Electrochemical applications of net-benefit analysis via Bayesian probabilities

Abstract

By means of three specific applications to electrochemical science, this paper demonstrates the usefulness of the net-benefit principle and Bayesian (posterior) probabilities in deciding whether equipment in an electrochemical laboratory or plant should be repaired or replaced.

Appendix

1.1 A brief illustration of Bayes’ theorem

For the sake of simplicity two mutually independent events: A ₁and B ₁with their complements A ₂ and B ₂ are considered; P(A ₁) + P(A ₂) =1, and P(B ₁) + P(B ₂) = 1. Bayes’ theorem yields the conditional probabilities of events A ₁and A ₂ occurring given that events B ₁and B ₂, respectively, have occurred. As shown by Equations (A.1) and (A.2), they depend on previously established A _i ; i = 1,2 – driven probabilities as

$$ P \langle A_{1} |B_{1} \rangle =\frac{P \langle B_{1} | A_{1}\rangle P(A_1 )}{P \langle B_{1}|A_{1}\rangle P(A_1 )+P \langle B_{1}| A_{2} \rangle P(A_{2} )} $$

(A.1)

and

$$ P \langle A_{2}|B_{2}\rangle =\frac{P \langle B_{2}| A_{2}\rangle P(A_2 )}{P \langle B_{2} | A_{1} \rangle P(A_1 )+P \langle B_{2}| A_{2}\rangle P(A_2 )} $$

(A.2)

Table 1 Establishment of likelihoods in the simplified analysis of electrode failure (Sect. 3.1)

Table 2 The effect of merit factor magnitudes on decision in Sect. 3.2; scale for f – vector elements: 0 (worst) – 10 (best)

Table 3 Summary of calculations for Sect. 3.3

with ${P \langle A_{2}|B_{1} \rangle = 1 -- P \langle A_{1}| B_{1} \rangle }$, and ${P \langle A_{1}| B_{2}\rangle = 1 - P \langle A_{2}|B_{2}\rangle }$ serving as shortcuts in lieu of further two equations similar to Equations (A.1) and (A.2). Proofs based on set – theoretic interpretations of probability can be found in a large variety of textbooks on probability and statistics.

A commercial potassium-ion selective electrode with a valinomycin membrane (active material [(C₁₀H₂₁0)₂PO ⁻₂ ] and 1 μmol dm⁻³ – 1 mol dm⁻³ range [10] serves for illustration. Major interferers with accurate indication are cesium and ammonium ions. The theorem applied to four events considered in Table 4 indicates a very high reliability of the instrument in the absence of the interfering species [${P \langle A_{2} | B_{2}\rangle \approx }$ 99.9%], but only a moderate reliability in their presence [${P \langle A_{1}| B_{1}\rangle \approx }$ 75.2%]. The very low conditional probabilities ${P \langle B_{2}|A_{1}\rangle }$ and ${P \langle A_{1}|B_{2} \rangle }$ support, however, the candidacy of this instrument for field use.

Table 4 Computations required by Bayes’ theorem (Appendix). Events: A ₁: interfering species (IS) present in the sample; A ₂: IS absent from the sample; B ₁: incorrect indication of potassium-ion content in sample; B ₂: correct indication of potassium-ion content in sample CIPIC: correct indication of potassium-ion content; IIPIC: incorrect indication of potassium-ion content