Consumidor y producto

3.2.1 Notation

Throughout the chapter, h·,·i and k · k denote the inner product and norm of the Hilbert spaceX. For two sequenceskj and hj of real numbers,kj hj means that

|kj|

|hj| is bounded from above and below asj → ∞,kj .hj means that

kj

hj is bounded from above as j → ∞, andkj ∼hj means that _hkj

j →1 as j → ∞. We will use M to denote a constant which is different from occurrence to occurrence.

Let {φj}∞j=1 denote an orthonormal basis in X. Then we can express an

element u ∈ X as u =

∞

P

j=1

ujφj where uj = hu, φji. For γ ≥ 0 we define the Sobolev-like spaces Hγ ={u∈ X : ∞ X j=1 j2γu2_j <∞},

with normk · kHγ given by

kuk2 Hγ := ∞ X j=1 j2γu2_j.

Forγ <0, we define the spaces Hγ _{by duality: H}γ _{= (H}−γ₎∗_.

In the following we consider random variables drawn from Gaussian distri- butions in X, denoted by N(θ,Σ) where the mean θ is an element of X and the covariance operator Σ is a positive definite, self-adjoint, trace class, linear operator inX. The operator Σ possesses an infinite set of eigenfunctions{φj}j∈N which cor- respond to positive eigenvalues{σj}j∈Nand which form an orthonormal basis of X. One can express a drawx from N(θ,Σ) using the Karhunen-Loeve expansion as

x=θ+ ∞ X j=1 √ σjξjφj, (3.2.1)

whereξj are independent and identically distributedN(0,1) real random variables, [17, 78]. In particular, the expansion coefficientsxj =θj+

√

σjξj areN(θj, σj) real random variables and it is straightforward to see that Ex

2 =θ 2 + Tr(Σ) and

that for any bounded linear operatorT inX,T xis distributed asN(T θ, TΣT∗). It is also straightforward to check that if θ = 0 and σj = j−2r for some r ∈ R, then

x∈ Hγ _{almost surely , for any γ < r−}1 2.

3.2.2 Bayesian setting and informal charaterization of the posterior In this subsection we describe the assumptions underlying the Bayesian formulation of the linear inverse problem. Furthermore we provide informal calculations which motivate the expressions for the posterior mean and covariance. These will be made precise in Section 3.3.

We place a scaled Gaussian prior on the unknown u of the form µ0 :=

N(0, τ2C₀), where τ > 0 is a scale parameter and C₀ is a self-adjoint, positive- definite, trace class, linear operator onX. We assume Gaussian observational noise in (3.1.1) which is independent ofu. In particular, we model the data as

y=Ku+√1 nη, (3.2.2) that is we have η = √1 nη in (3.1.1), where 1 √

n is a scale parameter modelling the noise level andη is a random variable independent ofuand distributed asN(0,C₁). The linear operatorC1 is assumed to be self-adjoint, positive-definite, bounded, but

not necessarily trace class onX. This allows for the possibility of having irregular noise which is not inX. For example, the case whereη is white noise corresponds to

C1 =I, and can be viewed as a Gaussian random variable inH−r forr > 1₂. Under

these assumptions, the conditional distribution ofy|u, called the data likelihood, is the translation ofN(0,C1) byKu, which is also Gaussian:

N(Ku,1

nC1). (3.2.3)

In finite dimensions the density of the posterior distribution, that is the conditional distribution of u|y, is found from Bayes rule to be proportional to exp(−J(u;y)), where J(u;y) = n 2kC −1 2 1 (y−Ku)k 2₊ 1 2τ2kC −1 2 0 uk 2_{. (3.2.4)}

This suggests that in our infinite dimensional setting, the posterior distribution is Gaussian, µy _{:= N(m,C), where the mean m and covariance C can be informally}

derived from (3.2.4) using completion of the square: C−1_=nK∗_C−1 1 K+ 1 τ2C −1 0 , (3.2.5) and 1 nC −1_m=K∗_C−1 1 y. (3.2.6)

Observe that the posterior meanmis the minimizer of the functionalJ(u;y). If we defineJ0(u;y) = _n1J(u;y) and denote

λ:= 1

nτ2, (3.2.7)

thenm also minimizes the functionalJ0(u;y), that is, m= arg min u J0(u;y), (3.2.8) where J0(u;y) = 1 2kC −1 2 1 (y−Ku)k2+ λ 2kC −1 2 0 uk2.

Thus the posterior mean is a Tikhonov-Phillips regularized solution in the classi- cal sense (in fact J0 is almost surely infinite and we should really consider Ψ0 =

J₀−1 2 C −1 2 1 y 2

which is finite; the minimizer is unaffected). This reveals the close connection between Bayesian and classical regularization for inverse problems. In the deterministic framework,λis called the regularization parameter which is care- fully chosen in order to balance consistency and stability. Similarly, for given inverse noise leveln, the scale parameterτ introduced in the prior can be judiciously chosen to guarantee a small error between the posterior mean and the true solution, as we will see in Section 3.4.

Posterior consistency refers, in statistical inverse problems, to studying the relationship between the result of the statistical analysis and the truth which un- derlies the data in either the small noise or large data limits; we concentrate on the small noise limit. We consider the standard Bayesian variant on frequentist posterior consistency [20, 28] for our severely ill-posed inverse problem. To this end we consider observations which are perturbations of the image of a fixed element

u†∈ X by a scaled Gaussian additive noise, that is, we have datay=yn† of the form

y_n† =Ku†+√1

nη (3.2.9)

distribution as µy †

n

λ,n := N(m

†_{,C), where C is given by (3.2.5) and m}† _{is given by}

(3.2.6) with y = yn†. Similar to the practice in the deterministic framework, we assume a-priori known regularity of the true solution and identify contraction rates of the posteriorµy

†

n

λ,n to a Dirac measure centered on the true solution, as the noise disappears (n→ ∞).

3.2.3 Model assumptions

In this subsection we present our assumptions on the operators appearing in our framework, that is, on the forward operatorK, the prior covariance operatorC₀and the noise covariance operatorC1.

Assumption 3.2.1. The operators K, C0 and C1 commute with one another, so

that K∗K, C₀ and C₁ have the same eigenfunctions {φj}∞j=1. The corresponding

eigenvalues{l2_j}∞_j=1,{c0j}∞_j=1and{c1j}∞_j=1 ofK∗K,C0 andC1 are assumed to satisfy lj exp(−sjb), c0j =j−2α, c1j =j−2β, (3.2.10)

fors >0, b >0, α > 1₂, β≥0. Furthermore, the fixed true solution u† belongs to Hγ

for some γ >0.

Remark 3.2.2.As is well known in finite dimensions, in the current infinite dimen- sional separable Hilbert-space setting, if K, C₀ and C₁ commute with one another, thenK∗K, C₀ andC₁ have the same eigenfunctions {φj}∞j=1 [51, 75].

Remark 3.2.3. One can relax the assumptions on the eigenvalues ofC₀ and C₁ to

c0j j−2α and c1j j−2β without affecting any of the subsequent results.