VAE:Lower bound of logP(x) logPe(x)= q(zlx)logP(x)dz g(x)can be any distribution =4(zx )1og (》=awg P(z,x)q(zlx) q(zlx)P(zlx) dz q(zlx) ( + q(zlx)log dz P(zlx) ≥0 KL(q(zx)P(zx)) ≥∫eog dz =Eg()llog P(x,z) g(zx) lower bound EncoderVAE: Lower bound of 𝑙𝑜𝑔𝑃 𝑥 𝑙𝑜𝑔𝑃𝜃 𝑥 = න 𝑧 𝑞 𝑧|𝑥 𝑙𝑜𝑔𝑃 𝑥 𝑑𝑧 𝑞 𝑧|𝑥 can be any distribution = න 𝑧 𝑞 𝑧|𝑥 𝑙𝑜𝑔 𝑃 𝑧, 𝑥 𝑃 𝑧|𝑥 𝑑𝑧 = න 𝑧 𝑞 𝑧|𝑥 𝑙𝑜𝑔 𝑃 𝑧, 𝑥 𝑞 𝑧|𝑥 𝑞 𝑧|𝑥 𝑃 𝑧|𝑥 𝑑𝑧 = න 𝑧 𝑞 𝑧|𝑥 𝑙𝑜𝑔 𝑃 𝑧, 𝑥 𝑞 𝑧|𝑥 𝑑𝑧 + න 𝑧 𝑞 𝑧|𝑥 𝑙𝑜𝑔 𝑞 𝑧|𝑥 𝑃 𝑧|𝑥 𝑑𝑧 𝐾𝐿 𝑞 𝑧|𝑥 ||𝑃 𝑧|𝑥 ≥ න 𝑧 𝑞 𝑧|𝑥 𝑙𝑜𝑔 𝑃(𝑧,𝑥) 𝑞 𝑧|𝑥 𝑑𝑧 ≥ 0 = E𝑞 𝑧|𝑥 [𝑙𝑜𝑔 𝑙𝑜𝑤𝑒𝑟 𝑏𝑜𝑢𝑛𝑑 𝑃 𝑥, 𝑧 𝑞 𝑧|𝑥 ] Encoder