2. An Introduction to Bayesian Inference

Classical inferential models do not permit the introduction of prior knowledge into the calculations. For the rigours of the scientific method, this is an appropriate response to prevent the introduction of extraneous data that might skew the experimental results. However, there are times when the use of prior knowledge would be a useful contribution to the evaluation process.

Assume a situation where an investor is considering purchasing some sort of exclusive franchise for a given geographic territory. Her business plan suggests that she must achieve 25% of market saturation for the enterprise to be profitable. Using some of her investment funds, she hires a polling company to conduct a randomized survey. The results conclude that from a random sample of 20 consumers, 25% of the population would indeed be prepared to purchase her services. Is this sufficient evidence to proceed with the investment?

If this is all the investor has to go on, she could find herself on her break-even point and could just as easily turn a loss instead of a profit. She may not have enough confidence in this survey or her plan to proceed.

Fortunately, the franchising company has a wealth of experience in exploiting new markets. Their results show that in 20% of cases, new franchises only achieve a 25% market saturation, while in 40% of cases, new franchises achieve a 30% market saturation. The entire table of their findings appears below:

Table 2: Percent of New Franchises achieving a given Market Saturation
Percent of New Franchises achieving a given Market Saturation" Percent of Franchises (Relative Frequency)
0.10 0.05
0.15 0.05
0.20 0.20
0.25 0.20
0.30 0.40
0.35 0.10
  Total = 1.00

Our investor's question is simple: "What is the probability that my population will achieve a market saturation of greater than 25% given the poll conducted and the results found in other places?" In effect, she needs to determine the probability that her population will one of the 70% of cases where market saturation is greater than or equal to 25%. She now has the information she needs to make a Bayesian inference of her situation.