About Bayes' theorem

  • LaPlace seems to have been the first person to explicitly state Bayes' theorem mathematically, but Bayes wrote down the concept earlier than LaPlace and he gets most of the credit. The important thing to remember about Bayes' theorem is that it provides a way to estimate inverse probability. For example, given the probability that elevated gas flux results from magma ascent, what is the probability that magma is ascending given elevated gas flux? Bayes' theorem lets us study these "inverse" probabilites.
  • In this example, "RSAM is high, or RSAM is not high" is an observation. "Magma is ascending, or not" is a hidden variable. We do not know. We are mostly interested in how the probability of magma ascent changes by making observations about whether RSAM is high or low.
  • We have to know, or guess, about the probability that RSAM is high, given magma is ascending. This estimate might be based on past experience, or data on pre-eruptive events. Similarly, we need to estimate the probability that RSAM is not high, given that magma is not ascending. This could be a problem because there are potentially other sources of high RSAM, such as frequent rockfall or other slope instability not associated with magma ascent, or tectonic earthquakes.
  • Wait a minute! If we have to guess what the probability of magma ascent is, what have we gained? There is no free lunch with Bayes' theorem. We still need an estimate of the probability of magma ascent based on expert judgment and independent of our geophyscial observations. This initial guess is sometimes called the a priori belief in the transition probability of the node of an event tree. In this case we update the belief by including geophysical information.
  • We gain a sense of uncertainty in our event tree transition probability by calculating the the probability magma is not ascending given we observe RSAM is high, and the probability magma is ascending given that we observe RSAM is not high. This might be the most important part of application of Bayes' theorem in eruption forecasting and event trees!

Useful Links

Forecasting magma ascent with Bayes' theorem

How can the transition probability in an event tree be calculated from geophysical observations? The most common approach is Bayes' theorem. Bayes' theorem can be used to estimate the probability of an event (e.g., magma ascent) given an observation (e.g., high seismic energy release based on RSAM data). Consider a universe (actually a universal set) in which magma is ascending (\(A\)), or not (\(\neg A\)); RSAM is high (\(B\)), or not (\(\neg B\)).

Assume \(P[A]\) 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:

The formulas:

\(P[A]\) 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: $$P[\neg B] = 1 - P[B] = P[\neg B|A] P[A] + P[\neg B| \neg A] P[\neg A]$$

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: $$P[A| \neg B] = \frac{P[\neg B|A] P[A] }{P[\neg B]}$$

Some References

Aspinall et al., 2003 discuss the use of Bayes' theorem during eruption crisis, using a Galeras post mortem as an example.

Marzocchi et al., 2006 explain the use of Bayes' rule in event trees, further developed in Marzocchi et al., 2008.

Anderson and Poland (2016) use Bayes' theorem to constrain eruption scenarios at Kilauea.