## A domino formula for Bayes factors

This afternoon, when working on ABC with Jean-Marie Cornuet, I came up with a domino formula for Bayes factors. It goes like this

$B_{12}(x_1,\ldots,x_n) = \dfrac{\mathbb{E}^{\pi_1}[f_1(x_n|\theta_1)|x_1,\ldots,x_{n-1}]}{\mathbb{E}^{\pi_2}[f_2(x_n|\theta_2)|x_1,\ldots,x_{n-1}]} \times$

$\times \cdots\times\dfrac{\mathbb{E}^{\pi_1}[f_1(x_2|\theta_1)|x_1]}{\mathbb{E}^{\pi_2}[f_2(x_2|\theta_2)|x_1]} \times\dfrac{\mathbb{E}^{\pi_1}[f_1(x_1|\theta_1)]}{\mathbb{E}^{\pi_2}[f_2(x_1|\theta_2)]}$

with hopefully clear if implicit notations. I wonder if this has been exploited previously for computational purposes as each expectation is taken under the “previous” posterior: it could have an appeal from a sequential Monte Carlo perspective.

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