Foundations — Model retraining pipelines

Before you can read the parent note Model retraining pipelines, you must be fluent in a small pile of symbols and pictures. This page builds every one of them from nothing. Read top to bottom; each block only uses ideas already defined above it.

1. What is a "distribution" $P$? (the shape of a pile of data)

Everything in this topic rests on one word: distribution. If that word is fuzzy, nothing else lands.

Look at the figure. The horizontal axis is the thing we measured (age). The height of each bar is how big a slice of the crowd falls in that age band. The bars must add up to $1$ (i.e. $100\%$) — because every customer is somewhere. That "adds to 1" rule is what makes it a distribution and not just a bar chart.

2. The subscripts: $P_{\text{train}}$ vs $P_{\text{serve}}$, and $X$ vs $Y$

Now we decorate $P$ with labels so we can talk about which pile and which variable.

Why two different objects, $P(X)$ and $P(Y\mid X)$? Because the world can move in two independent ways, and the topic's central classification depends on telling them apart:

You cannot understand the parent's "two kinds of drift" box without these two symbols.

3. The model $f_{\theta }$ (the frozen photograph itself)

4. Sum notation $\sum$ (how to add up many pieces)

The drift score adds one number per bin. You must be able to read the "add-them-all" symbol.

Here $i$ is the bin number and $k$ is the number of bins — we chop a feature's range into $k$ buckets and add one contribution per bucket.

5. Fractions $a_i$ and $b_i$ (two piles, same bins)

In the figure the blue bars ($a_i$, training) and orange bars ($b_i$, production) sit side by side per bin. Drift is literally the gap between blue and orange. If they overlap perfectly, nothing moved.

6. The logarithm $\ln$ (why this tool for measuring "how many times bigger")

The drift formula uses $\ln (b_i/a_i)$. Why a logarithm and not just $b_i-a_i$?

Read the figure: $\ln$ crosses zero at $x=1$, is negative for shrinking ratios, positive for growing ones — the sign of $\ln (b_i/a_i)$ tells you which direction the slice moved.

7. The sign-matching trick (why every drift term is $\ge 0$)

Now every piece of the parent's PSI formula is defined: $\text{PSI}={\sum }_{i=1}^k(b_i-a_i)\ln \frac{b_i}{a_i}$

Here is the one clever observation that makes it a valid "distance-like" score:

8. The comparison symbols $>$, $⟺$, $\delta$ (the promote gate)

The parent's promotion rule is $\text{promote}⟺(\text{metric}(\text{challenger})-\text{metric}(\text{champion}))>\delta .$

How these foundations feed the topic

Every arrow says "you need this to understand that." Trace any path top to bottom and you have re-derived the parent note's spine.

Related vault prerequisites

• Data drift and concept drift — the two ways $P$ can move.
• Model monitoring and observability — how $b_i$ gets measured live.
• Feature stores — where the columns $X$ actually live.
• Data versioning and lineage — pinning which pile trained which $\theta$.
• KL divergence — the ancestor of the PSI formula.

Equipment checklist

Cover the right side; you are ready for the parent note only if you can say each answer out loud.