$$ \def\ba{\mathbf{a}} \def\bb{\mathbf{b}} \def\bc{\mathbf{c}} \def\bd{\mathbf{d}} \def\be{\mathbf{e}} \def\bf{\mathbf{f}} \def\bg{\mathbf{g}} \def\bh{\mathbf{h}} \def\bi{\mathbf{i}} \def\bj{\mathbf{j}} \def\bk{\mathbf{k}} \def\bl{\mathbf{l}} \def\bm{\mathbf{m}} \def\bn{\mathbf{n}} \def\bo{\mathbf{o}} \def\bp{\mathbf{p}} \def\bq{\mathbf{q}} \def\br{\mathbf{r}} \def\bs{\mathbf{s}} \def\bt{\mathbf{t}} \def\bu{\mathbf{u}} \def\bv{\mathbf{v}} \def\bw{\mathbf{w}} \def\bx{\mathbf{x}} \def\by{\mathbf{y}} \def\bz{\mathbf{z}} \def\bA{\mathbf{A}} \def\bB{\mathbf{B}} \def\bC{\mathbf{C}} \def\bD{\mathbf{D}} \def\bE{\mathbf{E}} \def\bF{\mathbf{F}} \def\bG{\mathbf{G}} \def\bH{\mathbf{H}} \def\bI{\mathbf{I}} \def\bJ{\mathbf{J}} \def\bK{\mathbf{K}} \def\bL{\mathbf{L}} \def\bM{\mathbf{M}} \def\bN{\mathbf{N}} \def\bO{\mathbf{O}} \def\bP{\mathbf{P}} \def\bQ{\mathbf{Q}} \def\bR{\mathbf{R}} \def\bS{\mathbf{S}} \def\bT{\mathbf{T}} \def\bU{\mathbf{U}} \def\bV{\mathbf{V}} \def\bW{\mathbf{W}} \def\bX{\mathbf{X}} \def\bY{\mathbf{Y}} \def\bZ{\mathbf{Z}} \def\balpha{\boldsymbol{\alpha}} \def\bbeta{\boldsymbol{\beta}} \def\bgamma{\boldsymbol{\gamma}} \def\bdelta{\boldsymbol{\delta}} \def\bepsilon{\boldsymbol{\epsilon}} \def\bvarepsilon{\boldsymbol{\varepsilon}} \def\bzeta{\boldsymbol{\zeta}} \def\btheta{\boldsymbol{\theta}} \def\bvartheta{\boldsymbol{\vartheta}} \def\biota{\boldsymbol{\iota}} \def\bkappa{\boldsymbol{\kappa}} \def\blambda{\boldsymbol{\lambda}} \def\bmu{\boldsymbol{\mu}} \def\bnu{\boldsymbol{\nu}} \def\bxi{\boldsymbol{\xi}} \def\bpi{\boldsymbol{\pi}} \def\brho{\boldsymbol{\rho}} \def\bsigma{\boldsymbol{\sigma}} \def\bvarsigma{\boldsymbol{\varsigma}} \def\btau{\boldsymbol{\tau}} \def\bupsilon{\boldsymbol{\upsilon}} \def\bphi{\boldsymbol{\phi}} \def\bvarphi{\boldsymbol{\varphi}} \def\bchi{\boldsymbol{\chi}} \def\bpsi{\boldsymbol{\psi}} \def\bomega{\boldsymbol{\omega}} \def\bGamma{\boldsymbol{\Gamma}} \def\bDelta{\boldsymbol{\Delta}} \def\bTheta{\boldsymbol{\Theta}} \def\bLambda{\boldsymbol{\Lambda}} \def\bXi{\boldsymbol{\Xi}} \def\bPi{\boldsymbol{\Pi}} \def\bSigma{\boldsymbol{\Sigma}} \def\bUpsilon{\boldsymbol{\Upsilon}} \def\bPhi{\boldsymbol{\Phi}} \def\bPsi{\boldsymbol{\Psi}} \def\bOmega{\boldsymbol{\Omega}} \def\bba{\mathbb{a}} \def\bbb{\mathbb{b}} \def\bbc{\mathbb{c}} \def\bbd{\mathbb{d}} \def\bbe{\mathbb{e}} \def\bbf{\mathbb{f}} \def\bbg{\mathbb{g}} \def\bbh{\mathbb{h}} \def\bbi{\mathbb{i}} \def\bbj{\mathbb{j}} \def\bbk{\mathbb{k}} \def\bbl{\mathbb{l}} \def\bbm{\mathbb{m}} \def\bbn{\mathbb{n}} \def\bbo{\mathbb{o}} \def\bbp{\mathbb{p}} \def\bbq{\mathbb{q}} \def\bbr{\mathbb{r}} \def\bbs{\mathbb{s}} \def\bbt{\mathbb{t}} \def\bbu{\mathbb{u}} \def\bbv{\mathbb{v}} \def\bbw{\mathbb{w}} \def\bbx{\mathbb{x}} \def\bby{\mathbb{y}} \def\bbz{\mathbb{z}} \def\bbA{\mathbb{A}} \def\bbB{\mathbb{B}} \def\bbC{\mathbb{C}} \def\bbD{\mathbb{D}} \def\bbE{\mathbb{E}} \def\bbF{\mathbb{F}} \def\bbG{\mathbb{G}} \def\bbH{\mathbb{H}} \def\bbI{\mathbb{I}} \def\bbJ{\mathbb{J}} \def\bbK{\mathbb{K}} \def\bbL{\mathbb{L}} \def\bbM{\mathbb{M}} \def\bbN{\mathbb{N}} \def\bbO{\mathbb{O}} \def\bbP{\mathbb{P}} \def\bbQ{\mathbb{Q}} \def\bbR{\mathbb{R}} \def\bbS{\mathbb{S}} \def\bbT{\mathbb{T}} \def\bbU{\mathbb{U}} \def\bbV{\mathbb{V}} \def\bbW{\mathbb{W}} \def\bbX{\mathbb{X}} \def\bbY{\mathbb{Y}} \def\bbZ{\mathbb{Z}} \def\cA{\mathcal{A}} \def\cB{\mathcal{B}} \def\cC{\mathcal{C}} \def\cD{\mathcal{D}} \def\cE{\mathcal{E}} \def\cF{\mathcal{F}} \def\cG{\mathcal{G}} \def\cH{\mathcal{H}} \def\cI{\mathcal{I}} \def\cJ{\mathcal{J}} \def\cK{\mathcal{K}} \def\cL{\mathcal{L}} \def\cM{\mathcal{M}} \def\cN{\mathcal{N}} \def\cO{\mathcal{O}} \def\cP{\mathcal{P}} \def\cQ{\mathcal{Q}} \def\cR{\mathcal{R}} \def\cS{\mathcal{S}} \def\cT{\mathcal{T}} \def\cU{\mathcal{U}} \def\cV{\mathcal{V}} \def\cW{\mathcal{W}} \def\cX{\mathcal{X}} \def\cY{\mathcal{Y}} \def\cZ{\mathcal{Z}} \def\ta{\textnormal{a}} \def\tb{\textnormal{b}} \def\tc{\textnormal{c}} \def\td{\textnormal{d}} \def\te{\textnormal{e}} \def\tf{\textnormal{f}} \def\tg{\textnormal{g}} \def\th{\textnormal{h}} \def\ti{\textnormal{i}} \def\tj{\textnormal{j}} \def\tk{\textnormal{k}} \def\tl{\textnormal{l}} \def\tm{\textnormal{m}} \def\tn{\textnormal{n}} \def\to{\textnormal{o}} \def\tp{\textnormal{p}} \def\tq{\textnormal{q}} \def\tr{\textnormal{r}} \def\ts{\textnormal{s}} \def\tt{\textnormal{t}} \def\tu{\textnormal{u}} \def\tv{\textnormal{v}} \def\tw{\textnormal{w}} \def\tx{\textnormal{x}} \def\ty{\textnormal{y}} \def\tz{\textnormal{z}} \def\tA{\textnormal{A}} \def\tB{\textnormal{B}} \def\tC{\textnormal{C}} \def\tD{\textnormal{D}} \def\tE{\textnormal{E}} \def\tF{\textnormal{F}} \def\tG{\textnormal{G}} \def\tH{\textnormal{H}} \def\tI{\textnormal{I}} \def\tJ{\textnormal{J}} \def\tK{\textnormal{K}} \def\tL{\textnormal{L}} \def\tM{\textnormal{M}} \def\tN{\textnormal{N}} \def\tO{\textnormal{O}} \def\tP{\textnormal{P}} \def\tQ{\textnormal{Q}} \def\tR{\textnormal{R}} \def\tS{\textnormal{S}} \def\tT{\textnormal{T}} \def\tU{\textnormal{U}} \def\tV{\textnormal{V}} \def\tW{\textnormal{W}} \def\tX{\textnormal{X}} \def\tY{\textnormal{Y}} \def\tZ{\textnormal{Z}} \newcommand{\defeq}{\overset{\text{def}}{=}} \newcommand{\smallR}{\mbox{\tiny R}} \newcommand{\smallF}{\mbox{\tiny F}} \newcommand{\softmax}{\mathrm{softmax}} \newcommand{\sigmoid}{\sigma} \newcommand{\KL}{D_{\mathrm{KL}}} \newcommand{\Var}{\mathrm{Var}} \newcommand{\Cov}{\mathrm{Cov}} $$

The Essence of Bayesian Flow Networks

September 20, 2024

For easy understanding, the reader can treat the variables as discrete variables. Without loss of generality, the formulation can be easily extended to continuous variables by swapping the summation with integration. This post introduces the Bayesian Flow Networks¹ (BFN) in a more simple language.

Alex Graves, Rupesh Kumar Srivastava, Timothy Atkinson, Faustino Gomez. “Bayesian flow networks”, arXiv 2023. ↩