Parameters in the Acquisition Model

Finally, we assume (θp, θlr) to be known. The vector of unknown model parameters is given as θla =



aw, ψw, σ2d|r



. In general, the dimension of θla is dim (ψw) + 2. As for

the response likelihood model, the dimension of ψ_w is in general unknown, however we assume a parametric acquisition convolution kernel. We assume the convolution kernel to be defined by a normalized powered exponential,

w(h; χ, δ) ∝ exp ( − h χ δ) . (4.24)

In Eq. (4.24), χ and δ are respectively the range and shape parameter. By assuming a parametric acquisition convolution kernel we reduce the dimensionality of the approximate likelihood function, which entails an optimization of a lower dimension

The k-th order MMLE is given as ˆ θ(k)_l a,mml = arg max θla − log p(k)_{(d; θ} la) , (4.25)

and similar for the MAP estimate. The MAP estimate depends on a vector of hyperpa- rameters, τla. Eq. (4.25) constitues a hard optimization problem, but in can be evaluated as in Section 4.3.2. A study of parametric acquisition convolution kernels is found in Lindberg and Omre (2014a).

Chapter 5

MAP Case Studies

We compare the truncation and projection based likelihood approximations for various orders of k. The one dimensional reference profile, κ is displayed in Fig. 5.1, and it is assumed to be of length n = 100. It contains three different classes, {light-grey, dark- grey, black}. From our reference profile we generate various response models r, given κ. Conditioned on r, we generate observations, d, through the acquisition model. We study different response and acquisition models. In particular, we vary the spatial correlation function and class response variance in the response model. The apparent convolution kernel is assumed to be either a powered exponential, second order exponential, or Ricker function with different kernel widths.

We compare the MAP and MMAP predictors for the likelihood approximations for var- ious k, and estimate the similarity measure, α. The similarity measure is a measure of similarity between the approximate and exact posterior models, see Section 3.3. Higher values of α indicate that the approximate posterior, p(k)(κ|d), is a good approxima- tion of the correct posterior, p (κ|d). The distance measures, D[p(k)(κ|d) , p (κ|d)] and DKL[p(k)(κ|d) , p (κ|d)] are also estimated.

Sequences of 100,000 realizations from the correct posterior models, p(κ|d), are gener- ated, using an independent proposal McMC MH-algorithm. We discard the 10,000 first realizations as a burn-in period. The McMC MH-algorithm is initiated with the MAP predictor of the approximate posterior model.

The model parameters are assumed to be fixed and known in this case study.

5

Figure 5.1: Reference profile, κ.

5.1

Model Specification

The reference profile, κ, with K = 3, is generated from a prior with the symmetric transition matrix Pκ =   0.8 0.2 0.0 0.2 0.6 0.2 0.0 0.2 0.8  , (5.1)

having stationary distribution 1/3 × (1, 1, 1)>

. We see that the light-grey class and black class are not allowed to be neighbours. The time-reversed Markov chain is distributed identically to the original Markov chain since the marginal distribution is uniform, see Eq. (2.8).

The class response means are fixed to µ_r|κ0 = (−1, 0, 1)>, and remain unchanged through- out this chapter. The variances, σ2

r|κ0, are varied throughout this chapter. The test cases are defined from a spatial correlation function, ρr(h; ξ), and either an apparent convolution

kernel, wA_{, or an acquisition convolution kernel, w.}

We sort the various test cases by name dependent on their apparent convolution kernel, apparent kernel width, response model variances, and spatial correlation range. The name conventions are listed in Tab. 5.1. Each test case is uniquely defined by its name, and we define SE/MK/MV/MC to be the reference case. That is, the case with a second order exponential acquisition kernel, medium kernel width, medium variances in the response model, and a medium spatial correlation range.

Table 5.1: Name conventions for the MAP test case studies.

Name Abbreviation

Apparent convolution kernel type

Powered exponential PE Second order exponential SE Ricker exponential RE Apparent convolution kernel width Short kernel SK Medium kernel MK Long kernel LK

Class response variance

Low variance LV Medium variance MV High variance HV Spatial correlation range Short correlation SC Medium correlation MC Long correlation LC The observational error is assumed to be σ2

d|r= 10

−4 _{throughout this chapter. Since the}

observational error is assumed to be fixed, we define the associated signal-to-noise ratio to be S/Ndef= Tr W A
N ×P κ0_∈Ω κσ 2 r|κ0ps(κ0) , (5.2)

where Tr(·) denotes the trace of a matrix. A high signal-to-noise ratio assures the obser- vations to be a good read off from the response profile. For each i = 1, . . . , K, we define

5.1. MODEL SPECIFICATION 39 the misclassification rates as in Lindberg (2010),

li = PN n=11{κ r n= i}p(k)(κn= i|d) PN n=11{κrn = i} ui = PN n=1p (k)_(κ n= i|d) PN n=11{κrn = i} , (5.3) where κr

n is n-th value of the reference profile. We refer to li as the lower part, and

it represents the ability for the approximate posterior model to correctly predict the reference profile. Similarly, ui is the upper part, and it is defined to be the ratio between

how much the posterior favors class i, compared to the reference model. Indeed, we have li ≤ ui. If a predictor is good, both li and ui are close to unity.