Should I use Bayes' formula to generate simple data about business competition and the market place?

Question

I run a small business, and I'm keen on understanding my market a little more. Market surveying and similar tools are not available to me, mainly due to time and budget. I don't require rock-solid data, just something that allows me to add a "pinch of salt" to my reasoning once in a while.

Is it appropriate to use Bayes' theorem to determine belief that my customers will increase as the market (population) increases, even as my competition increases? This assumes homogenous geographic distribution of customer populations and competition, which is inaccurate, and assumes that all customers are drawn equally to all businesses, which is again (probably) inaccurate. But at least it gives me a starting point beyond my ill-informed hunches.

I am not a statistician, which no doubt will be very evident, and have encountered several questions while trying to make sense of this approach. Apologies in advance for the number of questions here.

In my limited understanding, Bayes' theorem might help me modulate my beliefs regarding my competition, the market place, and so on.

I have a number of questions:

1. is this an appropriate use of Bayes' theorem?

   Step 1: Constructing some formulae

   Let's say:

   $ P(C) = \text{probability customer numbers will increase} \\$ $ P(M) = \text{probability market population will increase} \\$ $ P(K) = \text{probability competition will increase} $

   ...then I presume I should be able to construct something like the following:

   $ P(C|M) = \frac{P(M|C)P(C)}{P(M)} $ and $ P(C|K) = \frac{P(K|C)P(C)}{P(K)} $

2. Am I doing it right? :)

   Step 2: Using appropriate data

   The numbers I have are as follows:

   • Professionals in my field, in the UK: approx 4,500.
   • Growth in my field: approx 1,000 per year.
   • Shrinkage in my field (due to retirement etc): approx 500 per year.
   • Number of people in my city, and the rate at which it's growing.
   • Percentage of a typical population that uses my profession's services: 10%.
   • Month-on-month customer data in my business.

   I can probably get more precise figures regarding the number of professionals in my field.

   Regarding customer data, I have 18 months of past figures in a spreadsheet. In 9 of 17 completed months, customer numbers are higher than the previous month. This suggests to me that:

   $ P(C) = \frac{9}{17} = 0.53 $

   Regarding the market (population), let's say local government projects growth from 1,000,000 to 1,100,000 in the next year. This suggests to me that:

   $ P(M) = \frac{1100000}{1000000} = 1.1 $

   Regarding the competition, I'd assume:

   $ P(K) = \frac{4500 + 1000 - 500}{4500} = 1.\dot{3} $

3. The figures may be accurate, but am I using them appropriately?

   Step 3: The results

   $ P(C|M) = \frac{P(M|C) \cdot 0.53}{1.1} $

   This is where I run out of steam. How do I determine $P(M|C)$, the probability of the market increasing given that my customers also increase? Since my business has negligible effect on market size should I simply consider that to be 100%, i.e. 1? My business will also have minimal effect on competition (or so I believe, although maybe new local businesses are deterred by its presence) so I'd assume that similarly $ P(K|C) = 1 $.

4. Is that a sensible approach? Somehow I suspect not.

   If that approach is right, the answers become:

   $ P(C|M) = \frac{1 \cdot 0.53}{1.1} = \frac{0.53}{1.1} = 0.4\dot{8}\dot{1} $

   $ P(C|K) = \frac{1 \cdot 1.\dot{3}}{1.1} = \frac{1.\dot{3}}{1.1} = 1.\dot{2}\dot{1} $

5. How should I then combine these two beliefs that customer numbers will go up as the market increases but competition also increases? Since they're probabilities, can I simply multiply them? That would suggest:

   $ 0.4\dot{8}\dot{1} \cdot 1.\dot{2}\dot{1} = 0.584 $

Many thanks for any help you can offer, and apologies again for asking a lot of questions. I'm perfectly happy to be pointed toward useful reading materials, although my statistics and mathematics abilities are (clearly) limited, so novice-friendly info would be appreciated.

Answer 1

The general idea is an interesting--and good--one, but your actual execution has gone off the rails in a few places.

• Step 1 is, indeed, an accurate version of Bayes' Law.
• Step 2, however, has some problems. Probabilities must be between zero and one, which 1.1 and 1.3 are, umm...not. It looks like the numbers you calculated are something like growth rates, not probabilities. Think about $P(A)$ as indicating the probability that an event A occurs. It cannot occur less than never (i.e., $P(A)=0$), and it can't occur more than always (i.e., $P(A)=1$. Unfortunately, you can't recover the probability of the market growing from the two pieces of information you have. You would need to know something like the range of projected values for next year's population/retirement/etc.

• Step 3 inherits some problems from step 2. However, it seems like you could actually estimate $P(C|M)$ directly from some data--do your customer numbers go up when the population increases? You're right that if $A$ and $B$ are independent, then $P(A|B) = P(A)$, but if that were true, this whole exercise would be somewhat pointless.

• Step 4 correctly applies the formula from step 1, though the numbers are off.

• Step 5 is reasonable, assuming that you think $M$ and $K$ are independent.

However, It's not clear to me that knowing whether the market/competition/customer base increases or decreases is particularly valuable, without knowing amount of change. One more customer is unlikely to materially change your life, but a 10-fold increase would probably be fantastic. In that light, I'd suggest thinking about this as a regression problem instead, which might let you predict actual values for # of customers/market size/etc. You could potentially write down a model like: $$ C(t) = \beta_0 + \beta_1 M(t) + \beta_2 K(t) + \beta_3 M(t) K(t) +\textrm{<other potentially relevant factors>}$$ and then fit it with your historical data. I think that would give you a much better picture of your business environment and would let you explore various scenarios.