Reminders reversibility 1 Today Default setting Markov X...
Transcript of Reminders reversibility 1 Today Default setting Markov X...
Today : Reminders about reversibility ,finishing with mixing times �1�
Default setting : Markov chain X=CXn,
no ),
State space V. Trans.matrix P
,Stat
.
dist. it
Reversibility
Chain reversible if Tupou= I pun for all a. VEV
.
Def : Random walk on weighted graph G= ( V. E. w ),
w( e) e [ a - ) is MC.
with
puu:
- w(uD/§ WC e).
Can always assume E={ { Uv } :u,VeV } by using some 0 - weightsUEE
Pap
Therandom walk on G- ( U
,E. w ) has Stat
.
distribution tu= E wluv )Vi UVEE
Proof :Let µyµ,ueV ) be µu=u§⇐w/uv)
.2£Ew=Then (µP)x = §yµupux= E §w⇐w( uvlpux = E § wluvl . wld: UEV UEV WEE E W (e)
Us ( c. E UKEE n' CEE e :ueE
= §e.
WM " ) =µ " . Soµ
is a stationary measure .
To make it a distribution divide through by u§µ=§§u4e1=2§ew(e)( each edge has 2 endpoints
.
).
�2�
In particular : for an unsighted graph G=CU , E),
Stat.
dist.
ofsrw is to ) =d{gC¥, .
Corollary Random walk on G = ( V. E. w ) is reversible .
Proof : Tupuv =(§a⇐w(e) )'w(uv)/§⇐[w( e) = wluv ) = want Trpvu .
Conversely , suppose X is reversible . Setting duh =
tupuv then X is RW on
G= CV, Ec ) where Et{ { uv } : a ,ueV }
. 0
so Reversible M.C. s
= RWS on graphs .
NB : This connection requires the existence of a stationary distribution.
Example: Consider the graph 12 with edge weights wli,
IH )=2i for all i.
TqheuRW on Z with weights w has pine , =z?z÷
. , =§ .
( Drift tonight)
No reasonable def. of reversible should include this Me
.
Warning : Random walks on weighted graphs are different in discrete and continuous time
-30Natural continuous generalization : for a graph G= ( V
,E
,w ) ,
w ( uv ) is rate of jumps from u to v on v to U .
Consider, e.g. ,
star with n leaves.
÷¥4,
;→ All weights 1
.
CTRW-
leaves hub at rate n,
returns at rate 1.
Stationary dist.
uniform on V
Discrete RW → itlhubkt,
I ( i ) =
zlg for all it, .
→n
.
In general CTRW on a weighted finite graph always has it = unit
.distribution
But there are obviously others. E.
g.
.
§¥O with @bT¥0 for all i .
More closely"
mimics" discrete chain
.
Total Variation . Recap .
( All this works in either discrete on continuous time.)
�4�
Total variation distance is Hn -
Nowmax { IMAI - VA ) 1 : Ac V }
=L,§v1µm - Hal
= inf { PCXH ' ) :( X. Y ) coupling ofµ
and v }.
DHK IF
HEPI- it Htv IHK
,;yqgHER - of Pttkv
.
(distribution of X , given that X. =x
.
Fact : DH ) EDTH for all t.
Seen last week .
Atso : d- H ) =
,;gqg11 dcpt - of Ptlkv €, ;gqgHEPt- till
,+ Hit . of Ptlkv ) = Zdtt )
.
Prop : For all s,t > o ,
at ( stt ) e d- ( s ) . at ( t ).
proof : Fix States y,zeV
.
Let (Ys,
Zs) be an optimal coupling ofdy Ps anddzps
.
In other words,
P(Ys=x) - (
ofPs )x
,
PCZs=x) = (
ofPs )
. ,PC Ystz ) = 11
ofPs -
ftp.ltr.
= Py( Xs=x ) =P.
IXs=x )
Note : For all uv ,n
,
Pu ( Xn=v ) = Phu = @up"
)v.
�5�
Next. Psgttw - § Ps
,.
B.tw = § PlYs=x ) Pxtw = ElPIs ,w ]Likewise ,
PIHW = E [ Pts,w ]
so Psjtw - PIHW = El PI, ] - E (Pta,if = IE (PIs; Ptzsw )
Take absolute values,
sum over w to gett.E.lpj.tw- PIII = 's EIIE(BB ; Pta,w ) l '
IEIEIBBIPtzswl
n=Et El RBIPtzswl
ygypstt.gzpstty2
ev
= IEHokspioieptyFinally ,IE Hok
,PicksPty = § IPCH, Zskcy ,
⇒tlldjpt- ofPtyEV
£ § ,
Pllktskcy,
⇒ ) . d- H ) = Post Zs ) - d- H ) =d- Gidtt )
.
o
�6�
Mixingtime is In :X =I* (f) = inf ( t :dltl ' t )
NB : If t i. Tmix then
a ( t , Idtt) =ak¥÷¥I"±@ t.DK e C zdH*Dt= It
So have exponential rate of convergence to station anity on this scale.
In preceding proof we used a coupling ( Ys,
E) of GPs and AP ? This coupled
the locations of two copies of the ME.
at a fixed time.
But can also couple copies
of the entire ME. ( we did this in proving the convergence theorem for a pained .
MCs )
Def : A cguplidgof MC
.s with transition matrix P is a process ( ( Ys
,
Zs),
530 ) st.
( Ys) s soand ( Zs ) s >
o are each MC. s with transition matrix P
.
Given any coupling ( ( Ys,
Zs),
530 ) ,the corresponding stickyopting is (01 it's )
,s > o )
where £s= Z V. s,
Is :{ If,
5 , t E- min ( t : YEZD
This is again a coupling ( exercise )
�7�
theorem : For any coupling ( ( Ys,
Zs),
530 ) of MC.
s with transition matrixB any
t 'o,
andany y ,zeV,
11 ofPt -
ofPtlkue Plt >
TKYO, E) =y,z )
Poot "
Note PC Fey1 ( Y. , E) = ( y ,⇒ ) =
BLXEU) =
B.tnVa
and PLEEV1 1 Yo, E) = c y ,⇒ ) = PdXev ) =BY V.v
So ( It,Et) is a coupling of cryptand of Pt .
Thus
11 orypt - Jeptlltv =P ( It ±
£+10. ,ZHy ,
#=P ( t > t 1 Ho
, -201=4,7)).
o