how to use a single Gaussian distribution to approximate mixture Gaussian distribution?

[ShellingFord221]

Hi, in Class BayesianLinear, the weight is sampled from a unimodal Gaussian distribution

self.weight = Gaussian(self.weight_mu, self.weight_rho)

, and the prior of weight is sampled from a mixture Gaussian

self.weight_prior = ScaleMixtureGaussian(PI, SIGMA_1, SIGMA_2)

But is it possible to use a single Gaussian distribution to approximate mixture Gaussian distribution theoretically? It seems meaningless...