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Startup Hypothesis Testing [Bayesian Inference
Posted on March 2, 2017 @ 09:44:00 AM by Paul Meagher

In my last lean startup blog on measurement, I talked about using a Minimal Viable Product (MVP) to test hypothesis derived from leap of faith assumptions contained in the startup vision.

In the case of Joe's lemonade stand (see previous blog), the first leap of faith was that the customer would buy the lemonade. Customers purchased the lemonade but not in the amounts he was looking for (10 customers). Joe then tested the price people would pay for it starting off at a premium price of $1.50 a glass. He measured sales volume at that price and another price ($1.00 a cup) and concluded that it was better to sell at the lower price because the volume more than compensated for the lower price. Joe is making progress towards operating a successful lemonade stand.

In this blog I want to look at the process of hypothesis testing in more detail and see how it maps onto some of the terms we have been using.

Let H be a class of hypothesis and h be a specific hypothesis.

Let M be a class of measurement outcomes and m be a specific measurement outcome.

We can use Modus Ponens (latin for "the affirming mode") to draw conclusions about whether our hypothesis is true:

If H=h Then M=m
M=m
-------------
H=h

We can also use Modus Tollens (latin for "the denying mode") to draw conclusions about whether our hypothesis is true:

If H=h Then M=m 
Not M=m 
--------------
Not H=h

So far we are in the realm of formal logic and these two forms of inference are foundational in guiding automated forms of inference.

We can cross over into the realm of informal logic by using the P( ) operator around all our assertions, where P stands for "the probability of".

So Modus Ponens now looks like this:

P(If H=h Then M=m)
P(M=m)
------------------
P(H=h)

And Modus Tollens now looks like this:

P(If H=h Then M=m)
P(Not M=m)
------------------
P(Not H=h)

It is this form of Modus Ponens and Modus Tollens that we are dealing with when we test our startup assumptions. The application of scientific methods to startup hypothesis does not necessarily yield clear cut answers, but answers where one hypothesis might seem be better supported by the evidence than another hypothesis, without being able to completely rule out an alternative hypothesis.

In the case of the learning platform company Grockit (see previous blog), they were adding new peer-learning features to their learning platform and not seeing any effects on their metrics. They concluded that the learner only wanted peer-learning up to a point, then the learner wanted to engage in solo mode learning. A logical possibility was also that Grockit didn't zone in on the proper peer-learning approach yet. The alternative hypothesis is not completely ruled out by testing and measurement, but made sufficiently implausible that a pivot was deemed necessary.

We will be getting into the topic of pivoting in the next blog, but it is important to note here that deciding when to pivot or not is made difficult by the fact that the original and alternative hypothesis may each have merit making it difficult to decide what to do.

Recognizing that probabilities are involved can be helpful in deciding what decision making framework you want to use in your startup hypothesis testing. If P(H=h) is .6 perhaps that is enough certainty to go by in situations of irreducible uncertainty (you don't have the time or resources to achieve greater certainty).

You could examine formula-laden articles on sequential A/B testing and bayesian A/B testing to try to figure out when to stop collecting data and what to conclude (which I recommend reading), but I'm also interested in a more practical approach based on using informal logic to evaluate the probability of the premises P and the probability of the inference (i.e., P(if P Then C)) to arrive at a probability of the conclusion C of an argument.

P(If P Then C)
P(P)
---------------
P(C)

The evaluation of the premises and the inferences is based upon informal logic techniques appropriate to criticizing scientific arguments, combined with common sense, to assign probabilities to each premise. The evaluation of the premises and the inferences of the argument determines the evaluation you assign to the conclusion. Bayesian forms of informal logic may also involve assigning a prior probability to the conclusion so that the posterior probability of the conclusion can be evaluated.

P(If P Then C)
P(P)
P(C)
---------------
P(C)

Whether these probabilities are to be combined additively or multiplicatively to yield the posterior conclusion is worth thinking about, although multiplicative combination tends to used more often and to work better. Informal logic nowadays often involves creating a graphical representation of the argument. Below is how we might graphically express this Bayesian approach to evaluating arguments (where hypothesis testing is just one type of argument). The premises (e.g., the measurements and other assumptions) appear at the top with lines connecting them to the conclusion. The lines are your inferences (if P1 then C, if P2 then C). The prior probability of the conclusion C (based on previous knowledge) appears next to the premises as a separate contribution to the posterior conclusion probability C. The posterior probability of the conclusion at the bottom is what you get when you combine your prior probability of C and a likelihood estimate (the left side of the argument below).

The purpose of this blog was to dig a bit deeper into what startup hypothesis testing might involve from an formal and informal logic perspective. I am not a practicing logician and this is not a peer reviewed discussion so you may or may not find this a useful framework to use when approaching the problem of testing the leaps of faith that your startup vision implies.

Inspiration for this blog and the argument evaluation diagramming comes form my undergraduate mentor Wayne Grennan and his book Informal Logic (1997).

Ian Flemming in his excellent book Lean Logic (2016) has this to say about the relationship between informal and formal logic.

It sounds banal, but the syllogisms of formal logic are the building blocks of reasoning, which - in combination with a series of conditions, affirmed or denied in sequence and in parallel - can develop into a problem-solving capacity of great complexity, used as the logical structure on which artificial intelligence is based.

Informal logic is, of course, the junior partner in all this, since it depends on the reasoning of formal logic, and its mixing up of logic and content is exactly what you cannot do with formal logic. On the other hand, without content, logic has no purpose. Formal logic is the road, informal logic is the journey. ~ p. 165

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