The author of this passage is proposing a very simple *hidden variable model*. This model has two variables:
* $A$ — whether the patient has certain unknown genes (this variable is hidden)
* $B$ — whether the treatment succeeds (this variable is observed)
And the model proposes that $A$ influences $B$. Mathematically, this model predicts that $P(B) = P(B|A)P(A)$. The author observes that an observed value of $P(B)$ (42%) is insufficient to infer either $P(B|A)$ or $P(A)$ by themselves, assuming this model.
The two examples the author gives correspond to assuming that $P(A)$ is 1 (everyone's genes are good enough for the treatment—in which case $P(B|A) = P(B) = 0.42$) and that $P(B|A)$ is 1 (the treatment is always effective if the right genes are present—in which case $P(A) = P(B) = 0.42$). If your hidden variable model contains probabilities that identically equal 1, you have an opportunity to eliminate variables from the model (both of these examples represent opportunities to reduce the number of variables from 2 to 1). This process might be called dimensionality reduction, though that usually refers to lossy simplifications of models from more to fewer variables, and eliminating identical variables is lossless.
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2023-05-13T01:07:29Z (12 months ago)