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Statistical Analysis6 min read

Introduction to Bayesian Statistics: An Intuitive Understanding of Prior and Posterior

This article provides a highly practical approach to decision-making when faced with insufficient evidence, such as when a CPK of 1.08 causes uncertainty about whether a machine or measurement instrument is at fault. It explains how past experiences or 'gut feelings' represent statistical prior probabilities, enabling a more scientific method to identify root causes rather than relying on guesswork.

The day the CPK report came out, the whole room fell silent for three seconds

Remember when the CPK report for the new process batch came out before the last shipment? The PM's face turned green because the CPK was only 1.08. Although it 'barely' passed, it was a long way from our target of 1.33. The boss's face immediately fell, asking, "Is it the machine that has a problem, or has the measurement instrument drifted?" You know, in situations like this, everyone tends to blame instrument drift first, and only then start to suspect the machine. But the question is, how do you know which 'possibility' is greater?

Where is the problem?

To put it plainly, this situation means you don't have enough 'concrete evidence' when making a judgment. You might have a hunch, feeling that a certain machine has often acted up before, or that the instrument felt strange after the last calibration. These 'feelings' or 'past experiences' are actually what statistics calls 'Prior Probability.' It represents your 'belief' or 'guess' about something happening before you see any new data.

For example, if our past experience shows that the probability of a machine malfunctioning is 70%, and the probability of instrument drift is 30%. This is your prior. But now, you have new CPK data of 1.08. After this new data comes out, will you adjust your view on 'machine malfunction' or 'instrument drift'? Yes, you will! This adjusted view is the 'Posterior Probability.'

So the key point is that Bayesian statistics teaches you how to use 'new evidence' (such as CPK 1.08) to update your 'old beliefs' (your view on the machine or instrument), making your judgments more accurate. It's not about overturning old experiences, but about using new information to revise them.

How to actually do it?

Let's return to the CPK 1.08 example. Suppose, based on past records, you have the following prior information:

1. The probability that the machine truly has a problem is 70%.
2. The probability that the measurement instrument has drifted is 30%.

Next, you also need to know the likelihood of encountering 'CPK 1.08' data under two scenarios:

1. If the machine truly has a problem, the probability of CPK 1.08 appearing might be 60%. (Because problems usually lead to worse outcomes, but sometimes they just barely fall on the edge.)
2. If the measurement instrument has drifted, the probability of CPK 1.08 appearing might be 80%. (Instrument drift usually causes data to worsen, and often drifts just to the edge.)

With this, you can use Bayes' Theorem to calculate the 'Posterior Probability.' It will help you calculate:

  • The probability that 'the machine truly has a problem' given 'seeing CPK 1.08'.
  • The probability that 'the measurement instrument has drifted' given 'seeing CPK 1.08'.

Simply put, it's about multiplying the 'prior probability' by the 'probability of this event occurring under various circumstances,' and then normalizing. This way, you can find out which scenario's probability has increased after seeing CPK 1.08. You will find that after calculation, the posterior probability of instrument drift will be significantly higher than that of a machine problem. This means you should prioritize checking the instrument!

The Most Common Pitfall

Let me tell you, the most common pitfall is 'preconceived notions.' Many people receive data and have already decided who is to blame. For example, if the yield of a certain shift suddenly drops, with DPMO changing from 6210 to 9800, the immediate gut feeling is that the new operator (OP) messed up. This is acting purely on 'feeling,' without evaluating the 'prior probability' and 'the probability of various possibilities occurring when seeing this DPMO value.'

What was the result? A lot of time was spent scrutinizing the new hire, only to eventually discover that an engineer from the previous shift had casually adjusted a process parameter during material change, and failed to record it! If we had calmed down then and used Bayesian thinking to evaluate the prior probabilities of 'new hire error' and 'parameter change,' as well as the respective probabilities of these two events occurring given a DPMO of 9800, we might have identified the parameter issue much sooner.

Therefore, don't let your 'intuition' completely dominate your judgment; it's important, but it also needs data for correction.

One Thing You Can Do Today

Next time you encounter a problem, first consider what your 'first impression' of the cause is? That is your 'prior'.

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