Fixed Dimension Reassessment The log-likelihood function is
p ( y ∣ x ) = ∑ i = 1 n log { x ( y i ) } − ∫ 0 L x ( t ) d t = ∑ j = 1 k m j log h j − ∑ j = 0 k h j ( s j + 1 − s j ) \begin{aligned}
p(y\mid x) &= \sum_{i=1}^n\log\{x(y_i)\}-\int_0^Lx(t)dt\\
&= \sum_{j=1}^km_j\log h_j-\sum_{j=0}^kh_j(s_{j+1}-s_j)
\end{aligned} p ( y ∣ x ) = i = 1 ∑ n log { x ( y i ) } − ∫ 0 L x ( t ) d t = j = 1 ∑ k m j log h j − j = 0 ∑ k h j ( s j + 1 − s j ) where m j = # { y i ∈ [ s j , s j + 1 ) } m_j=\#\{y_i\in[s_j,s_{j+1})\} m j = # { y i ∈ [ s j , s j + 1 ) } .
choose one of h 0 , h 1 , … , h k h_0,h_1,\ldots,h_k h 0 , h 1 , … , h k at random, obtaining h j h_j h j
propose a change to h j ′ h_j' h j ′ such that log ( h j ′ / h j ) ∼ U [ − 1 2 , 1 2 ] \log(h_j'/h_j)\sim U[-\frac 12, \frac 12] log ( h j ′ / h j ) ∼ U [ − 2 1 , 2 1 ] .
the acceptance probability is
α ( h j , h j ′ ) = p ( h j ′ ∣ y ) p ( h j ∣ y ) × J ( h j ∣ h j ′ ) J ( h j ′ ∣ h j ) = p ( y ∣ h j ′ ) p ( y ∣ h j ) × π ( h j ′ ) π ( h j ) × J ( h j ∣ h j ′ ) J ( h j ′ ∣ h j ) . \begin{aligned}
\alpha(h_j, h_j') &=\frac{p(h_j'\mid y)}{p(h_j\mid y)}\times \frac{J(h_j\mid h_j')}{J(h_j'\mid h_j)}\\
&= \frac{p(y\mid h_j')}{p(y\mid h_j)}\times\frac{\pi(h_j')}{\pi(h_j)}\times\frac{J(h_j\mid h_j')}{J(h_j'\mid h_j)}\,.
\end{aligned} α ( h j , h j ′ ) = p ( h j ∣ y ) p ( h j ′ ∣ y ) × J ( h j ′ ∣ h j ) J ( h j ∣ h j ′ ) = p ( y ∣ h j ) p ( y ∣ h j ′ ) × π ( h j ) π ( h j ′ ) × J ( h j ′ ∣ h j ) J ( h j ∣ h j ′ ) . Note that
log h j ′ h j = u , \log\frac{h_j'}{h_j}= u \,, log h j h j ′ = u , it follows that the CDF of h j ′ h_j' h j ′ is
F ( x ) = P ( h j ′ ≤ x ) = P ( e u h ≤ x ) = P ( u ≤ log ( x / h ) ) = log ( x / h ) + 1 / 2 F(x) = P(h_j'\le x) = P(e^uh\le x) = P(u\le \log(x/h)) = \log(x/h) + 1/2 F ( x ) = P ( h j ′ ≤ x ) = P ( e u h ≤ x ) = P ( u ≤ log ( x / h ) ) = log ( x / h ) + 1 / 2 and
f ( x ) = F ′ ( x ) = 1 x , f(x) = F'(x) = \frac{1}{x}\,, f ( x ) = F ′ ( x ) = x 1 , so
J ( h j ′ ∣ h j ) = 1 h j ′ . J(h_j'\mid h_j) = \frac{1}{h_j'}\,. J ( h j ′ ∣ h j ) = h j ′ 1 . Then we have
α ( h j , h j ′ ) = p ( y ∣ h j ′ ) p ( y ∣ h j ) × ( h j ′ ) α exp ( − β h j ′ ) h j α exp ( − β h j ) = likelihood ratio × ( h j ′ / h j ) α exp { − β ( h j ′ − h j ) } \begin{aligned}
\alpha(h_j,h_j') &= \frac{p(y\mid h_j')}{p(y\mid h_j)}\times \frac{(h_j')^{\alpha}\exp(-\beta h_j')}{h_j^\alpha\exp(-\beta h_j)}\\
&=\text{likelihood ratio}\times (h_j'/h_j)^\alpha\exp\{-\beta(h_j'-h_j)\}
\end{aligned} α ( h j , h j ′ ) = p ( y ∣ h j ) p ( y ∣ h j ′ ) × h j α exp ( − β h j ) ( h j ′ ) α exp ( − β h j ′ ) = likelihood ratio × ( h j ′ / h j ) α exp { − β ( h j ′ − h j ) } Draw one of s 1 , s 2 , … , s k s_1,s_2,\ldots,s_k s 1 , s 2 , … , s k at random, obtaining say s j s_j s j .
Propose s j ′ ∼ U [ s j − 1 , s j + 1 ] s_j'\sim U[s_{j-1}, s_{j+1}] s j ′ ∼ U [ s j − 1 , s j + 1 ]
The acceptance probability is
α ( s j , s j ′ ) = p ( y ∣ s j ′ ) p ( y ∣ s j ) × π ( s j ′ ) π ( s j ) × J ( s j ∣ s j ′ ) J ( s j ′ ∣ s j ) = likelihood ratio × ( s j + 1 − s j ′ ) ( s j ′ − s j − 1 ) ( s j + 1 − s j ) ( s j − s j − 1 ) \begin{aligned}
\alpha(s_j,s_j') &=\frac{p(y\mid s_j')}{p(y\mid s_j)}\times \frac{\pi(s_j')}{\pi(s_j)}\times \frac{J(s_j\mid s_j')}{J(s_j'\mid s_j)}\\
&=\text{likelihood ratio}\times \frac{(s_{j+1}-s_j')(s_j'-s_{j-1})}{(s_{j+1}-s_j)(s_j-s_{j-1})}
\end{aligned} α ( s j , s j ′ ) = p ( y ∣ s j ) p ( y ∣ s j ′ ) × π ( s j ) π ( s j ′ ) × J ( s j ′ ∣ s j ) J ( s j ∣ s j ′ ) = likelihood ratio × ( s j + 1 − s j ) ( s j − s j − 1 ) ( s j + 1 − s j ′ ) ( s j ′ − s j − 1 ) since
π ( s 1 , s 2 , … , s k ) = ( 2 k + 1 ) ! L 2 k + 1 ∏ j = 0 k + 1 ( s j + 1 − s j ) \pi(s_1,s_2,\ldots,s_k)=\frac{(2k+1)!}{L^{2k+1}}\prod_{j=0}^{k+1}(s_{j+1}-s_j) π ( s 1 , s 2 , … , s k ) = L 2 k + 1 ( 2 k + 1 ) ! j = 0 ∏ k + 1 ( s j + 1 − s j ) Choose a position s ∗ s^* s ∗ uniformly distributed on [ 0 , L ] [0,L] [ 0 , L ] , which must lie within an existing interval ( s j , s j + 1 ) (s_j,s_{j+1}) ( s j , s j + 1 ) w.p 1.
Propose new heights h j ′ , h j + 1 ′ h_j', h_{j+1}' h j ′ , h j + 1 ′ for the step function on the subintervals ( s j , s ∗ ) (s_j,s^*) ( s j , s ∗ ) and ( s ∗ , s j + 1 ) (s^*,s_{j+1}) ( s ∗ , s j + 1 ) . Use a weighted geometric mean for this compromise,
( s ∗ − s j ) log ( h j ′ ) + ( s j + 1 − s ∗ ) log ( h j + 1 ′ ) = ( s j + 1 − s j ) log ( h j ) (s^*-s_j)\log(h_j') + (s_{j+1}-s^*)\log(h_{j+1}')=(s_{j+1}-s_j)\log(h_j) ( s ∗ − s j ) log ( h j ′ ) + ( s j + 1 − s ∗ ) log ( h j + 1 ′ ) = ( s j + 1 − s j ) log ( h j ) and define the perturbation to be such that
h j + 1 ′ h j ′ = 1 − u u \frac{h_{j+1}'}{h_j'}=\frac{1-u}{u} h j ′ h j + 1 ′ = u 1 − u with u u u drawn uniformly from [ 0 , 1 ] [0,1] [ 0 , 1 ] .
The prior ratio, becomes
p ( k + 1 ) p ( k ) 2 ( k + 1 ) ( 2 k + 3 ) L 2 ( s ∗ − s j ) ∗ ( s j + 1 − s ∗ ) s j + 1 − s j × β α Γ ( α ) ( h j ′ h j + 1 ′ h j ) α − 1 exp { − β ( h j ′ + h j + 1 ′ − h ) } \frac{p(k+1)}{p(k)}\frac{2(k+1)(2k+3)}{L^2}\frac{(s^*-s_j)*(s_{j+1}-s^*)}{s_{j+1}-s_j}\times \frac{\beta^\alpha}{\Gamma(\alpha)}\Big(\frac{h_j'h_{j+1}'}{h_j}\Big)^{\alpha-1}\exp\{-\beta(h_j'+h_{j+1}'-h)\} p ( k ) p ( k + 1 ) L 2 2 ( k + 1 ) ( 2 k + 3 ) s j + 1 − s j ( s ∗ − s j ) ∗ ( s j + 1 − s ∗ ) × Γ ( α ) β α ( h j h j ′ h j + 1 ′ ) α − 1 exp { − β ( h j ′ + h j + 1 ′ − h ) } the proposal ration becomes
d k + 1 L b k ( k + 1 ) \frac{d_{k+1}L}{b_k(k+1)} b k ( k + 1 ) d k + 1 L and the Jacobian is
( h j ′ + h j + 1 ′ ) 2 h j \frac{(h_j'+h_{j+1}')^2}{h_j} h j ( h j ′ + h j + 1 ′ ) 2 If s j + 1 s_{j+1} s j + 1 is removed, the new height over the interval ( s j ′ , s j + 1 ′ ) = ( s j , s j + 2 ) (s_j',s_{j+1}')=(s_j,s_{j+2}) ( s j ′ , s j + 1 ′ ) = ( s j , s j + 2 ) is h j ′ h_j' h j ′ , the weighted geometric mean satisfying
( s j + 1 − s j ) log ( h j ) + ( s j + 2 − s j + 1 ) log ( h j + 1 ) = ( s j + 1 ′ − s j ′ ) log ( h j ′ ) (s_{j+1}-s_j)\log(h_j) + (s_{j+2}-s_{j+1})\log(h_{j+1}) = (s_{j+1}'-s_j')\log(h_j') ( s j + 1 − s j ) log ( h j ) + ( s j + 2 − s j + 1 ) log ( h j + 1 ) = ( s j + 1 ′ − s j ′ ) log ( h j ′ ) The acceptance probability for the corresponding death step has the same form with the appropriate change of labelling of the variables, and the ratio terms inverted.
The histogram of number of change points is
And we can get the density plot of position (or height) conditional on the number of change points k k k , for example, the following plot is for the position when k = 1 , 2 , 3 k=1,2,3 k = 1 , 2 , 3
