Evaluación clínica del paciente diabético

The following section deals with the interpretation of the marginal posterior probability densities computed with the NPS. The possible effects of invalid prior assumptions and data outliers will be treated in the first part, whereas in the second part, the interpretation of the posterior based on credible volumes will be discussed, since it was used often in this work.

Invalid prior and data

In any experiment, one should make sure that the assumed conditions apply, and that the acquired data does not contain any outliers or systematic errors. In the case of NPS, besides the failure of Dale-Eisinger-Blumberg theory discussed already in section 6.7.2, several other sources of systematic errors are possible.

A possible error source that affects only the prior assumptions is an incorrect crystal structure. One or more satellite position priors would then be shifted away from their

“true” position, and probably also the shape of the “true” and simulated accessible volume would differ. It is difficult to predict how strong such erroneous satellite priors would influence the inferred antenna positions, since the effect depends on the FRET network structure and the number of satellites.

Even when the structures of the individual macromolecules in the complex are correct, all complexes or just a fraction might be assembled in a wrong way. While the latter case results in a possible misassignment of peaks in FRET efficiency or FRET anisotropy his- tograms, invalid data is definitely acquired in the first case. In any of the scenarios above, the result of the inference might not reflect a real position of the antenna fluorophore. Similarly, like in the case of incorrectly assumed satellite positions, it is hard to predict the final localization accuracy.

Incorrectly assembled complexes are an even greater problem in bulk measurements, since they cannot be easily detected. In particular, such errors might affect the bulk flu- orescence anisotropies when fluorophores exhibit different anisotropies in different com- plexes or complex constituents. Time-resolved fluorescence anisotropies should be hence determined in single-molecule measurements, if possible.

Presumably, systematic errors will have in general a larger impact when the FRET network is analyzed globally. At least in the case of FRET efficiencies and anisotropies, such errors should be easier to detect with the consistency check proposed (section5.3.6), since it is more likely to produce a contradiction in the result. Yet, the consistency check will work only, when there is enough valid data and when a sufficient amount of satellite positions is assigned correctly (section6.2.2). These conditions could be established by control measurements between different satellite attachment sites, which has other positive effects and will be discussed later (section6.7.4).

One could also try to account for outliers and invalid prior assumptions in the likelihood and prior assignment. Instead of being purely Gaussian, each likelihood factor could be composed of a constant “background” and a Gaussian. The “background” would model possible outliers in the data, while the Gaussian would describe a correct measurement. Similarly, the satellite position priors could be assigned to be less informative and in that way, incorrect satellite positions could be modeled.

Yet, since the influence of the prior was found to be strong in all experiments shown in this work (sections 6.2, 6.4 and 6.6), a less informative satellite position prior and likelihood would lead to an even less peaked marginal antenna position posterior. This would express the general distrust in the data and the prior assumptions rather than help to detect and eliminate inconsistencies automatically. However, when more informative data is available, such modifications of the prior and the likelihood might be useful.

In addition, the prior assumptions and the measured data should be verified also by other methods than FRET, if possible.

Credible volumes

Even when the data and prior assumptions are correct, one has to be cautious when a probability density is interpreted in terms of credible intervals. This applies to all marginal posterior distributions, and in particular to the marginal fluorophore position posterior densities computed with NPS.

A credible interval is defined as the smallest volume that contains a certain probability, the credibility level (see section 2.4.4). When one would analyze many random FRET networks, and when the influence of the prior on the shape of the posterior would be negligible, the frequency of the “true” parameters being inside the volume should approx- imately correspond to the credibility level.

6.7 General discussion

Figure 6.43: Misunderstandings in

marginal posterior interpreta- tion. In the case of a strange shaped posterior, p(x, y|data, I) (grayscale), the “true” set of parameters (cross) might lie far away from its maximum (circle) but may still have reasonable probability density. The positions corresponding to the maxima of the marginalized density (squares) and the average values ofxandy (triangles) are indicated by arrows and might have very low probabil- ity values. Credible intervals, like the 68% credible interval shown here (dashed lines), are just a way to display the posterior and should not be overinterpreted.

On the other hand, one could define an infinite number of other credible volumes (i.e. not the smallest one), and still, the above argument would be valid. Because of that, one has to be comfortable with the plain interpretation of the surfaces of credible volumes as a useful way to display iso-levels of the posterior. It is the complete posterior probability distribution that is, by definition, the result of a Bayesian parameter estimation.

Nevertheless, the credible volumes were used in order to be comparable with standard methods of data analysis. For example, a 68% and 95% credible interval corresponds to an error margin of one and two standard deviations, respectively, when the posterior is normal distributed.

In the same way, one should be aware of pitfalls that might occur when the proximity of the “true” parameters to the maximum of a posterior probability density is interpreted, especially when the posterior density was marginalized. In figure 6.43, a peculiar case is shown, in which the “true” parameters exhibit a fairly high posterior probability density and lie within the 68% credible volume, while being outside of the 68% credible interval of one of the marginalized posterior densities. Due to the special shape of the posterior also the point defined by the maxima of the marginal posterior densities as well as the center of mass of the posterior exhibit very low posterior density values and cannot be used to summarize the posterior in a few numbers. It is likely that exactly this effect is observed in the test calculations of section6.3.3, in which only FRET efficiency data was analyzed. There, the simulated antenna positions were found at asymmetrical locations in the marginal posterior.

However, if there is enough informative data to concentrate the posterior density in a single spot in the parameter space, the posterior might be approximated well by a multivariate Gaussian. In that case, the effect discussed above will not occur, since the posterior maximum coincides with the point defined by the maxima of the marginalized posteriors, which are also described approximately by Gaussians.

As discussed in the previous part, it is desirable that the posterior probability is well localized in the parameter space. In the next section, a discussion of the FRET network architecture will be given in respect thereof.

P P P P_P Nsat Nant _{1 2 3 4 5 6 7 8 9 10 11 12 13 14 15} 1 -2 -4 -6 -8 -10 -12 -14 -16 -18 -20 -22 -24 -26 -28 -30 2 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12 -13 -14 -15 3 0∗ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 5 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 6 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 7 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 8 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 9 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 10 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105 11 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120 12 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 13 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 14 11 22 33 44 55 66 77 88 99 110 121 132 143 154 165 15 12 24 36 48 60 72 84 96 108 120 132 144 156 168 180

Table 6.4:Determinacy Nobs −Nunk for FRET efficiency-only networks with fluorescence

anisotropies equal 0 in dependence of number of satellitesNsat and number of antennas Nant. Negative and zero determinacies are printed on a dark gray and light gray back- ground, respectively. The minimal overdetermined networks is denoted by a star.