Bayesian Model and Problem Formulation

Let_A consist of two disjoint homogeneous regions_A0:={x|E[f(x)] =µf0,Var[f(x)] =

σ2

f0,x ∈ A}, and A1 := {x|E[f(x)] = µf1,Var[f(x)] = σf12 ,x ∈ A}, giving rise to a

hidden label field z := [z(˜x1), . . . , z(˜xNg)]

>

∈ {0,1_}Ng _{with binary entries z(˜x}

i) = k if

˜

xi ∈ Ak ∀i, and k = 0,1. The two separate regions can be used to model heteroge-

neous environments. For instance, if _A corresponds to an urban area, _A1 may include

densely populated regions with buildings, while _A0 withµf0 < µf1 may capture the less

obstructive open spaces. In such a paradigm, we model the conditional distribution of

f(˜xi) as

f(˜xi)|z(˜xi) =k∼ N(µfk, σ 2

while the Ising prior [85], which is a binary version of the discrete MRF Potts prior [38], is assigned tozin order to capture the dependency among spatially correlated labels. By the Hammersley-Clifford theorem [34], the Ising prior of z follows a Gibbs distribution

p(z|β) = 1 C(β)exp  β Ng X i=1 X j∈N(˜xi) δ(z(˜xj)−z(˜xi))   (4.3)

where_N(˜xi) is a set of indices associated with 1-hop neighbors of ˜xi on the rectangular grid in Fig. 4.1, β is a granularity coefficient controlling the degree of homogeneity in

z,δ(·) is Kronecker’s delta, and

C(β) := X z∈Z exp  β Ng X i=1 X j∈N(˜xi) δ(z(˜xj)−z(˜xi))   (4.4)

is the partition function with Z := {0,1}Ng_{. By assuming conditional independence}

of _{f(˜xi)} Ng

i=1 given z, the resulting model is referred to as the Gauss-Markov-Potts

model with two labels. The Gauss-Markov-Potts model for channel-gain cartography is depicted in Fig. 4.2 with the measurement model in (2.2).

To describe priors of other parameters, let νt be independent and identically dis-

tributed (i.i.d.) Gaussian with zero mean and varianceσ2

ν, andθ denote a hyperparam-

eter vector comprising σ2

ν, β, and θf := [µf0, µf1, σf02 , σ2f1]>. The weight matrix Wt∈

RNg×t_{is formed with columnsw}(n,n 0₎ τ := [w(xn(τ),xn0_(τ),x˜₁), . . . , w(x_n(τ),x_n0_(τ),xN˜ g)] >_:= [wτ,1, . . . , wτ,Ng] > ∈ RNg _{of the link x}

n(τ)–xn0_(τ) for τ = 1, . . . , t. Assuming the inde-

pendence among entries of θ,p(θ) can be expressed as

p(θ) =p(σ_ν2)p(β)p(µfk)p(σ 2 fk) (4.5) with p(µfk) = p(µf0)p(µf1) and p(σ 2 fk) = p(σ 2

f0)p(σ2f1), where the individual priors

p(σ2

ν),p(β), p(µfk), and p(σ 2

fk) are specified next.

1) Granularity coefficient β. To cope with the variability of β in accordance with structural patterns of the propagation medium, β is viewed as an unknown random variable that is to be estimated together with f and z under the Bayesian framework.

Figure 4.1: Four-connected MRF with z(˜xi) marked red and its neighbors in N(˜xi)

marked blue.

Figure 4.2: The Gauss-Markov-Potts model with Ising prior for channel-gain cartogra- phy, together with the measurement model for sensors located at (xn,xn0).

Similar to e.g., [72], the uniform distribution is adopted for the prior of β as

p(β) =_U(0,βmax)(β) :=    1/βmax, ifβ∈[0, βmax] 0, otherwise. (4.6) 2) Noise variance σ2

ν. In the presence of the additive Gaussian noise with fixed mean,

it is common to assign a conjugate prior toσ2

ν, which reproduces a posterior distribution

forσ2 ν ∈R+ as follows: p(σ_ν2) =IG(aν, bν) := baν ν Γ(aν) (σ_ν2)−aν−1_exp − bν σ2 ν (4.7)

where aν is referred to as the shape parameter, bν as the scale parameter, and Γ(·)

denotes the gamma function.

3) Hyperparameters of the SLF θf. While the prior forµfk is assumed to be Gaus-

sian with meanmk and varianceσ_k2 ∈R+(see also [5]), the inverse Gamma distribution

parameterized by _{ak, bk} is considered for the prior ofσ2fk:

p(µfk) =N(mk, σ 2

k), k= 0,1 (4.8)

p(σ_f2_k) =IG(ak, bk), k= 0,1. (4.9)

Such choice of the conjugate priors in (4.8) and (4.9) provides analytical tractability for estimatingµfk. Note that a truncated Gaussian prior forµfk also can be adopted when

the support of µfk is known a priori.

Together with the priors for _{f,z,θ_}, our joint posterior becomes

p(f,z,θ|ˇst)∝p(ˇst|f, σ2_ν)p(f|z,θf)p(z|β)p(θ) (4.10) where p(ˇst|f, σν2)∼ N(W>t f, σν2It) is the data likelihood with the weight matrixWt∈

RNg×t_{formed from columnsw}(n,n 0₎ τ := [w(xn(τ),xn0_(τ),x˜₁), . . . , w(x_n(τ),x_n0_(τ),xN˜ g)] >_:= [wτ,1, . . . , wτ,Ng] > _∈RNg _{of the linkx}

n(τ)–xn0_(τ)forτ = 1, . . . , t. Note that Fig. 4.3 sum-

marizes the proposed hierarchical Bayesian model for {ˇst,f,z,θ}as a directed acyclic graph, where the dependency between (hyper) parameters is indicated with an arrow.

We will pursue the the conditional MMSE estimator

ˆ

fMMSE:=E[f|z = ˆzMAP,ˇst] (4.11)

where the marginal MAP estimate is

ˆ

zMAP:= arg max

Figure 4.3: Graphical representation of the hierarchical Bayesian model with Ising prior for (hyper) parameters (those in boxes are fixed).

Furthermore, the marginal MMSE estimates of the θ entries are found as

c σ2 νMMSE:=E[σν2|ˇst] (4.13) b βMMSE:=E[β|ˇst] (4.14) c µfkMMSE:=E[µfk|ˇst], k= 0,1 (4.15) c σ2 fkMMSE:=E[σ 2 fk|ˇst], k= 0,1. (4.16)