Nhoj Morley

Nhoj Morley

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Joined 22-02-2005

11 October 2018 07:51

Bayesian stuff sounds so genteel. What is the probability of a given post being the last word?

mapadofu

mapadofu

Total Posts: 626

Joined 20-07-2017

11 October 2018 17:57

So I got around to reading the Goodman paper https://courses.botany.wisc.edu/botany_940/06EvidEvol/papers/goodman2.pdf

I believe that what we had been talking about is this problem’s analog of this observation:

“The hypothesis with the most evidence for it has the maximum mathematical likelihood, which means that it predicts the observed data best. If we observe a 10% difference between the cure rates of two treatments, the hypothesis with the maximum like- lihood would be that the true difference was 10%. In other words, whatever effect we are measuring, the best-supported hypothesis is always that the un- known true effect is equal to the observed effect”

I.e., the maximum likelihood hypothesis for unknown true state of the world is the one that matches what was reported.

For various reasons this paper structures the problem such that the likelihood factors tend to be <1 (are always?) but this is a function of choosing to put the null (no effect) hypothesis in the numerator. But note “Whether the null hypothesis likelihood is in the top or bottom of the ratio depends on the context of use.”. If Sam had said something along the lines of “testifying that an accomplice was present decreases the likelihood of false testimony”, then I would have been talking about p(r=1| false, a=null)/p(r=1|true, a=1), which is (afaict) always less than one, and all of the complaints about needing a likelihood ratio <1 would have never come up.

TheAnal_lyticPhilosopher

TheAnal_lyticPhilosopher

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12 October 2018 06:15

[Gasp].

Since you are making *new* mistakes using *my *citation that could throw off readers who might stand to learn something from this thread…

But note “Whether the null hypothesis likelihood is in the top or bottom of the ratio depends on the context of use.” If Sam had said something along the lines of “testifying that an accomplice was present decreases the likelihood of false testimony”, then I would have been talking about p(r=1| false, a=null)/p(r=1|true, a=1), which is (afaict) always less than one, and all of the complaints about needing a likelihood ratio <1 would have never come up.

Setting aside the fact that logically Sam said just that, so nothing should change, on your new formulation the inverse problem would come up: the Bayes Factor would be fixed at* less* than 1 and therefore the “model” would not even be a test. The first principle—the very first, representing the core logic of a Bayesian hypothesis test—is that the Bayes Factor take on *any* value greater than 0, with a value of 1 meaning equal probability between the two hypothesis, a value greater >1 indicating the numerator, and a value < 1 indicating the denominator. Switching the null to the bottom only means < 1 indicates the null instead of >1. But on your modeled test, the null will still always be affirmed, never the alternative, hence nothing is* tested* and nothing new determined.

If you are calling p(r=1| false, a=null) the null and putting it the numerator, then you’ll always affirm the alternative and never affirm the null, and so on.

(The other two errors—*a priori* structuring the sample space with a specific distribution via a “probabilistic model” and the contradiction built into the model you construct—still apply, no matter which hypothesis you call the null, or where you put the null once specified.)

Your fundamental error here is this ignorance: “I don’t know where this requirement on likelihood ratios spanning from<1 to >1 comes from.” This can only be because you do not really know what a Bayesian hypothesis test is, even though you say here: “This is pretty straightforward Bayesian hypothesis testing stuff.”

I regret pointing this out so directly, I honestly do. You seem to be working in good faith. But you are working in error—a horribly misguided error. I don’t mean this personally, and I hope you don’t take it that way, but I just can’t let such a horribly misguided error slip by and still entertain the hope that future readers might learn something from this exchange—not when you use my citation to make it. I’ve put some thought into these posts, and I’d hate to see that effort diminished. In any case, as patronizing as it sounds (and I don’t mean it that way), I suggest you take the time to reconsider your model in light of the criticisms I’ve made, then re-approach the Ford-Kavanaugh question in* bona fide* Bayesian terms.

Since I think informal Bayesian reasoning is how people approach issues like this, any opportunity to explore that for myself would be welcome.

mapadofu

mapadofu

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12 October 2018 09:32

I the class of problems considered by Goodman, how does one get a likelihood ratio >1?

As far as I can tell, In the problem formulated there, you always end up with a likelihood ratio <=1 between the null (no difference) hypothesis and the ML hypothesis (true difference = observed sample difference). Same “one sidedness” problem as you criticize in my model.

The structure of clinical medical trials are a limited subset of the range of problems that comprise Bayesian hypothesis testing, so I don’t see the need for all possible observables across all possible models would need to conform to the from <1 to >1 criterion you have adopted. Even Goodman’s case doesn’t satisfy that.

TheAnal_lyticPhilosopher

TheAnal_lyticPhilosopher

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14 October 2018 01:11

As far as I can tell, in the problem formulated there, you always end up with a likelihood ratio <=1 between the null (no difference) hypothesis and the ML hypothesis (true difference = observed sample difference). Same “one sidedness” problem as you criticize in my model.

You are reading Goodman too narrowly. He says quite specifically that the “the Bayes factor is not a probability itself but a ratio of probabilities, and it can vary from zero to infinity.” He focuses only on BF < 1 because of the conventions of experimentation, specifically in relating it to non-Bayesian methods (i.e. frequentists) that rely on low p-values for rejecting the null. In experiments in frequentist circles—his audience—one focuses on rejecting the null, never on affirming it. This focus is even clearer in the first article of the pair, with us referring here to the second of the two.

in the class of problems considered by Goodman, how does one get a likelihood ratio >1?

Goodman offers minimum Bayes Factors for various p-values, like p= .1 is BF .26, or p=.05 is BF .15. These minimum BFs are what would act as evidence *against* the null in any hypotheses test. For evidence *for* the null, one would use the same basic formula for the minimum BF to get the maximum BF. So, for instance, for p-values > .5 one would get p=.9 as a BF 3.8, and for p=.95 a BF 6.8—i.e. BF >1. Goodman probably doesn’t do these computations because he’s interested in the standard focus on* rejecting* the null, not *affirming* it, the latter being a fallacy in frequentist circles—his target audience. But for* any *given hypothesis test with a p >.5, the BF would be >1. And since *any * test can have any p-value from 0 < p < 1, the full range of BF from <1 to >1 (or 1 for p=.5) will *always *be the case when comparing the null and the alternative in *any* hypothesis test, even for Goodman.

From another article I have: “The fact that Bayes factor analyses can communicate evidence either in favor of or against null hypotheses means that these analyses are an informative method for communicating statistical findings in a wider range of conditions than are NHST [null hypothesis statistical test] analyses.” Note either *for * or* against* the null, meaning BF <1 (against) and BF >1 (for) for any given test (or vice versa, if the null is in the denominator). The way you structure your test, it is *always *and can *only* be *for *the null, no matter what the results are; therefore it’s either an invalid or non-informative test, or both.

The structure of clinical medical trials are a limited subset of the range of problems that comprise Bayesian hypothesis testing, so I don’t see the need for all possible observables across all possible models would need to conform to the from <1 to >1 criterion you have adopted.

This is an unnecessary qualification based on a fundamental misunderstanding of what Bayesian hypothesis testing is and does. A Bayesian hypothesis test is a Bayesian hypothesis test, whatever the field. Your fixation on these *apriori* models with p (D|H)=1 for the null, then testing an alternative against that null, and thereby restricting *apriori* the proper range of BF values is the statistical equivalent of modeling the orbit of the earth after denying the heliocentricity of the solar system. Absent the proper range of BF from <1 to >1, the “model” and “test” is completely uninformative, for a hypothesis test that by stipulation can *only affirm one hypothesis *is thereby meaningless *as a test*.

I can think of no case in any field where models that only affirm one of two hypotheses, *regardless of the data*, would be useful.

mapadofu

mapadofu

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14 October 2018 07:57

“Another paper” — reference please.

“Goodman offers minimum Bayes Factors for various p-values, like p= .1 is BF .26, or p=.05 is BF .15. These minimum BFs are what would act as evidence against the null in any hypotheses test. For evidence for the null, one would use the same basic formula for the minimum BF to get the maximum BF. So, for instance, for p-values > .5 one would get p=.9 as a BF 3.8, and for p=.95 a BF 6.8—i.e. BF >1. ”

Can you show your math to get theses results. You seem to be saying BF(p) =1/BF(1-p); do you have a proof of this? This is inconsistent with the expression for the minimal BF =exp[-Z^2/2] < 1 (for Gaussian statistics) presented in Goodman’s paper.

[ F(p) = p/(1-p) satisfies F(p)=1/F(1-p). But, following Goodman here, the Bayes factor is

P(data | H0)/p(data| Htest). It is not the case that p(data|H0) = 1- p(data|Htest). So if that is where you came up with your expressions for the BF when p>1/2, that is wrong.]

“And since any test can have any p-value from 0 < p < 1”. Nope; or at least not when you are applying Bayesian techniques to inference problems that you don’t design. Some real world problems do not even admit p values, e.g. inferring discrete categorical values so that the concept of a CDF, and thus p values, doesn’t even apply.

You don’t like it, or can’t understand it, but the situation we’re talking about is exactly like Goodman’s minimal Bayesian factors.

Indeed, getting more abstract, it is just a fact of life. We’re doing Bayesian hypothesis testing so we have p(r |h_n), the probability of observing result r conditioned on the hypothesis h0, h1… hn. (They might be discrete, they might have continuous parameters in them whatever). For any given r there exists at least one h_{ML}(r) that maximizes the conditional probability. It is alway the case that p(r | h_{ML}(r))/p(r | h0) >= 1 (I’ve picked h0 a particular null hypothesis);

Maybe you missed these features of the model. Note that in the Ford testimony model p(r=1| true, a=0)/p(r=1|false,a=null) =0 < 1. So across the e span of hypotheses for truthfully actual number of accomplices some are <1 and one is >1. Similarly, if you fix the number of accomplices and look at how the LR varies as a function of the number accomplices reported, it will vary from <1 to >1 under reasonable assumptions for the “lying” distribution (mainly, I figure, it’s monotonically decreasing, with enough spread to cover really wacky/delusional lies). These kinds of considerations seem like what you are talking about in the most recent post.

Your hang up seems to be that, in the case of considering Ford’s testimony, this ML hypothesis involves her telling the truth. I see this as a natural consequence of there being many ways to lie, but only one way to tell the truth being correctly represented when you formally model the problem in a Bayesian manner. Despite the fact that, according to you, almost any other model would be better, you have yet to offer any improvement.

TheAnal_lyticPhilosopher

TheAnal_lyticPhilosopher

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14 October 2018 08:50

“Another paper” — reference please.

“Goodman offers minimum Bayes Factors for various p-values, like p= .1 is BF .26, or p=.05 is BF .15. These minimum BFs are what would act as evidence against the null in any hypotheses test. For evidence for the null, one would use the same basic formula for the minimum BF to get the maximum BF. So, for instance, for p-values > .5 one would get p=.9 as a BF 3.8, and for p=.95 a BF 6.8—i.e. BF >1. ”

Can you show your math to get theses results. You seem to be saying BF(p) =1/BF(1-p); do you have a proof of this? This is inconsistent with the expression for the minimal BF =exp[-Z^2/2] < 1 (for Gaussian statistics) presented in Goodman’s paper.

[ F(p) = p/(1-p) satisfies F(p)=1/F(1-p). But, following Goodman here, the Bayes factor is

P(data | H0)/p(data| Htest). It is not the case that p(data|H0) = 1- p(data|Htest). So if that is where you came up with your expressions for the BF when p>1/2, that is wrong.]“And since any test can have any p-value from 0 < p < 1”. Nope; or at least not when you are applying Bayesian techniques to inference problems that you don’t design. Some real world problems do not even admit p values, e.g. inferring discrete categorical values so that the concept of a CDF, and thus p values, doesn’t even apply.

You don’t like it, or can’t understand it, but the situation we’re talking about is exactly like Goodman’s minimal Bayesian factors.

Indeed, getting more abstract, it is just a fact of life. We’re doing Bayesian hypothesis testing so we have p(r |h_n), the probability of observing result r conditioned on the hypothesis h0, h1… hn. (They might be discrete, they might have continuous parameters in them whatever). For any given r there exists at least one h_{ML}(r) that maximizes the conditional probability. It is alway the case that p(r | h_{ML}(r))/p(r | h0) >= 1 (I’ve picked h0 a particular null hypothesis);

Maybe you missed these features of the model. Note that in the Ford testimony model p(r=1| true, a=0)/p(r=1|false,a=null) =0 < 1. So across the e span of hypotheses for truthfully actual number of accomplices some are <1 and one is >1. Similarly, if you fix the number of accomplices and look at how the LR varies as a function of the number accomplices reported, it will vary from <1 to >1 under reasonable assumptions for the “lying” distribution (mainly, I figure, it’s monotonically decreasing, with enough spread to cover really wacky/delusional lies). These kinds of considerations seem like what you are talking about in the most recent post.

Your hang up seems to be that, in the case of considering Ford’s testimony, this ML hypothesis involves her telling the truth. I see this as a natural consequence of there being many ways to lie, but only one way to tell the truth being correctly represented when you formally model the problem in a Bayesian manner. Despite the fact that, according to you, almost any other model would be better, you have yet to offer any improvement.

Nope, mapadofu, I’m done. Just done. Anything else would be a pointless waste of my time. See you around.

mapadofu

mapadofu

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Joined 20-07-2017

14 October 2018 13:50

Yep, not providing references to the articles from which you pull quotes wastes all of our times. Plus now you are constructing mathematical fictions in trying to support your position. At least the Goodman paper was a useful read for me.

TheAnal_lyticPhilosopher

TheAnal_lyticPhilosopher

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16 October 2018 02:39

Although I’m done trying to pierce the ignorance behind pseudo-mathematized mumbo-jumbo, I thought, belatedly, this topic might still be interesting to readers who have followed along this far. Despite the obvious frustrations, it remains highly interesting to me.

This sub-thread began with a criticism of Harris’s suggestion that adding a “witness” to her account—in fact an accomplice, not a witness—improves the likelihood that Ford is telling the truth. From Harris’ statement mapadofu has constructed a “Bayesian model” that captures Harris’s intuition. But Bayesian statistics otherwise applied suggests a correction to both Harris’ intuition and mapadofu’s approach.

I think—stress *think*, without being entirely sure—that the following approach updates the probability that Ford is lying, given that she’s named multiple assailants. Its main point is to accommodate the relative rarity of sexual assaults with multiple assailants, given both the prior probability that Ford is lying and that she has fingered two assailants in her accusation—presuming, that is, that Judge would have partaken in, not just observed, the assault, had it gone further (or that in any case he is legally classified as an assailant).

Since we don’t know the prevalence of attempted rapes, but we can estimate the prevalence of rape, we can use rape statistics for this sexual assault problem.

For instance, we know that 1 in 5 women will be raped at some point in their lives, or 20%. This frequency includes all kinds of rape—multiple assailants or not. We also know that approximately 6% of these rapes are committed by more than one assailant. This means that any given woman’s chance of being raped by multiple assailants ‘in her lifetime’ is .012%. For tractability, let’s just assume Ford’s case qualifies as “a lifetime.” Let’s also say that 5% of rape accusations are false, whatever the accusation, for a prior odds of falsely claiming rape at .052—the middle range of the estimates of the false reports by women.

With this information in hand, a Bayes factor for lying can be computed using p (rape of any kind| false report) and p (rape by more than one assailant | false report). Specifically, one can compare the probability that a woman will experience some form of rape, even if lying about a particular rape, to the probability that she will raped specifically by multiple men, even if lying about a particular rape. In other words, provided the probability of rape and assuming the possibility of lying, one can update the probability of lying given a report accusing multiple rapists, relative to one naming just a single assailant.

Using the rape statistics above, the computation is straightforward: BF= p (R| F)/ p (R_multiple rapists| F), or .2/.012, for a value of 16.7. Taken with the prior odds, that gives a posterior odds of Ford lying at .87, or 46%. So, in light of the relative rarity of rapes with multiple rapists, the prior probability that Ford is lying increases from 5% to a much higher 46%, given that she has fingered multiple assailants.

This, of course applies to rapes, not sexual assaults, which are certainly far more frequent and may involve multiple assailants like the Kavanaugh/Judge situation more often than in 6% of cases. Using numbers that might be more probable for sexual assaults (1 in 3 women for assaults generally, with 10% of those like what Ford describes), the probability that she is lying only increases to 34%.

Whatever the exact numbers, this way of formulating the problem takes into account that sexual assaults with multiple assailants are quite rare—very rare, in fact—and intuitively, we know something like this knowledge needs to be taken into account when assessing Ford’s testimony. For intuitively, if someone told you they won the Mega Millions jackpot (odds of winning 1 in 260,000,000), or they told you they won a prize in the McDonald’s Monopoly game (odds of winning are 1 in 4), and you had to guess which statement was more likely to be a lie—knowing that people lie, say, 5% of the time— obviously you’d pick the Mega Millions claim as the likely lie. The above Bayesian updating attempts to formalize this intuition using the Bayes factor to update the probability that Ford is lying about an attempted sexual assault, given she’s accused Kavanaugh and Judge of a quite rare event among sexual assaults as such. Thus the *relative* believability of her statement can be assessed. Notably, it is significantly less believable than had she said only one assailant, provided the relatively rare probability of a multiple-assailant assault.

Whatever its defects or merits, this alternative to Harris’ intuition and mapadofus’ “model” avoids the rather absurd consequence that the more accusers Ford names, the *more* likely it is she is telling the truth, with 10 plus accused, say—a virtually non-existent event in the US—being *more* likely than a single accused (a distressingly common event). And anecdotally, two of the biggest rape-lie scandals in recent memory involved women naming multiple accusers (>6), suggesting evidence for this alternative approach. Or alternatively, the absurdity that, say, 25 rapists (a non-existent event in the US) makes her claim almost *unassailably* true suggests the above approach is more sensible than either Harris’ intuition or mapadofu’s model.

In any case, on this Bayesian reasoning who to believe between Kavanaugh and Ford becomes, given rape statistics, more or less a coin flip. Given reasonable approximations for sexual assaults, it leans somewhat more in Ford’s favor, but still not decisively so. Other factors, then, in their testimony would have to be decisive for determining who is telling the truth (assuming there even can *be* decisive factors at this point). But given the relative rarity of the event Ford describes, the possibility that she is lying increases, not decreases, with her reporting an accomplice.

TheAnal_lyticPhilosopher

TheAnal_lyticPhilosopher

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16 October 2018 02:45

Incidentally, for posterity….

Note for the Goodman paper that 1/e^-x = e^x. From this trivial property one can “prove” the various Bayes factors for affirming the null, since for affirming the null one wants the reciprocal of the minimum Bayes factor that serves as evidence against it, or 1/e^-(Z^2/2). From this it follows that the minimum Bayes factor as evidence *for *the null is e^(Z^2/2).

Goodman doesn’t do this simple inverse of rejecting the null with a minimum Bayes factor because given the context of the article and the conventions of statistical hypothesis testing, he’s not interested in convincing frequentists to *affirm* the null (a fallacy to them), only to use Bayes factors to reject it (as opposed to simply relying on p-values). He’s fighting one battle at a time, presumably. But the minimum Bayes factor for affirming the null with some specified ‘degree of confidence’ corresponding to a p-value > .5 can be easily computed from his appendix, leaving open the difficulty of integrating its logic into traditional null hypothesis testing terms. But in terms of affirming the null, as the observed difference or point estimate approaches 0, high p-values will increasingly serve as evidence for it in *Bayesian* terms, with a BF >1.

BF=1 for p=.5 follows from Z=0 in either the minimum Bayes factor formula for rejecting the null, or its inverse for affirming it. In z-score terms, when Z=0, p=.5 (and e^0=1). In both point estimation and comparative tests, p=.5 (BF=1) suggests both the null and the alternative are “equally likely” in both frequentist and Bayesian terms.

Justifying a citation that a Bayesian hypothesis test relies on a Bayes factor 1< or =1 or >1 to either reject or affirm “the null” is like justifying a citation for the heliocentricity of the solar system. But if anyone wants to check, here’s a representative *introductory* course from MIT, for which, as far as I know, there are no published exceptions (specifically see p. 6). mapadofu would have readers believe that fixing the Bayes factor at < 1 or >1 still *tests* two hypothesis, when everyone else in the world knows that fixed in either direction the Bayes factor can only *affirm *one of two hypothesis, not test them against each other in terms of a likelihood ratio. That Goodman suggests no such fixing follows from the universally acknowledged properties of statistical hypothesis testing and the simple conversion of its p-values into Bayes factors, as offered above.

Both of these points invoke difficult conceptual issues in statistical hypothesis testing, all of which are fascinating but none of which needs to be resolved in order to appreciate their elementary and uncontroversial basis.

Jan_CAN

Jan_CAN

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16 October 2018 08:40

I have no mathematical or statistical expertise, but it appears that this type of attempt to determine who is more likely to be telling the truth in this case (or others) would be unlikely to be useful. There are just too many assumptions that have to be made and variables that cannot be measured.

In addition, care should be taken in how terms like ‘sexual assault’ and ‘rape’ are used and viewed. Too often it seems, one will be considered worse than the other based on the type of sexual act committed rather than the level of violence, threat of violence, fear or harm done to the victim.

GAD

GAD

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Joined 15-02-2008

16 October 2018 08:49

I have no mathematical or statistical expertise, but it appears that this type of attempt to determine who is more likely to be telling the truth in this case (or others) would be unlikely to be useful. There are just too many assumptions that have to be made and variables that cannot be measured.

In addition, care should be taken in how terms like ‘sexual assault’ and ‘rape’ are used and viewed. Too often it seems, one will be considered worse than the other based on the type of sexual act committed rather than the level of violence, threat of violence, fear or harm done to the victim.

Yet that didn’t stop you from judging.

mapadofu

mapadofu

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Joined 20-07-2017

16 October 2018 10:15

Analytic, what are the hypotheses you are comparing? The ratio you wrote down is conditioned on “false” (false accusation), so it only tells us about the relative likelihood of different lies, but nothing about the relative likelihood of lying vs truthful testimony.

Normally the BF is p(x|h0)/p(x|h1) (or its inverse) x is what you’ve observed or measured, h0,h1 are two hypotheses under consideration. Note that we condition on two different hypotheses between the numerator and denominator, but evaluate the probability of the observation keeping that observation fixed. The expression you are using is, abstractly, p(r |x)/p(r’|x) (with r’ a subset of r). This isn’t something that you can just multiply into a prior likelihood ratioin order to get a posterior likelihood ratio.

What are (plausible) P(r|true) and p(r multiple | true) for your model?

I also find it odd that you are computing the likelihood of any report to a subset of that domain; I’m more used to thinking about LRs between disjoint outcomes.

Another point, you seem to have derived the probabilities assuming that the statistics for false accusations exactly match the true (or at least what we have as estimates of the true) statistics for actual events. That might be OK, especially as a starting point, but I’d expect some differences between the distribution of false reports and that of actual events.

Your discussion on extreme events doesn’t take into account the priors. Though the LR p(r=n|true, a=n)/p(r=n|false, a=none) may increase with increasing n (I’d say it will for plausible models) the prior LR p(true, a=n)/p(false, a=none) goes down as function of n, and I’d argue that the prior drops faster than the evidence LR increases (people can make up pretty extreme lies, so the tail of p(r=n|true, a=none) it fatter than that of p(true, a=n)). This is one of the issues with focusing on or only talking about just the evidence LR.

aside. is there a name for the variable in front of the conditioning bar? You can call the variables behind it “conditioning variables”, but I don’t know of an analogous term for the other).

mapadofu

mapadofu

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16 October 2018 10:37

Example 1 on page 6 of the lecture notes https://math.mit.edu/~dav/05.dir/class12-prep.pdf seems very similar to what we are discussing here. In that example there was type O blood, along with type AB at the scene, and Oliver is known to have type O. So that the probability of getting type O, given Oliver was present p(type O at crime scene|Oliver) =1 (this example problem stipulates that two people were at the crime scene, and each left traces of blood.). This is spelled it in words as “The data says that both type O and type AB blood were found. If Oliver was at the scene then ‘type O’ blood would be there. So P(D|S) is the probability that the other person had type AB blood. We are told this is .01, so P(D|S) = 0.01.“ and the computation is, implicitly, p(O | Oliver)*p(AB|random person)=1*0.01.

Note that by virtue of having a blood type that was present at the scene the LR for implicating him is >1 no matter what the type of the other blood is.

Conversely, observing type O means that the LR for any individual with type O is >1.

mapadofu

mapadofu

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16 October 2018 10:51

When I looked up “maximal Bayes factor” all that I found was discussions where the likelihood ratio was flipped, I.e. the null hypothesis in the denominator, but otherwise the same logic as Goodman’s paper. That’s just the context dependence mentioned before.

I get the sense that your maximal Bayes factor as evidence /for/ the null is something different. Is this widely used in the medical statistics community?

I’ve lost track with what you are talking about. First it looked like you were defining some Q(p)=1/BF(1-p), now you’re just talking about 1/BF(z) (p=p(z) for the types of problems Goodman discussed). I can’t tell if these are the same things or you are throwing around different ideas. Precise mathematical statements of what the maximal Bayes factor and how it relates to evidence for the null would be helpful.