Let the score vector be \(g=(1,0)\) and start at \((1/2,1/2)\). The likelihood at time \(t\) is \((e^t,1)\), so the first posterior coordinate is
The odds \(p_0/p_1\) grow exponentially, while the probabilities remain inside the simplex. At \(t=\log 2\) the posterior is exactly \((2/3,1/3)\).
Reading the drawing
The solid curve rises from one half while the dashed curve falls from one half. The expressions on the right give the probabilities and their differential equation. The note distinguishes conservation of total mass from preservation of mixtures.
Drawing labels and data 14 labels
Read the original labels as selectable text. Tables preserve grid relationships; other short labels are grouped compactly.
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- time \(t\)
- probability
- \(p_0(t)=\dfrac{e^t}{1+e^t}\)
- \(p_1(t)=\dfrac{1}{1+e^t}\)
- \(\dot p_0=p_0(1-p_0)\)
Mass stays one.
Mixtures are not preserved.
Evidence note from the source edition
The solid and dashed curves sum to one. The curvature is caused by the shared evidence denominator, not by a nonlinear likelihood multiplication law.
Two properties that must not be conflated
If \(a>0\) multiplies the odds, the first-coordinate map is
Direct algebra gives \(\phi_a(\phi_b(x))=\phi_{ab}(x)\). Thus \(a=e^t\) gives a nonlinear semigroup. Nevertheless, \(\phi_3(1/2)=3/4\), while the mean of \(\phi_3(0)\) and \(\phi_3(1)\) is \(1/2\). Superposition fails.
Differentiation of the normalized flow yields the centered score equation
For the displayed model, \(\dot p_0=p_0(1-p_0)\). The negative average-score term is what keeps the total derivative zero. Omitting the derivative of the normalizer would not preserve mass.
Likelihood composition also preserves the evidence identity: the first evidence times the second conditional evidence equals the combined evidence. Normalization cancels in the posterior, but that factor still matters when comparing data models.
Read this as
A semigroup is a composition law in time. It does not imply that the maps are linear or that they preserve mixtures.