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:
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.
Bayes' Theorem, developed by the Rev. Thomas Bayes, an 18th century mathematician and theologian, was first published in 1763.[4] [1] Mathematically it is expressed as:
(eq.6)
where we can update our belief in hypothesis H given the additional evidence E and the background context c. The left-hand term, P(H|E,c) is known as the "posterior probability," or the probability of H after considering the effect of E on c. The term P(H|c) is called the "prior probability of H given c alone. The term P(E|H,c) is called the "likelihood" and gives the probability of the evidence assuming the hypothesis H and the background information c is true. Finally, the last term P(E|c) is independent of H and can be regarded as a normalizing or scaling factor.
In the case of our investor, P(H|c) is already known to be 0.40 so the Bayesian equation resolves to:
(eq. 7)
It is important to note that all of these probabilities are conditional. They specify the degree of belief in some proposition or propositions based on the assumption that some other propositions are true. As such, the theory has no meaning without prior resolution of the probability of these antecedent propositions.
Let us return the example of the investor. From theory of binomial distributions, if the probability of some event occurring on any one trial is p, then the probability of x such events occurring out of n trials is expressed as:
For example, the likelihood that 5 out of 20 people will support her enterprise should her location actually fall into the category where 20% of franchises actually achieve 25% saturation is:
The likelihood of the other situations can also be determined:
Event (Market Saturation) p_{i} |
Prior Probability P_{0}(p_{i}) |
Likelihood of Situation P(x=5|p_{i}) |
Joint Probability of Situation |
Posterior Probability |
---|---|---|---|---|
0.10 | 0.05 | 0.03192 | 0.001596 | 0.00959 |
0.15 | 0.05 | 0.10285 | 0.005142 | 0.00309 |
0.20 | 0.20 | 0.17456 | 0.034912 | 0.20983 |
0.25 | 0.20 | 0.20233 | 0.040466 | 0.24321 |
0.30 | 0.40 | 0.17886 | 0.071544 | 0.43000 |
0.35 | 0.10 | 0.12720 | 0.012720 | 0.07645 |
Totals | 1.00 | 0.81772 | 0.166381= P(x=5) |
0.99997 |
The sum of all the Joint Probabilities provides the scaling factor found in the denominator of Bayes Theorem and is ultimately related to the size of the sample. Had the sample been greater than 20, the relative weighting between prior knowledge and current evidence would be weighted more heavily in favour of the latter. The Posterior Probability column of Table 4 shows the results of the Bayesian theorem for this case. By adding up the relative posterior probabilities for market shares >=25% and those <25%, our investor will see that there is a 75% probability that her franchise will make money--definitely a more attractive situation on which to base an investment decision.
Links
[1] https://niedermayer.ca/node/31#fn3
[2] https://niedermayer.ca/user/login?destination=node/33%23comment-form
[3] https://niedermayer.ca/node/31#fn4