Paper VI visual guide · Visual 10 of 42 · SUFFICIENCY / A COMPUTABLE FAILURE WITNESS

Two priors that fool the coarse model

Traditional Part I: Theorem 8.2 and the one-packet exception.

Keep the same likelihood \(\lambda=(4,1,1,1)^{\mathsf T}\) and partition A\(=\{0,1\}\), B\(=\{2,3\}\). Compare

\[ p^{(a)}=(1/2,0,1/2,0)^{\mathsf T},\qquad p^{(b)}=(0,1/2,1/2,0)^{\mathsf T}. \]

Both have coarse prior \((1/2,1/2)^{\mathsf T}\). A procedure given only that two-entry vector cannot distinguish them.

Two priors with the same coarse masses produce different coarse posteriors. — Original drawing. Read the description below; expand the labels and data for selectable text.

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The left panel gives their common two-packet prior. The middle and right panels give the two updated packet distributions. The note above attributes the difference to redistribution hidden inside the first packet.

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Both coarse priors
A

\(\frac{1}{2}\)

From \(p^{(a)}\)
A

\(\frac{4}{5}\)

\(\frac{1}{5}\)

From \(p^{(b)}\)
A

\(\frac{1}{2}\)

Hidden redistribution changes the packet-A posterior from \(1/2\) to \(4/5\).

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Evidence note from the source edition

After the identical fine likelihood is applied, the coarse posteriors are \((4/5,1/5)\) and \((1/2,1/2)\). A single deterministic update of the common coarse prior cannot produce both.

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Why this witnesses the necessity of blockwise constancy

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The A evidence for \(p^{(a)}\) is \(2\), whereas it is \(1/2\) for \(p^{(b)}\). Their B evidence is \(1/2\) in both cases. The posterior odds therefore distinguish the hidden distribution inside A.

For strictly positive fine and coarse likelihoods, and at least two coarse blocks, the manuscript proves the all-prior criterion

\[ C\mathcal B_\lambda(p)=\mathcal B_{\bar\lambda}(Cp)\quad\hbox{for every }p \quad\Longleftrightarrow\quad \lambda_i=c\bar\lambda_a\quad(i\in I_a) \]

for one global \(c>0\). The global scale is invisible after normalization. A different arbitrary scale for each packet is not invisible: it changes posterior odds.

If all states are merged into a single packet, every normalized posterior coarsens to \((1)\). That degenerate commutation holds for every likelihood and cannot identify any within-packet information. It is explicitly excluded from the nontrivial criterion.

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Read this as

A prior-dependent effective likelihood can always summarize the update. A fixed likelihood that works for every prior is a much stronger requirement.

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