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RJMCMC

Bayesian Model Choice

Green's algorithm

Fixed Dimension Reassessment

RJMCMC for change points

The log-likelihood function is
p(yx)=i=1nlog{x(yi)}0Lx(t)dt=j=1kmjloghjj=0khj(sj+1sj)\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}
where
mj=#{yi[sj,sj+1)}m_j=\#\{y_i\in[s_j,s_{j+1})\}
.

H Move

  1. 1.
    choose one of
    h0,h1,,hkh_0,h_1,\ldots,h_k
    at random, obtaining
    hjh_j
  2. 2.
    propose a change to
    hjh_j'
    such that
    log(hj/hj)U[12,12]\log(h_j'/h_j)\sim U[-\frac 12, \frac 12]
    .
  3. 3.
    the acceptance probability is
    α(hj,hj)=p(hjy)p(hjy)×J(hjhj)J(hjhj)=p(yhj)p(yhj)×π(hj)π(hj)×J(hjhj)J(hjhj).\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}
    Note that
    loghjhj=u,\log\frac{h_j'}{h_j}= u \,,
    it follows that the CDF of
    hjh_j'
    is
    F(x)=P(hjx)=P(euhx)=P(ulog(x/h))=log(x/h)+1/2F(x) = P(h_j'\le x) = P(e^uh\le x) = P(u\le \log(x/h)) = \log(x/h) + 1/2
    and
    f(x)=F(x)=1x,f(x) = F'(x) = \frac{1}{x}\,,
    so
    J(hjhj)=1hj.J(h_j'\mid h_j) = \frac{1}{h_j'}\,.
    Then we have
    α(hj,hj)=p(yhj)p(yhj)×(hj)αexp(βhj)hjαexp(βhj)=likelihood ratio×(hj/hj)αexp{β(hjhj)}\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}

P Move

  1. 1.
    Draw one of
    s1,s2,,sks_1,s_2,\ldots,s_k
    at random, obtaining say
    sjs_j
    .
  2. 2.
    Propose
    sjU[sj1,sj+1]s_j'\sim U[s_{j-1}, s_{j+1}]
  3. 3.
    The acceptance probability is
    α(sj,sj)=p(ysj)p(ysj)×π(sj)π(sj)×J(sjsj)J(sjsj)=likelihood ratio×(sj+1sj)(sjsj1)(sj+1sj)(sjsj1)\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}
    since
    π(s1,s2,,sk)=(2k+1)!L2k+1j=0k+1(sj+1sj)\pi(s_1,s_2,\ldots,s_k)=\frac{(2k+1)!}{L^{2k+1}}\prod_{j=0}^{k+1}(s_{j+1}-s_j)

Birth Move

Choose a position
ss^*
uniformly distributed on
[0,L][0,L]
, which must lie within an existing interval
(sj,sj+1)(s_j,s_{j+1})
w.p 1.
Propose new heights
hj,hj+1h_j', h_{j+1}'
for the step function on the subintervals
(sj,s)(s_j,s^*)
and
(s,sj+1)(s^*,s_{j+1})
. Use a weighted geometric mean for this compromise,
(ssj)log(hj)+(sj+1s)log(hj+1)=(sj+1sj)log(hj)(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
hj+1hj=1uu\frac{h_{j+1}'}{h_j'}=\frac{1-u}{u}
with
uu
drawn uniformly from
[0,1][0,1]
.
The prior ratio, becomes
p(k+1)p(k)2(k+1)(2k+3)L2(ssj)(sj+1s)sj+1sj×βαΓ(α)(hjhj+1hj)α1exp{β(hj+hj+1h)}\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)\}
the proposal ration becomes
dk+1Lbk(k+1)\frac{d_{k+1}L}{b_k(k+1)}
and the Jacobian is
(hj+hj+1)2hj\frac{(h_j'+h_{j+1}')^2}{h_j}

Death Move

If
sj+1s_{j+1}
is removed, the new height over the interval
(sj,sj+1)=(sj,sj+2)(s_j',s_{j+1}')=(s_j,s_{j+2})
is
hjh_j'
, the weighted geometric mean satisfying
(sj+1sj)log(hj)+(sj+2sj+1)log(hj+1)=(sj+1sj)log(hj)(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.

Results

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
kk
, for example, the following plot is for the position when
k=1,2,3k=1,2,3

References