Hypothesis testing: Defining H0, HA hypotheses so that ( H_A)_A' makes sense

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Discussion Overview

The discussion revolves around the definition and implications of null and alternative hypotheses (H0 and HA) in hypothesis testing, particularly focusing on the mapping of these hypotheses into decision sets and the potential for alternative operators. The scope includes theoretical considerations and comparisons between frequentist and Bayesian approaches.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants propose that the standard operator _A maps the null hypothesis H0 into a decision set, questioning the sense of (HA)_A given that H0 and HA are not exhaustive.
  • Others challenge the assertion that H0 and HA are not exhaustive, asking for clarification on why this would be the case.
  • One participant explains that in a frequentist framework, hypotheses cannot be accepted, only concluded as not rejected or rejected, which complicates the notion of exhaustiveness.
  • A later reply illustrates a specific example of hypotheses in a placebo-controlled test, arguing that while the hypotheses may be mutually exclusive and collectively exhaustive, the decisions made regarding those hypotheses are not exhaustive.
  • Another participant highlights the distinction between frequentist and Bayesian statistics, noting that frequentist methods focus on P(D|H0), while Bayesian methods allow for the calculation of P(H0|D).

Areas of Agreement / Disagreement

Participants express differing views on the exhaustiveness of H0 and HA, with some agreeing that the decision-making process is not exhaustive while others question this premise. The discussion remains unresolved regarding the implications of these differing perspectives.

Contextual Notes

The discussion touches on limitations in understanding the implications of hypothesis testing frameworks, particularly the differences between frequentist and Bayesian interpretations, and the complexity of decision outcomes in hypothesis testing.

WWGD
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TL;DR
The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive,
The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive, can we find an alternative operator, _A' , so that ( H_A)_A' makes sense? Isn't Pearson Neyman related to this? Hope I'm making sense.
Edit: I was motivated by a superficial similarity of the idea with double transposition of matrices M, with ## (M^{T})^{T}=M##, and just wanted to see if it made sense to talk about the " Alternative of the Alternative " returned the initial hypothesis.
 
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WWGD said:
TL;DR: The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive,

Since H0, HA aren't exhaustive
Why wouldn’t they be exhaustive?
 
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Dale said:
Why wouldn’t they be exhaustive?
Well, at least in the Frequentist set up , we don't get to actually accept, just conclude we don't have enough evidence to reject or not reject. I don't understand the Bayesian approach to tell.
 
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Ah, I see your question and my confusion.

Let’s say that you are doing a placebo controlled randomized test for a medicine to reduce the duration of a cold with ##\mu_p## being the mean duration of a cold with the placebo and ##\mu_d## being the mean duration of a cold with the drug.

Usually the hypotheses would be ##H_0: \ \mu_p=\mu_d## and ##H_A: \ \mu_p\ne \mu_d##. So those are exhaustive. No matter the actual values of ##\mu_d## and ##\mu_p## exactly one of those two hypotheses will be true. So the hypotheses are mutually exclusive and collectively exhaustive.

But you are looking not at the hypotheses but the decisions. And I agree with you there.

So you could make a decision to reject a hypothesis, fail to reject a hypothesis, accept a hypothesis, or fail to accept a hypothesis. So that is four possibilities for each of two hypotheses for a total of sixteen possible outcomes. Maybe accept both is not a possibility so maybe it is fifteen. But certainly reject and fail to reject the null hypothesis is not exhaustive.

Is that more or less what you meant?
 
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With frequentist statistics you cannot speak of the probability of hypotheses, so you can only calculate ##P(D|H_0)##. With Bayesian statistics the big difference is that you can calculate ##P(H_0|D)##
 
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