A Perfect Time to Binge Read the (Strong) Likelihood Principle

.

An essential component of inference based on familiar frequentist notions: p-values, significance and confidence levels, is the relevant sampling distribution (hence the term sampling theory, or my preferred error statistics, as we get error probabilities from the sampling distribution). This feature results in violations of a principle known as the strong likelihood principle (SLP). To state the SLP roughly, it asserts that all the evidential import in the data (for parametric inference within a model) resides in the likelihoods. If accepted, it would render error probabilities irrelevant post data.

SLP (We often drop the “strong” and just call it the LP. The “weak” LP just boils down to sufficiency)

For any two experiments E₁ and E₂ with different probability models f₁, f₂, but with the same unknown parameter θ, if outcomes x* and y* (from E₁ and E₂ respectively) determine the same (i.e., proportional) likelihood function (f₁(x*; θ) = cf₂(y*; θ) for all θ), then x* and y* are inferentially equivalent (for an inference about θ).

(What differentiates the weak and the strong LP is that the weak refers to a single experiment.)

Violation of SLP:

Whenever outcomes x* and y* from experiments E₁ and E₂ with different probability models f₁, f₂, but with the same unknown parameter θ, and f₁(x*; θ) = cf₂(y*; θ) for all θ, and yet outcomes x* and y* have different implications for an inference about θ.

For an example of a SLP violation, E₁ might be sampling from a Normal distribution with a fixed sample size n, and E₂ the corresponding experiment that uses an optional stopping rule: keep sampling until you obtain a result 2 standard deviations away from a null hypothesis that θ = 0 (and for simplicity, a known standard deviation). When you do, stop and reject the point null (in 2-sided testing).

The SLP tells us  (in relation to the optional stopping rule) that once you have observed a 2-standard deviation result, there should be no evidential difference between its having arisen from experiment E₁, where n was fixed, say, at 100, and experiment E₂ where the stopping rule happens to stop at n = 100. For the error statistician, by contrast, there is a difference, and this constitutes a violation of the SLP.

———————-

Now for the surprising part: Remember the chestnut from my last post where a coin is flipped to decide which of two experiments to perform?  David Cox (1958) proposes something called the Weak Conditionality Principle (WCP) to restrict the space of relevant repetitions for frequentist inference. The WCP says that once it is known which E_i produced the measurement, the assessment should be in terms of the properties of the particular E_i. Nothing could be more obvious.     

The surprising upshot of Allan Birnbaum’s (1962) argument is that the SLP appears to follow from applying the WCP in the case of mixture experiments, and so uncontroversial a principle as sufficiency (SP)–although even that has been shown to be optional, strictly speaking. But this would preclude the use of sampling distributions. J. Savage calls Birnbaum’s argument “a landmark in statistics” (see [i]).

Although his argument purports that [(WCP and SP) entails SLP], we will see that data may violate the SLP while holding both the WCP and SP. Such cases also directly refute [WCP entails SLP].

(E,z) ⇒ Infr_E(z) is to be read “the inference implication from outcome z in experiment E” (according to whatever inference type/school is being discussed).
If E is within error statistics, for example, it is necessary to know the relevant sampling distribution associated with a statistic. If it is within a Bayesian account, a relevant prior would be needed.

[iv] I’ve blogged these links in the past; please let me know if any links are broken.