**3.1 Example 1: Drake equation**

Since 1959, SETI has yet to find an alien signal. Two questions arise as a result what is the probability of there being life in the galaxy, and why have not we received a response to our transmissions? In 1959, a US astronomer, Frank D. Drake, a NASA employee who carried out the first SETI radio telescope experiments, outlined an equation for finding communicable civilizations [40, 41]:

$$N = RL \tag{1}$$

Which was later expanded to : *R f <sup>p</sup> ne fi ft f <sup>c</sup> L*.

where *N* is the number of communicable civilizations, *R* is the rate at which stars are born in the galaxy, *fp* is the fraction of stars with planetary systems, *ne* is the number of planets that might hold life, *ft* is the fraction of planets with life, *fi* is the fraction of planets with life that have evolved, *fc* is the number of civilizations of evolved civilizations with the ability to communicate, and *L* is the length of time over which the communication is possible. An additional factor named *C* is a recent suggestion for colonization [42]. Bloetscher [3] suggested that the factor *ne* actually comprises four factors: planet size (PS), presence of a moon (M), location within the "Goldilocks" or habitable zone (HZ), and the correct star type (ST), creating four unknowns from one. None of the 7–12 factors is fully known, so no specific answer on the likelihood of intelligent life on another planet communicating with Earth is possible. However, hierarchical Bayesian methods can be used to investigate the probability of intelligent life on another planet communicating with Earth. This approach involves the assignment of probability distributions to the underlying factors and using those to develop an MCMC protocol to determine the final predictive solution [3]. Subjective data, shown in **Table 1**, were included when little or no data were available to specify the parameters of these distributions [3]. Then probability distributions were assigned to the prior parameters within the initial distributions to determine the location and scale parameters of the factor distribution (**Table 2**). The subjective information serves to create these prior distributions until such time as real data are developed or become available [3].




**Table 1.**

*Summary of Drake parameters used in Bayesian calculations and comparison to prior estimates.*

*Applications of Hierarchical Bayesian Methods to Answer Multilayer Questions with Limited… DOI: http://dx.doi.org/10.5772/intechopen.104784*


#### **Table 2.**

*Monte Carlo results for parameters used in MCMC.*

For the Drake equations, a distribution for *N* was developed through using the Hierarchical Monte Carlo distributions for the factors of the equation run 10,000 times (see **Figure 1**). A series of Hierarchical Monte Carlo algorithms were developed for each parameter, and the means were inserted into a Monte Carlo Markov Chain program that uses a Metropolis-Hastings algorithm with a Gibbs sampler to develop a final probabilistic result [43–49] that was solved for *N*. Based on suggestions by Glade et al. [50] and Maccone [51], the target MCMC distribution was proposed to be lognormal. Given the uncertainly involved, the standard deviation used for the target distribution was assumed to be the square root of 6, after Wu et al. [52]. Given a multivariate distribution, like the example above, Gibbs sampling breaks down the problem by drawing samples for each parameter directly from that parameter's

**Figure 1.** *Development of HPB for Drake.*

#### **Figure 2.**

*Results for the probability of* N *using the Monte Carlo calculation of the factors placed into a MCMC predictive Bayesian MATLAB calculation. Note that the number of other planets is constantly small despite three options as noted above (reproduced from Bloetcher [3]).*

*conditional distribution* or the probability distribution of a parameter *given* a specific value of another parameter [53].

The solution is shown in **Figure 2** (red data points). Of importance, there is nearly a 50% probability that we are alone in the galaxy. The graph indicates that there is a 95% probability that there are less than 100 communicating civilizations concurrent with Earth, and a 99% that there are 1000 such civilizations.
