Leave

- $ (E, E math, lambda) $ Y $ (E & # 39 ;, mathcal E & # 39 ;, lambda & # 39;) $ be measuring spaces
- $ I $ be a finite set not empty
- $ varphi_i: E & # 39; a E $ be $ ( mathcal E & # 39 ;, mathcal E) $-measurable with $$ lambda & # 39; circ varphi_i ^ {- 1} = q_i lambda tag1 $$ for $ i in I $
- $ p, q_i: E to (0, infty) $ be $ mathcal E $-measurable with $$ int p : { rm d} lambda = int q_i : { rm d} lambda = 1 $$
- $ w_i: E a (0,1) $ be $ mathcal E $-measurable, $ w_i & # 39 ;: = w_i circ varphi_i $, $$ p_i & # 39 ;: = begin {cases} frac p {q_i} circ varphi_i & text {on} left {q_i circ varphi_i> 0 right } \ 0 & text {on} left {q_i circ varphi_i = 0 right } end {cases} $$ Y $$ f_i & # 39 ;: = begin {cases} frac f {q_i} circ varphi_i & text {on} left {q_i circ varphi_i> 0 right } \ 0 & text {on} left {q_i circ varphi_i = 0 right } end {cases} $$ for $ i in I $
- $ zeta $ denote the counting measure in $ (I, 2 ^ I) $ Y $$ nu & # 39 ;: = w & # 39; p & # 39; ( zeta otimes lambda & # 39;) $$

Leave $ f in L ^ 2 ( lambda) $. Assume $$ {q_i = 0 } subseteq {w_ip = 0 }, tag2 $$ $$ {p = 0 } subseteq {f = 0 } tag3 $$ Y $$ {pf ne0 } subseteq left { sum_ {i in I} w_i = 1 right }. tag4 $$

Leave $ ((T_n, X_n & # 39;)) _ {n in mathbb N} $ be the Markov chain (supposed to be in stationarity) generated by the Metropolis-Hastings algorithm with objective distribution $ nu & # 39; $ Y $$ A_n: = frac1n sum_ {i = 0} frac {f & # 39;} {p & # 39;} (T_i, X_i & # 39;) ; ; ; text {for} n in mathbb N. $$ I want to minimize asymptotic variance $$ sigma ^ 2: = lim_ {n to infty} n operatorname {Var} A_n $$ With respect to $ w_i $. How can we do that?

I know that yes $ (Y_n) _ {n in mathbb N_0} $ is any homogeneous Markov chain in time, $ mu: = math L (Y_0) $, $ g in L ^ 2 ( mu) $ Y $ B_n: = frac1n sum_ {i = 0} ^ {n-1} g (Y_i) $, so $ operatorname {Var} B_n = frac1n left ( operatorname {Var} _ mu (g) +2 sum_ {i = 1} ^ {n-1} left (1- frac in right) operatorname {Cov} (f (Y_0), f (Y_i)) right) $. Furthermore, if $ L ^ 2_0 ( mu): = left {h in L ^ 2 ( mu): int h : { rm d} mu = 0 right } $, $ mathcal D (G): = left {h_0 in L ^ 2_0 ( mu): left ( sum_ {i = 0} ^ n kappa ^ ih_0 right) _ {n in mathbb N_0} text {is convergent} right } $, $$ Gh_0: = sum_ {n = 0} ^ infty kappa ^ nh_0 ; ; ; text {for} h_0 in mathcal D (G), $$ Y $ g_0: = g- int g : { rm d} mu in mathcal D (G) $, so $$ n operatorname {Var} B_n xrightarrow {n to infty} 2 langle Gg_0, g_0 rangle_ {L ^ 2 ( mu)} – operatorname {Var} _ mu (g) tag5. $$ In particular, leaving $ mathcal L: = – (1- kappa) $, we can consider the spectral gap of $ math L $, $$ operatorname {gap} mathcal L = inf _ { substack {h in L ^ 2 ( mu) setminus {0 } \ 1 : perp : h}} frac { langle- mathcal Lh, h rangle_ {L ^ 2 ( mu)}} { left | h right | _ {L ^ 2 ( mu)} ^ 2} = 1- left | kappa right | _ { mathfrak L (L ^ 2_0 ( mu))}, $$ where do we consider $ kappa $ as a non-negative self-attachment operator in $ L ^ 2 ( mu) $. With this definition, the right side of $ (5) $ it is at most $ left ( frac2 { operatorname {gap} mathcal L} -1 right) operatorname {Var} _ mu (g) $.