This topic is close to my heart, as I named my company Bayes Actuarial Solutions after teaching Bayesian credibility at a university.
Bayesian statistics is named after Reverend Thomas Bayes who is credited with founding this approach many centuries ago, and it was rediscovered by Pierre Simon Laplace.
The Philosophy
Statistics can be split into two schools - frequentist and Bayesian. Most of the statistics that we learn in our education pathway is frequentist, even in actuarial education. We touch upon Bayesian statistics a little in Credibility Theory.
The frequentist approach assumes there is a true population result, and we try to estimate the true result by gathering data. We gather data by sampling, and study the samples to infer information about the population.
In Bayesian statistics the data is more relevant and important than the population. That is, the population information is changing and variable, the data provides the most relevant information for us. So we update our expectations based on every data point gathered.
An overview
Here's a high level overview of the approach.
We start with an initial understanding of the population, called the prior. The prior can be a value or a probability distribution, and is set using judgment, pre-existing knowledge, information from other similar risks or regions, or other relevant sources.
We then gather data and calculate the posterior distribution, by using the Bayes Theorem. In lay terms, the posterior is our updated understanding of the risk given the data observed.
As we gather more data, the posterior keeps getting updated too.
This posterior distribution is then used to assess the risk, make predictions, etc.
Why is Bayesian statistics important for Actuaries?
First, judgment and business acumen is a cornerstone of our skill set. The Bayesian approach requires our acumen as the starting point, to set the prior.
This results in a far more efficient process, especially when data is scarce in the beginning.
Secondly, as Actuaries we are far more interested in analysing the risk rather than making predictions. The posterior distribution allows us to analyse the volatility/spread of risk, tail risk (a fancy term for worst case outcomes) and the impact of mitigations.
Thirdly, when we reflect, the Bayesian thinking is more efficient in VUCA (volatile, uncertain, complex, ambiguous) settings. Due to the inherent complexity and uncertainty, the underlying population would be too complex and changing too quickly for us to try and estimate it.
Rather, the data that we gather reflects the customers we attract, the risk that is most relevant to our business and the conditions that are most recent.
Hence the Bayesian approach of giving more importance to the data rather than the underlying population is more sensible in this light.
Hurdles in the use of Bayesian statistics
Calculating the posterior distribution unfortunately is very complex in practice.
The approach requires doing multiple integrations, which gets very complex when we have many parameters (which is common in practice), and when the distribution is not straightforward (also common in practice).
Fortunately we have ways of overcoming this with the Markov Chain Monte Carlo (MCMC) technique, coupled with a suitable sampling technique. More on that another time.
Applications
Typically in Actuarial studies, Bayesian statistics is applied in credibility theory, where the posterior distribution is used to set the actuarial premium and we can assess the risk by analysing the tail of the posterior distribution.
There are many other applications of the Bayesian approach, in business and commerce for price optimisation and decision making, in marketing to decide on optimal approaches, in trading to model stock price movements, in weather prediction, in disease prediction and much more.
What does that mean for us Actuaries? This Bayesian approach allows us to venture far beyond our traditional domains.
That is the vision of Bayes Actuarial Solutions - to take actuarial approaches, modernise them, embrace latest advances and apply them in traditional actuarial domains and venture beyond to new domains.
Let us see where this journey takes us to!
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