tokens: halos and therefore have different IGM densities, velocities, and temperature structures. These 100 independent model realizations are necessary to properly characterize the stochasticity of the IGM absorption. We forward model observational effects like finite spectral resolution, signal-to-noise ratio, and continuum-fitting errors, to make the model outputs directly comparable to observed spectra. Each spectrum extends for ±65 cMpc around the foreground quasars position, and we bin the spectra in chunks of 1 cMpc length. For each chunk, we obtain a kernel density estimate (KDE) of the probability distribution (PDF) of the transmission values in the chunk, based on the 100 independent realizations of that model. This results in smooth functions, , that describe the probability for measuring an IGM transmission F n , m in bin n , m , given the model parameter . Here, the index n identifies the different background sightlines within the sightline pattern while m denotes a certain chunk along the sightline. As long as we only focus on the quasar age, the parameter vector has only one component . However, we still denote it as a vector, because this approach allows for easy and straightforward generalization of the inference to additional parameters like quasar obscuration or orientation. Following this, each model of the H i proximity effect is fully described by a set of transmission probability functions . 5.2. Likelihood Computation Due to the high dimensionality of the observable, , i.e., several thousand individual transmission measurements, determining the likelihood poses a very challenging task. The usual approach of approximating the likelihood as a multivariate Gaussian is inadequate for our problem, because the individual transmission PDFs are not described well by Gaussians. In addition, determining the significant correlations between all the elements of { F n , m } would likely require an excessive number of model realizations. Therefore, we follow a likelihood-free approach for which we never have to actually write down an analytic form for the likelihood function. In Schmidt et al. ( 2018 ), we solved a similar problem. There, however, the dimensionality of the problem was low (at most three transmission measurements and two model parameters), which made it feasible to simply map the full likelihood space by brute-force sampling. An approach like this would be completely impossible for our current case. A fully Bayesian treatment is only achievable if the dimensionality of the problem can be drastically reduced. We do this by first computing a so-called pseudo-likelihood , which acts in many ways like a proper likelihood—except that it ignores correlations between the { F n , m }. In a second step, we then map this pseudo-likelihood to determine a proper posterior probability distribution. Our approach is inspired in many ways by Alsing et al. ( 2018 ) and Davies et al. ( 2018b ), but is customized to the problem at hand and is
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