Probability Dynamics

Abstract

‘Probability dynamics’ (PD) is a second-order probabilistic theory in which probability distribution d ^X = (P(X ₁), . . . , P(X _m )) on partition U ^X _m of sample space Ω is weighted by ‘credence’ (c) ranging from −∞ to +∞. c is the relative degree of certainty of d ^X in ‘α-evidence’ α ^X=[c; d ^X] on U ^X _m . It is shown that higher-order probabilities cannot provide a theory of PD. PD applies to both subjectivist and frequentist theories. ‘Straight PD’ (SPD) produces associative and commutative mergers of evidence, in which evidences of positive credence are mutually reinforcing. ‘Offsetting PD’ (OPD) sets off conflicting evidences against each other. Subjectivist PD is a quantified second-order logic of action. Frequentist PD relates to descriptions of physical states of affairs. Acceptance of evidence α ^X₁ = [c ₁; d ^X₁ ] at t ₁ updates α ^X₀ = [c ₀; d ^X₀ ] at t ₀ into SPD-merger $\alpha_0^{X}\oplus \alpha_1^X$ or OPD-merger $\alpha_0^X\diamond \alpha_1^X$. Given ‘co-evidence’ $E_0^{XY} = [c_0;P_0(XY),P_0(X\overline{Y}),P_0(\overline{X}Y),P_0(\overline{XY})]$ at t ₀ < t ₁, ‘indirect’ PD accepts evidence $\widetilde{\alpha}_1^{Y} = [\widetilde{c}_1; \widetilde{d}_1^{Y}]$ at t ₁ and produces support $\hat{\alpha}_1^{X}$ for update α ^X₁₀ = [c₁₀;d ^X₁₀ α ^X₀ such that $\alpha_{10}^X = \alpha_{0}^X\oplus \hat{\alpha}_{10}^X$ in SPD and $\alpha_{10}^X= \alpha_{0}^X\diamond \hat{\alpha}_{10}^X$ in OPD. For binary X and Y, with α ^X₀ = [c ₀; P ₀(X)] at t ₀ (short-hand for $[c_0;P_0(X),P_0(\overline{X});X,\overline{X}]$) and $\widetilde{\alpha}_1^{Y}= [\widetilde {c}_1;\widetilde{P}_1(Y)]$ the accepted evidence, the support is $\hat{\alpha}_1^{X}=[\hat{c}_1; \widetilde{P}_1(Y)P_0(X|Y)+\widetilde{P}_1(\overline{Y})P_0(X|\overline{Y})]$; $\hat{c}_1=|\rho_0(X, Y)|c_0\widetilde{c}_1/[c_0+(1-|\rho_0(X, Y)|)\widetilde{c}_1]$; where ρ₀(X, Y) is the correlation coefficient of X and Y, and update α ^X₁₀ of α ^X₀ is $P_{10}(X)=[c_0P_0(X)+\hat{c}_1\hat{P}_1(X)]/ (c_0+\hat{c}_1); \quad c_{10} = \lambda(c_0+\hat{c}_1)$ with ‘accord’ λ = 1 in SPD and $\lambda=1-2|P_0(X)-\hat{P}_1(X)|\sqrt{c_0\hat{c}_1}/(c_0 + \hat{c}_1)$ in OPD.

As $\hat{c}_1/c_0\to \infty$, $\tilde{P}_1(Y)$ tends toward update P ₁(Y) of P ₀(Y), but P ₁₀(X) does not converge toward ‘ $P_1(Y)P_0(X|Y)+P_1(\overline{Y})P_0(X|\overline{Y})$’ of ‘probability kinematics’. Therefore PD is not compatible with probability kinematics. A process of ‘normalization’ interprets ‘β-evidence’ [[c ₁; q ₁]& . . . &[c _m ; q _m ]; U ^X_m ]] with [c _j ; q _j ] on $\{X_j, \overline{X}_j\}$ and m ≥ 3, as an α-evidence on {X ₁, . . . , X _m }. It is shown that SPD- and OPD-updates can be derived from updated cumulative functions. Time-biased updates are discussed. A PD-based theory of confirmation (PDCT) is presented