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](https://s0.wp.com/latex.php?latex=B_%7B12%7D%28x_1%2C%5Cldots%2Cx_n%29+%3D+%5Cdfrac%7B%5Cmathbb%7BE%7D%5E%7B%5Cpi_1%7D%5Bf_1%28x_n%7C%5Ctheta_1%29%7Cx_1%2C%5Cldots%2Cx_%7Bn-1%7D%5D%7D%7B%5Cmathbb%7BE%7D%5E%7B%5Cpi_2%7D%5Bf_2%28x_n%7C%5Ctheta_2%29%7Cx_1%2C%5Cldots%2Cx_%7Bn-1%7D%5D%7D+%5Ctimes&bg=000000&fg=B0B0B0&s=0&c=20201002)
![\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)]}](https://s0.wp.com/latex.php?latex=%5Ctimes+%5Ccdots%5Ctimes%5Cdfrac%7B%5Cmathbb%7BE%7D%5E%7B%5Cpi_1%7D%5Bf_1%28x_2%7C%5Ctheta_1%29%7Cx_1%5D%7D%7B%5Cmathbb%7BE%7D%5E%7B%5Cpi_2%7D%5Bf_2%28x_2%7C%5Ctheta_2%29%7Cx_1%5D%7D+%5Ctimes%5Cdfrac%7B%5Cmathbb%7BE%7D%5E%7B%5Cpi_1%7D%5Bf_1%28x_1%7C%5Ctheta_1%29%5D%7D%7B%5Cmathbb%7BE%7D%5E%7B%5Cpi_2%7D%5Bf_2%28x_1%7C%5Ctheta_2%29%5D%7D+&bg=000000&fg=B0B0B0&s=0&c=20201002)
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.
Xi'an's Og 
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