tokens: errors for are 3%–7%. Because there is no physical motivation to assume that the prior distribution of the pressures is uniform, we also tested the inversion with priors that are uniform in the logarithm of pressure. The results of this test, with identical data as above and the Gaussian regularizer included, are shown in the bottom panel of Figure 2 . The top panel of Figure 2 shows that assuming a uniform prior introduces a preference toward higher pressures. On the other hand, assuming that the prior is uniform in the logarithm of pressures biases the results toward lower pressures. The inversion is particularly sensitive to this bias in the low-mass/low-density region of the EoS, where we lack data. Our Gaussian regularizer helps reduce this bias, as shown in the middle and bottom panels of Figure 2 , in which the results are similar for the two types of priors when the regularizer is also included. In the absence of a regularizer, the freedom introduced by using five polytropes in the parameterization combined with this sensitivity to prior distributions could lead to significantly skewed results, or even the false inference of a phase transition with a perceived high statistical confidence. It is therefore important to use a Gaussian regularizer on the second derivative, with a characteristic scale , to avoid this sensitivity to over-fitting. In Figure 4 , we explore potential correlations between the inferred parameters for this test. Specifically, we show the correlations for the inversion using uniform priors and the Gaussian regularizer (i.e., the middle panel of Figure 2 ). There are slight anti-correlations between adjacent inferred pressures, due to fact that, even with the addition of the regularizer, we are still over-parameterizing our model while trying to fit the effectively single-polytrope EoS SLy. Overall, however, the pressures that are not adjacent are uncorrelated with one another. Özel Psaltis ( 2009 ) showed that, for a three-polytrope parameterization, parameterizing with the pressures as free parameters instead of the polytropic indices reduced the correlations between the inferred values. Figure 4 shows that the low levels of correlations are maintained here. Figure 4. Correlation plots for the inferred pressures at our five fiducial densities. The inversion was performed using uniform priors and a Gaussian regularizer (). The red diamonds mark the pressures of the true EoS (SLy) that we are trying to infer. There are slight anti-correlations between adjacent pressures (e.g., P 1 and P 2 ), but the non-adjacent pressures are relatively uncorrelated. The triangular shape of the P 4 vs. P 5 correlation is due to the causality requirement, which is shown as the red line. In order to ensure that the Bayesian inference works well for other underlying EoS and that the regularizer does not adversely limit our ability to detect potential phase transitions, we tested the method on a number of different EoS as well. As an
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