Paper VI: Earlier exact examples

R6 Earlier exact examples

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These five short examples are retained from the earlier human edition. Their numerical parameters are independent of the later visual guide's worked models.

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Example R6.1: One transfer between two packets

Let

\[ A= \begin{pmatrix} a&0\\ c&b \end{pmatrix} \]

and let the two coordinate axes be the packet spaces. If \(a\neq b\), then

\[ {\mathrm e}^{tA} = \begin{pmatrix} {\mathrm e}^{at}&0\\ \displaystyle c\frac{{\mathrm e}^{at}-{\mathrm e}^{bt}}{a-b}&{\mathrm e}^{bt} \end{pmatrix}. \]

The lower left block is produced by one transfer from the first packet to the second. Its expansion begins with \(ct\), in agreement with Corollary R1.3. When \(a=b\), the same block is \(ct{\mathrm e}^{at}\).

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Example R6.2: A four state lumpable chain

Partition four states into \(B_1=\{1,2\}\) and \(B_2=\{3,4\}\). For positive rates \(a,b,u,v\), set

\[ Q= \begin{pmatrix} -u-a&u&b/2&b/2\\ u&-u-a&b/2&b/2\\ a/2&a/2&-v-b&v\\ a/2&a/2&v&-v-b \end{pmatrix}. \]

The rates \(u\) and \(v\) move inside the coarse blocks. Every state in \(B_1\) leaves for \(B_2\) at total rate \(a\), and every state in \(B_2\) leaves for \(B_1\) at total rate \(b\). Hence

\[ \overline Q= \begin{pmatrix} -a&b\\ a&-b \end{pmatrix} \]

and \(CQ=\overline QC\). The internal rates disappear from the coarse generator even though they remain present in the fine evolution.

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Example R6.3: A visible likelihood loss

Let \(B_1=\{1,2\}\), let \(B_2=\{3\}\), and take

\[ p=(3/10,2/10,5/10), \qquad L=(1,3,2). \]

The coarse prior is \((1/2,1/2)\). The exact block likelihoods are

\[ \ell_1(p)=\frac{(3/10)+3(2/10)}{1/2}=\frac95, \qquad \ell_2(p)=2. \]

Thus the exact coarse posterior is \((9/19,10/19)\). If the first block is replaced by the fixed representative value \(2\), the approximate coarse posterior is \((1/2,1/2)\). The \(\ell^1\) defect is \(1/19\), while Theorem R3.3 gives the valid upper bound \(1/10\).

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Example R6.4: One hidden mode

Consider

\[ A= \begin{pmatrix} -a&c\\ d&-b \end{pmatrix}, \qquad a,b>0, \]

and retain the first coordinate. The memory kernel and self energy are

\[ {\mathcal K}(t)=cd{\mathrm e}^{-bt}, \qquad \Sigma(z)=\frac{cd}{z+b}. \]

The retained coordinate has an exact time local closure for every retained initial value precisely when \(cd=0\). When \(cd\neq0\), one auxiliary scalar state realizes the memory exactly.

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Example R6.5: Two sequential phases

Let

\[ A= \begin{pmatrix} -\lambda&0\\ \lambda&-\mu \end{pmatrix}, \qquad p_0= \begin{pmatrix}1\\0\end{pmatrix}, \]

where \(\lambda\neq\mu\) are positive. The first phase moves to the second at rate \(\lambda\), and the second is absorbed at rate \(\mu\). Direct calculation gives

\[ S(t)= \frac{\mu{\mathrm e}^{-\lambda t}-\lambda{\mathrm e}^{-\mu t}}{\mu-\lambda}, \]

\[ f(t)= \frac{\lambda\mu}{\mu-\lambda} \left({\mathrm e}^{-\lambda t}-{\mathrm e}^{-\mu t}\right). \]

The ratio \(f(t)/S(t)\) is generally not constant. A time local Markov model on two transient phases can therefore produce a nonconstant hazard after the phase coordinate is hidden.

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