A Likelihood-Free Approach for Characterizing Heterogeneous Diseases in Large-Scale Studies

Abstract

We propose a non-parametric approach for characterizing heterogeneous diseases in large-scale studies. We target diseases where multiple types of pathology present simultaneously in each subject and a more severe disease manifests as a higher level of tissue destruction. For each subject, we model the collection of local image descriptors as samples generated by an unknown subject-specific probability density. Instead of approximating the probability density via a parametric family, we propose to side step the parametric inference by directly estimating the divergence between subject densities. Our method maps the collection of local image descriptors to a signature vector that is used to predict a clinical measurement. We are able to interpret the prediction of the clinical variable in the population and individual levels by carefully studying the divergences. We illustrate an application this method on simulated data as well as on a large-scale lung CT study of Chronic Obstructive Pulmonary Disease (COPD). Our approach outperforms classical methods on both simulated and COPD data and demonstrates the state-of-the-art prediction on an important physiologic measure of airflow (the forced respiratory volume in one second, FEV1).

A Appendix: Non-parametric Inference

In this section, we first show that the unnormalized density f(x) has a closed-form using locally constant approximation. Then, we show why the second-order approximation is computationally expensive for our problem. Finally, we provide more detail on the approximation of the KL and HE divergences.

Assuming a locally constant function for $f(x) = \exp (a_0)$, we can compute a closed-form solution for $a_0$ by differentiating Eq. 4 with respect to $a_0$:

$$\begin{aligned} \frac{d \mathcal {L}_x (f_i)}{da_0} = \sum _{ v \in S_i } { w \left( \frac{x - \psi (v)}{h} \right) } - | S_i | \int { w \left( \frac{y - x}{h} \right) e^{a_0} dy} = 0 \end{aligned}$$

If we set $h \equiv \rho _{k, S_i } (x)$ and use the step window function ($w(x) = \mathbb {I}(\Vert x \Vert \le 1)$), the first term in the right hand-side becomes exactly k and the second term is the volume of a d-dimensional hyper-sphere with radius h which is $C_d h^d$, and we arrive at Eq. 5. For the Gaussian window function, the first term becomes a weighted sum k points in the vicinity of x and the second term has the same closed-form as the normalizer of the Gaussian distribution.

If we set h to a constant and use the Gaussian window function and the second-order polynomial, i.e., $\log f(u) |_x \approx a_0 + (u -x)^T a_1 + (u-x)^T a_2 (u-x)$, the local parameters have closed-forms [7, 11]:

$$\begin{aligned}&a_0 = \log (A_0 ) - \frac{ \Vert A_1 \Vert ^2}{A_0^2} -( d \log \sqrt{2\pi } + (d+1) \log n ), a_1 = \frac{1}{ h A_0 } A_1, \\&a_2 = \frac{1}{ 2h^2 } I_{d \times d} - \frac{A_0}{ 2h^2 } \left( A_2 - A_1 A_1^T \right) ^{-1} \end{aligned}$$

where $A_0 \equiv \sum _{v \in S_i} { \alpha _v (x) }$ and $\alpha _v (x) \equiv \text {exp}\left( - \frac{ \Vert \psi (v) - x \Vert ^2 }{ 2 h^2 } \right)$, for $D(x,v) \equiv \frac{1}{h} ( \psi (v) - x )$, $A_1 \equiv \sum _{v \in S_i} { \alpha _v (x) D(x,v) }$, and $A_2 \equiv \sum _{v \in S_i} { \alpha _v (x) D(x,v) D(x,v)^T }$. It is straightforward to see computing $a_2$ demands inversion of a $d\times d$ matrix ($O(d^3)$) which needs to be done for every patch hence it is computationally prohibitive.

The KL divergence is a straightforward substitution of Eq. 5. Our estimator for HE is proposed by Poczos et al. [14]. The HE estimator is also based on substitution. The minor adjustment (the term behind the summation in Eq. 6) makes sure that the estimator is unbiased.