Statistical physics of medical diagnostics: Study of a probabilistic model

We study a diagnostic strategy which is based on the anticipation of the diagnostic process by simulation of the dynamical process starting from the initial findings. We show that such a strategy could result in more accurate diagnoses compared to a strategy that is solely based on the direct implications of the initial observations. We demonstrate this by employing the mean-field approximation of statistical physics to compute the posterior disease probabilities for a given subset of observed signs (symptoms) in a probabilistic model of signs and diseases. A Monte Carlo optimization algorithm is then used to maximize an objective function of the sequence of observations, which favors the more decisive observations resulting in more polarized disease probabilities. We see how the observed signs change the nature of the macroscopic (Gibbs) states of the sign and disease probability distributions. The structure of these macroscopic states in the configuration space of the variables affects the quality of any approximate inference algorithm (so the diagnostic performance) which tries to estimate the sign-disease marginal probabilities. In particular, we find that the simulation (or extrapolation) of the diagnostic process is helpful when the disease landscape is not trivial and the system undergoes a phase transition to an ordered phase.

Figure 1

Interaction graph of disease variables (left circles) and sign variables (right circles) related by $M_a$ one-disease and $M_{ab}$ two-disease interaction factors (middle squares) in addition to interactions induced by the leak probability (right square) and the prior probability of diseases (left square). In general, an interaction factor $\alpha =a,ab$ is connected to $k_{\alpha }$ signs and $l_{\alpha }$ diseases [23].

Figure 2

Sign and disease probabilities in the homogeneous fully connected models vs the fraction of observed signs $N_O/N_S$. We assume that all the observed signs are positive. The prior disease probabilities and the leak sign probabilities are $P_0(D_a=1)=P(S_i=+1|\mathbf{0})=0.001$. Panels (a),(b) show the probability of observing a positive sign $P(S=1)$ and the probability of having a disease $P(D=1)$ for the D1S1 and D2S1 models ($K_{ij}^a=K_{ij}^{ab}=0$). Panels (c),(d) display the above probabilities for the D1S2 and D2S2 models ($K_{ij}^a=K_{ij}^{ab}=1$).

Figure 3

Comparing the accuracy (true positive plus true negative) of the diagnosis [panels (a),(b),(c)] and the relative gaps [panels (d),(e),(f)] in the disease probabilities (sorted by magnitude) $\mathrm{\Delta }{p}_{12}=(P_1-P_2)/P_1$ and $\mathrm{\Delta }{p}_{23}=(P_2-P_3)/P_2$ for the D1S1 and D2S1 models using an exhaustive inference algorithm. We consider the cases in which only one or two diseases are present $\left(\right|\mathbf{D}|=1,2)$. The prior disease probabilities are chosen such that $N_D{P}_0(D_a=1)=\left|\mathbf{D}\right|$. The model parameters are obtained from the exponential true model for different values of $\beta$. The filled circles in the top panels show the results after a simulation process of $T=N_O/2$ steps, where $N_O$ is the initial number of the observed signs with known true values. For the model structure we take a fully connected graph of $N_D=5,N_S=20$ variables with $M_a=5,M_{ab}=10$ interaction factors, and connectivities $k_a=k_{ab}=20$. The data are results of 2000 independent realizations of the problem.

Figure 4

Comparing the convergence probability [panels (a),(b),(c)] and accuracy (true positive plus true negative) of the diagnosis with and without extrapolation [panels (d),(e),(f)] for the D1S1 and D2S1 models using the MF approximation. We consider the cases in which only one or two diseases are present $\left(\right|\mathbf{D}|=1,2)$. The prior disease probabilities are chosen such that $N_D{P}_0(D_a=1)=\left|\mathbf{D}\right|$. The model parameters are obtained from the exponential true model. The filled circles show the results after a simulation process of $T=N_O/2$ steps, where $N_O$ is the initial number of the observed signs with known true values. For the model structure we take a random graph of $N_D=50,N_S=500$ variables with $M_a=50,M_{ab}=500$ interaction factors, and connectivities $k_a=k_{ab}=400$. The data are results from at least 100 independent realizations of the problem.

Figure 5

Improvement in the statistics of the first right and wrong diagnosis times $(T_R,T_W)$ after maximizing the objective function $\mathcal{E}\left[\mathbf{O}\right(T\left)\right]$ with the D1S1 and D2S1 models using the MF approximation. The algorithm starts from a random sequence of $T=N_O/2$ observations, where $N_O$ is the initial number of the observed signs with known true values. Panels (a),(b),(c) show the probabilities $P(\mathrm{\Delta }{T}_{R,W}<0)$ of decreasing the corresponding times by the algorithm. Panels (d),(e),(f) show the average values $\mathrm{\Delta }{T}_{R;W}$ of the changes in the corresponding times by the algorithm. We consider the cases in which only one or two diseases are present $\left(\right|\mathbf{D}|=1,2)$. The prior disease probabilities are chosen such that $N_D{P}_0(D_a=1)=\left|\mathbf{D}\right|$. The model parameters are obtained from the exponential true model. For the model structure we take a random graph of $N_D=50,N_S=500$ variables with $M_a=50,M_{ab}=500$ interaction factors, and connectivities $k_a=k_{ab}=400$. The data are results of at least 100 independent realizations of the problem.

Figure 6

Comparing the accuracy of the diagnosis with and without extrapolation for the D2S1 model using the MF approximation. We consider the cases in which only two diseases are present $\left(\right|\mathbf{D}|=2)$. The prior disease probabilities are chosen such that $N_D{P}_0(D_a=1)=\left|\mathbf{D}\right|$. The model parameters are obtained from the exponential true model. The filled circles show the results after a simulation process of $T=N_O/2$ steps, where $N_O$ is the initial number of the observed signs with known true values. For the model structure we take a random graph of $N_D=50,N_S=500$ variables with $M_a=50,M_{ab}=500$ interaction factors, and connectivities $k_a=k_{ab}=400$. The data are results of at least 100 independent realizations of the problem.

Figure 7

Free energy landscape as a function of the sign and disease probabilities in the homogeneous fully connected D2S1 model. We assume that all the observed signs are positive. Here $P(D=1)=x, P(S=+1)=(1+y)/2$, and $n_o=N_O/N_S$. The prior disease probabilities and the leak sign probabilities are $P_0(D_a=1)=P(S_i=+1|\mathbf{0})=0.001$.