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 h j h_j h j
propose a change to h j ′ h_j' h j ′ 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 ]
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 ′ ) . 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 ′
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 f ( x ) = F ′ ( x ) = 1 x , f(x) = F'(x) = \frac{1}{x}\,, f ( x ) = F ′ ( x ) = x 1 , 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 . 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 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 ]
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 ) 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 ∗ [ 0 , L ] [0,L] [ 0 , L ] ( s j , s j + 1 ) (s_j,s_{j+1}) ( s j , s j + 1 )
Propose new heights h j ′ , h j + 1 ′ h_j', h_{j+1}' h j ′ , h j + 1 ′ ( s j , s ∗ ) (s_j,s^*) ( s j , s ∗ ) ( s ∗ , s j + 1 ) (s^*,s_{j+1}) ( s ∗ , s j + 1 )
( 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 ) 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 [ 0 , 1 ] [0,1] [ 0 , 1 ]
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 )} 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 ( 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 ( 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 ) h j ′ h_j' 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') ( 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 ′ ) And we can get the density plot of position (or height) conditional on the number of change points k k k k = 1 , 2 , 3 k=1,2,3 k = 1 , 2 , 3
