending if gas flux is high: $$P[A|B2] = \frac{P[B2|A] P[A] }{P[B2]}$$
\(P[\neg A|B2]\) is the probability magma is not ascending if gas flux is high: $$P[\neg A|B2] = 1 - P[A|B2]$$
\(P[A|\neg B2]\) is the probability magma is ascending if gas flux is not high: $$P[A| \neg B2] = \frac{P[\neg B2|A] P[A] }{P[\neg B2]}$$
The probability that both gas flux and RSAM are high, given that magma is ascending: $$P[B1,B2 | A] = P[B1|A] P[B2|A]$$
The inverse probability is the probability that magma is ascending, given that both RSAM and gas flux are observed to be high: $$P[A|B1,B2] = \frac{P[B1,B2 | A] P[A]}{P[B1,B2 | A] P[A] + P[B1,B2 | \neg A] P[\neg A]}$$
Then the probability of eruption in the timeframe, given RSAM and gas flux are high is: $$P[C| A|B1,B2] = P[C] P[A|B1,B2]$$
Some References
Hincks et al., 2014 discuss Bayesian networks applied to volcano eruption forecasting.
Other examples of BBN in volcanic eruption forecasting are Christophersen et al., 2018 and Sheldrake et al., 2017.
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.