Suppose the times to failure, Xi, of identically manufactured components are identically and independently distributed with the survivor function The failure rate at age x r x, given
Trang 1Subject to certain regularity conditions q can take any form (providing the resultingMarkov chain is ergodic), which is a mixed blessing in that it affords great flexibility indesign It follows that the sequence X0 X1 is a homogeneous Markov chain with
p yx = x y q yx
for all x y∈ S, with x = y Note that the conditional probability of remaining in state x
at a step in this chain is a mass of probability equal to
S 1
Suppose x y < 1 Then according to Equation (8.6), y x= 1 Similarly, if x y= 1 then y x < 1 It follows from Equation (8.6) that for all x = y
x y f x q yx = y x f y q xy This shows that the chain is time reversible in equilibrium with
f x p yx = f y p xy
for all x y∈ S Summing over y gives
f x=
S
f y p xy dy
showing that f is indeed a stationary distribution of the Markov chain Providing thechain is ergodic, then the stationary distribution of this chain is unique and is also its limitdistribution This means that after a suitable burn-in time, m, the marginal distribution ofeach Xt t > m, is almost f , and the estimator (8.5) can be used
To estimate h, the Markov chain is replicated K times, with widely dispersed startingvalues Let Xitdenote the tth equilibrium observation (i.e the tth observation following burn-in) on the ith replication Let
ih =1n
n
t=1h
Trang 2is to plot a (several) component(s) of the sequence
Xt Another is to plot somefunction of Xt for t= 0 1 2 For example, it might be appropriate to plot
t= 1 2 Whatever choice is made, repeat for each of the K independent
replications Given that the initial state for each of these chains is different, equilibrium
is perhaps indicated when t is of a size that makes all K plots similar, in the sense
that they fluctuate about a common central value and explore the same region of the
state space A further issue is how many equilibrium observations, n, there should be
in each realization If the chain has strong positive dependence then the realization
will move slowly through the states (slow mixing) and n will need to be large in
order that the entire state space is explored within a realization A final and positive
observation relates to the calculation of x y in Equation (8.6) Since f appears in
both the numerator and denominator of the right-hand side it need be known only up
to an arbitrary multiplicative constant Therefore it is unnecessary to calculate P D in
chosen? Large step lengths potentially encourage good mixing and exploration of the state
space, but will frequently be rejected, particularly if the current point x is near the mode
of a unimodal density f Small step lengths are usually accepted but give slow mixing,
long burn-in times, and poor exploration of the state space Clearly, a compromise value
for is called for
Hastings (1970) suggested a random walk sampler; that is, given that the current point
is x, the candidate point is Y = x + W where W has density g Therefore
q yx = g y − x This appears to be the most popular sampler at present If g is an even function then such
a sampler is also a Metropolis sampler The sampler (8.7) is a random walk algorithm
with
Y = x + 1/2Zwhere 1/21/2
= and Z is a column of i.i.d standard normal random variables
An independence sampler takes q yx = q y, so the distribution of the candidate
point is independent of the current point Therefore,
x y= min
=
4 − 1 1 < < 15
4 2− 15 < ≤ 2
This is a symmetric triangular density on support 1 2 To sample from such a density,
take R1 R2∼ U 0 1 and put
= 1 +1
Trang 5It is also thought that the expected lifetime lies somewhere between 2000 and 3000 hours,depending upon the and values Accordingly, given that
PpriorX > x= Eg
exp
−
x
= 21
−
x
= 21
Trang 6In order to find a point estimate of Equation (8.14) for specified x, we will sample k
values i i from using MCMC, where the proposal density is simply the
joint prior, g This is therefore an example of an independence sampler The k
values will be sampled when the chain is in a (near) equilibrium condition The estimate
... is explored within a realization A final and positive
observation relates to the calculation of x y in Equation (8 .6) Since f appears in
both the numerator and denominator of the...
8.4 Single component Metropolis–Hastings and Gibbs
sampling
Single component Metropolis–Hastings in general, and Gibbs sampling in particular,... rate is increasing and indeed this must be the case with the Bayes
estimates (i.e the posterior marginal expectations of and ) These are bayes= 1131
and bayes=