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From a probabilistic point of view, the solution of the inverse problem corresponds to estimating the conditional distribution m∣d. The conditional probability P(m∣d) can be obtained using Bayes' theorem (Eqs. 1.8 and 1.25):

(1.52)

where P(d∣m) is the likelihood function, P(m) is the prior distribution, and P(d) is the marginal distribution. The probability P(d) is a normalizing constant that guarantees that P(m∣d) is a valid PDF.

In geophysical inverse problems, we often assume that the physical relation f in Eq. () is linear and that the prior distribution P(m) is Gaussian (Tarantola 2005). These two assumptions are not necessarily required to solve the Bayesian inverse problem, but under these assumptions, the inverse solution can be analytically derived. Indeed, in the Gaussian case, the solution to the Bayesian linear inverse problem is well‐known (Tarantola 2005). If we assume that: (i) the prior distribution of the model is Gaussian, i.e. , where μ_m is the prior mean and ∑_m is the prior covariance matrix; (ii) the forward operator f is linear with associated matrix F; and (iii) the measurement errors ε are Gaussian , with 0 mean and covariance matrix ∑_ε, and they are independent of m; then, the posterior distribution m∣d is also Gaussian with conditional mean μ_m∣d:

(1.53)

and conditional covariance matrix ∑_m∣d:

(1.54)

For the proof, we refer the reader to Tarantola (2005). This result is extensively used in Chapter 5 for seismic inversion problems.