Comments on <i>Divergence vs. Decision P‐values</i>

The distinction between the two uses of p-values described by Professor Greenland is related to distinct interpretations frequentist probability—that is, probability used describe a random event. I will illustrate with simple example. In North Carolina Pick-4 lottery, 10 ping pong balls labeled digits from 9 = 0 , 1 … $$ {I}_9=\left\{0,1,\dots, 9\right\} are mixed in clear container and opening door allows single ball be selected. Prior door, blown air mixes making equally likely selection each plausible. This repeated three identical containers obtain remaining digits. If winning ticket defined as one where sum four exceeds 28, state can charge $5 for $100 prize expect profit. There 330 4 1{0}^4 possible outcomes 28 so expected value . 033 × $ 100 3 30 0.033\times \$100=\$3.30 calculation requires no sampling but it natural interpret this long run. For an individual holder, all that required given equal chance selected drawing associated his ticket. holder does not need imagine sequence draws just cancer patient consider 5-year periods understand 30% survival. Using terminology Vos Holbert (2022), scope specific while generic. uniform distribution on 4-tuples {I}_9^4={I}_9\times {I}_9\times {I}_9 provides model data generation process. most inference applications, unknown population modeled rather than process generated data. We modify example inference. told lottery draw we infer whether came NC or A also has contains 8 labels 7 {I}_7=\left\{0,1,\dots, 7\right\} 29 other information given. reduction-to-contradiction argument establishes result lottery. Premise: produced our data; every belongs set \left\{0,1,\dots, 28\right\} ; set; conclusion: contradiction means impossible premise true. deductive work if 28. Logical certainty longer sums less still provide evidence, varying degrees, regarding which was used. reduction-to-incredibility above deduction 28; {8}^4 only produces large 4-tuple had being selected; / < 00025 1/{8}^4<0.00025 unlikely observation makes doubtful An important second step likely. Without say upper 0.025 percentile ordered their sum, cannot 0.00025. contrast, valid even hand-picked; randomization plays role. conclusion inductive argument, word “unlikely” refers stochastic obtaining “doubtful” describes degree-of-belief While these quantities—observations have occurred would create greater doubt—failure lead confusion, especially when numeric probability, 1/{8}^4 assign measure one's doubt absence any lotteries. p-value, obtained measurable function so, definition, variable. All functions variables. However, adjective use function, namely Random variables distributions relevant question. Although role definition distributions, relevance depend how observed sample population. As process, limiting relative frequency interpretation hypothetical imagining when, fact, actual may come That samples do population, did, infinite. more realistic dichotomous attribute say, high blood pressure (BP). pair frequencies attributes, ( − p pop ) \left(1-{p}_{\mathrm{pop}},{p}_{\mathrm{pop}}\right) {p}_{\mathrm{pop}} proportion BP. Bernoulli family \left\{\left(1-p,p\right),0<p<1\right\} models distribution. support \left\{0,1\right\} then n -fold convolution \left(1-p,p\right) binomial B B\left(n,p\right) placing mass y \left(\genfrac{}{}{0ex}{}{n}{y}\right){p}^y{\left(1-p\right)}^{n-y} rational considering (with replacement) size calculating sum.1 When probabilities. Convolution extends relationship case real number unit interval. key here part mathematics don't involve randomization. B\left(1,p\right) good sum. extreme tail B\Big(n,p necessarily mean should considered unlikely. saw example, fact area probability. same true general. Now replaced models, involves continuum null hypotheses arguments. p-value point premise. justifies areas Interpreting describing inclined variable, problematic hypotheses. infinite what give recognize there p-value. plethora such make confusing. My final comment concerns geometry. Generalized estimators Godambe's (1960) estimating useful studying inferential properties different methods defining p-values. These generalized form vector bundle over statistical manifold tangent bundle. Details (2022).