A. Putze et al.: An MCMC technique to sample transport and source parameters of Galactic cosmic rays. II. – r$_{h}$ , the size of the local bubble (kpc), which is most of the time set to be 0 (to compare with models in the literature that do not consider any local underdensity); – K$_{0}$ , the normalisation of the diffusion coefficient (in unit of kpc 2 Myr − $^{1}$); – δ , the slope of the diffusion coefficient; – V$_{c}$ , the constant convective wind perpendicular to the disc (km s − $^{1}$); – V$_{a}$ , the Alfv´ enic speed (km s − $^{1}$) regulating the reaccel- eration strength [see Eq. (5)]. and four transport ones The diffusion coefficient is taken to be The MCMC method, based on the Bayesian statistics, is used here to estimate the full distribution—the so-called conditional probability-density function (PDF)—given some experimental data (and some prior density for these parameters). We summarise below the salient features of the MCMC technique. A detailed description of the method can be found in Paper I. The issue of the efficiency, which was not raised in Paper I, is discussed in Appendix C. The Bayesian approach aims to assess the extent to which an experimental dataset improves our knowledge of a given theoretical model. Considering a model depending on m parameters we wish to determine the PDF of the parameters given the data, P ( θ | data). This so-called posterior probability quan- tifies the change in the degree of belief one can have in the m parameters of the model in the light of the data. Applied to the parameter inference, Bayes theorem reads where P (data) is the data probability (the latter does not depend on the parameters and hence, can be treated as a normalisation factor). This theorem links the posterior probability to the likelihood of the data L ( θ ) ≡ P (data | θ ) and the so-called prior probability, P ( θ ), which indicates the degree of belief one has before observing the data. The technically difficult point of Bayesian parameter es- timates lies in the determination of the individual poste- rior PDF, which requires an (high-dimensional) integration of the overall posterior density. Thus an efficient sampling method for the posterior PDF is mandatory. In general, MCMC methods attempt to studying any target distribution of a vector of parameters, here P ( θ | data), by generating a sequence of n points/steps (hereafter a chain) Each θ i is a vector of m components, e.g., as defined in Eq. (12). In addition, the chain is Markovian in the sense that the distribution of θ n +1 is influenced entirely by the value of θ $_{n}$. MCMC algorithms are developed to ensure We begin with the PDFs of the parameters based on the B/C constraint (dataset F) for the various classes of models (I, II, or III). The PDFs are shown in Fig. 2. The first important feature is that the marginal distri- butions of the transport parameters (diagonals) are mostly Gaussian. From the off-diagonal distributions, we remark that K$_{0}$ and δ are negatively correlated. This originates in the low-energy relation K ( E ) ∝ K$_{0}$R $^{δ}$, which should re- main approximately constant to reproduce the bulk of the data at GeV/n energy. The diffusion slope δ is negatively For stable species, the degeneracy between the normalisa- tion of the diffusion coefficient K$_{0}$ and the halo size of the Galaxy L prevents us from being able to constrain both pa- rameters at the same time. We choose to set L = 4 kpc (we also set r$_{h}$ = 0, i.e., standard DM). The free transport pa- rameters are { K$_{0}$, δ, V$_{c}$, V$_{a}$ } . The classes of models consid- ered are summarised in Table 1. The reference B/C dataset (denoted dataset F) used for the analysis is described in Appendix D.1. The chain analysis is based on the selection of a subset of points from the chains (to obtain a reliable estimate of the PDF). Some steps at the beginning of the chain are discarded (burn-in length). By construction, each step of the chain is correlated with the previous steps: sets of inde- pendent samples are obtained by thinning the chain (over the correlation length). The fraction of independent sam- ples measuring the efficiency of the MCMC is defined to be the fraction of steps remaining after discarding the burn- in steps and thinning the chain. The final results of the MCMC analysis are the target PDF and all marginalised PDFs. They are obtained by merely counting the number of samples within the related region of parameter space. that the time spent by the Markov chain in a region of the parameter space is proportional to the target PDF value in this region. Here, the prescription used to generate the Markov chains is the so-called Metropolis-Hastings algo- rithm, which ensures that the stationary distribution of the chain asymptotically tends to the target PDF. Table 1. Classes of models tested in the paper. Table 2. Most probable values for B/C data only ( L = 4 kpc). 4. Results for stable species (fixed halo size L ) 4.1. PDF for the transport parameters 5 { θ i } i =1 ,...,n ≡ { θ $_{1}$, θ $_{2}$, . . . , θ n } . (14) P ( θ | data) = P (data | θ ) · P ( θ ) P (data) , (13) θ ≡ { θ $^{(1)}$, θ$^{(2)}$, . . . , θ( m $^{)}$} , (12) K ( E ) = βK$_{0}$ R $^{δ}$. (11) 3. MCMC Model Transport parameters Description I { K 0 , δ, V c } Diffusion + convection II { K 0 , δ, V a } Diffusion + reacceleration III { K 0 , δ, V c , V a } Diff. + conv. + reac. Model K 0 × 10 2 δ V c V a Data (kpc $^{2}$Myr $^{$_{−}$1}$) (km s $^{$_{−}$1}$) (km s$^{$_{−}$1}$) I-F 0 . 42 +0 $^{.}$03 $_{−}$0 $^{.}$04 0 . 93 +0 $^{.}$02 $_{−}$0 $^{.}$03 13 . 5 +0 $^{.}$3 $_{−}$0 $^{.}$3 · · · II-F 9 . 7 +0 $^{.}$3 $_{−}$0 $^{.}$2 0 . 234 +0 $^{.}$006 $_{−}$0 $^{.}$005 · · · 73 +2 $_{−}$2 III-F 0 . 46 +0 $^{.}$08 $_{−}$0 $^{.}$06 0 . 86 +0 $^{.}$04 $_{−}$0 $^{.}$04 18 . 9 +0 $^{.}$3 $_{−}$0 $^{.}$4 38 +2 $_{−}$2