Paper VI: Bayesian conditioning as a normalized nonlinear operator

7 Bayesian conditioning as a normalized nonlinear operator

On this page · Visual guide: A dictionary with orientations · Permalink · Source line 723

Let \(p=(p_i)\) be a probability vector and let \(\lambda=(\lambda_i)\) be a nonnegative likelihood with positive evidence

\[ Z_\lambda(p)=\sum_i\lambda_ip_i. \]

Define

\[ \boxed{ {\mathcal B}_\lambda(p) = \frac{\lambda\odot p}{Z_\lambda(p)}. } \]

This map is nonlinear solely because of its scalar reconstruction factor.

Permalink · Source line 725

Theorem 7.1: Independent-likelihood composition

For nonnegative likelihoods \(\lambda\) and \(\mu\), assume \(Z_\lambda(p)>0\) and \(Z_{\lambda\odot\mu}(p)>0\). Equivalently, both successive updates have positive evidence. Then

\[ \boxed{ {\mathcal B}_\mu({\mathcal B}_\lambda(p)) = {\mathcal B}_{\lambda\odot\mu}(p). } \]

Read proof · Visual guide: Evidence changes shape and mass · Visual guide: A nonlinear semigroup of odds · Read statement and proof side by side · Permalink · Source line 739 · Claim record · Lean coverage · Code · Worked examples

Proof of Theorem 7.1

The first normalization appears in both numerator and denominator of the second update and cancels:

\[ \frac{\mu_i(\lambda_ip_i/Z_\lambda)}{\sum_j\mu_j\lambda_jp_j/Z_\lambda} = \frac{\mu_i\lambda_ip_i}{\sum_j\mu_j\lambda_jp_j}. \]

Back to statement · Read statement and proof side by side · Permalink · Source line 750 · Claim record · Lean coverage · Code · Worked examples

7.1 Packet evidence and reconstruction

On this page · Permalink · Source line 759

Let the state indices be partitioned into packets \(I_\alpha\). Define

\[ Z_\alpha=\sum_{i\in I_\alpha}\lambda_ip_i, \qquad m_\alpha=\frac{Z_\alpha}{\sum_\beta Z_\beta}, \qquad q_{\alpha,i}=\begin{cases} \lambda_ip_i/Z_\alpha,&Z_\alpha>0,\ i\in I_\alpha,\\ 0,&\text{otherwise}. \end{cases} \]

Permalink · Source line 761

Theorem 7.2: Bayesian packet reconstruction

\[ \boxed{ {\mathcal B}_\lambda(p) = \sum_\alpha m_\alpha q_\alpha. } \]

Only packets with positive evidence carry a conditional probability vector. A zero-evidence packet has zero posterior mass and contributes the zero vector. The nonlinear coupling between packets occurs only through the total evidence used to normalize the packet evidences.

Read proof · Visual guide: Evidence changes shape and mass · Read statement and proof side by side · Permalink · Source line 773 · Claim record · Lean coverage · Code · Worked examples

Proof of Theorem 7.2

If \(Z_\alpha=0\), nonnegativity implies \(\lambda_i p_i=0\) for every \(i\in I_\alpha\), so the reconstruction identity holds with both sides zero. For \(Z_\alpha>0\) and \(i\in I_\alpha\),

\[ m_\alpha q_{\alpha,i} = \frac{Z_\alpha}{Z}\frac{\lambda_ip_i}{Z_\alpha} = \frac{\lambda_ip_i}{Z}. \]

The packet supports are disjoint, so summing reconstructs the full posterior.

Back to statement · Read statement and proof side by side · Permalink · Source line 784 · Claim record · Lean coverage · Code · Worked examples

A finite prior and posterior split into two state packets. Evidence changes both the local packet shape and the global packet mass.

Theorem 7.3: Likelihood perturbation bound

Let \(p\) be fixed and let \(\lambda,\mu\ge0\) have positive evidences \(Z_\lambda,Z_\mu\). Then

\[ \boxed{ {\left\lVert{\mathcal B}_\lambda(p)-{\mathcal B}_\mu(p)\right\rVert}_1 \le \frac{2\sum_i|\lambda_i-\mu_i|p_i}{\min(Z_\lambda,Z_\mu)}. } \]

Read proof · Visual guide: Evidence changes shape and mass · Read statement and proof side by side · Permalink · Source line 803 · Claim record · Lean coverage · Code · Worked examples

Proof of Theorem 7.3

Add and subtract \(\mu\odot p/Z_\lambda\) and use the triangle inequality:

\[ {\left\lVert\frac{\lambda\odot p}{Z_\lambda}-\frac{\mu\odot p}{Z_\mu}\right\rVert}_1 \le \frac{{\left\lVert(\lambda-\mu)\odot p\right\rVert}_1}{Z_\lambda} + {\left\lVert\mu\odot p\right\rVert}_1\left|\frac1{Z_\lambda}-\frac1{Z_\mu}\right|. \]

The second term equals \(|Z_\lambda-Z_\mu|/Z_\lambda\) and is bounded by the same weighted \(L^1\) difference divided by \(Z_\lambda\). Interchanging \(\lambda\) and \(\mu\) gives the minimum-evidence form.

Back to statement · Read statement and proof side by side · Permalink · Source line 814 · Claim record · Lean coverage · Code · Worked examples

Source and conversion record

Source SHA-256: 736e615131f57daf341b6cb0600761d58950450947c85d433206148cca228333

Unsupported prose commands stop the build. The mathematical expression sequence is checked against the source, with manuscript macros expanded for display.

Conversion records

Read and explore

Proofs and implementation

About this work

Other resources