Illustration of the Gibbs sampling procedure applied to the binary MSP J1526-2744. A : photon phases are determined by the current timing model parameters, with probability weights indicated by the grey-scale. B: a short MCMC chain is produced over the template pulse profiles (faint black curves), given these photon phases, and the final step is chosen as the next sample for τ (solid red curve). C: the relative likelihoods as a function of phase for each Gaussian component in the template pulse profile are then used, alongside the photon weights, to determine the probabilities for assigning each photon to a single peak (or to the background). D: these assignments are performed randomly according to these probabilities, with the outcomes illustrated by the colours of photons in the panel corresponding to the relevant peak in the panel above (or the grey-scale for photons assigned to the background). E: a short MCMC chain, starting from the red square and following the blue path to the green triangle, is run targeting the marginal likelihood for the hyperparameters, illustrated by the grey-scale and red contour lines at the 1σ, 2σ, and 3σ levels. Here, a power-law timing noise model is assumed, with hyperparameters consisting of the log-amplitude log₁₀ A_TN and spectral index y_TN. The final sample in the chain is chosen as the next sample for I. F : a weighted least-squares fit is performed using Gaussian likelihoods for all photons assigned to peaks in the template, constrained by the prior distribution defined by the chosen sample of I. A single sample for the timing model (green curve) is chosen from the posterior distribution that results from this fit (illustrated by the faint black lines). Here, the sinusoidal curve whose amplitude grows with time indicates a significant detection of proper motion. The photon phases in panel A are then updated by these resulting phase shifts, and the process repeats.