Simple Probability Estimation with more than two outcomes

[Swammerdami]

In another thread, there seemed to be disagreement about how to calculate simple probabilities when there are three or more outcomes to consider.
It seemed best to start a math thread focusing on a very simple example depicting the confusion.

The three outcomes are S₁, S₂, and Not-S. But suppose we only care whether EITHER S₁ or S₂ is the outcome; we have no desire to distinguish between the two S cases.

I claim, with a simple demonstration as shown below, that to develop the best estimates we often need to handle the separate outcomes separately. But some in the thread seemed to think that "Bayesian analysis"^* would magically glean what can be gleaned.

* - The Bayes Theorem is a very trivial fact, trivially proven. While very useful it is so trivial an analyst will often invoke it without naming it. (When your hear a layman invoking "Bayesian analysis," as though with pride and awe, the grasp is too often over-confident.

I've prepared a table showing the probabilities needed for a computation. We are given prior probabilities for each of the three outcomes. We are given two clues, A and B, assumed to be independent and their probabilities conditioned on each outcome. I show the correct conclusions ("Swammi") and the inferior results ("Carrier"). I don't know or care whether Richard Carrier's grasp of simple math is really this bad, but in the other thread his method was described this way.

With all the net probabilities 50-50, when S₁ and S₂ are NOT separated, A and B provide us with NO useful information. "Carrier" doesn't even see the two final columns in the table. The posterior probabilities are the same as the priors. Note that each "50%" can be considered EXACT (50.0000%) -- there is no "range" of estimates that might mitigate the flaw.

When you look at S₁ and S₂ separately, you can see that S₁ increases the chance of A and decreases the chance of B. S₂ vice versa. If A were true and B false (or vice versa) the chance of Not_S would fall, but with both (or neither) true, Not_S is most likely.

There are many VERY smart posters at IIDB, but those with a good mathematical bent are in short supply! I hope every Infidel who feels qualified shows up and comments.

Outcomes	Not-S	S	S1	S2
Prior probabilities	50%	50%	25%	25%
P(A|.)	50%	50%	90%	10%
P(B|.)	50%	50%	10%	90%
A&B true --> Posterior probabilities (Carrier)	50%	50%	-	-
A&B true --> Posterior probabilities (Swammi)	73.5%	(26.5%)	13.2%	13.2%

 

[steve_bank]

A little clarification?

What do you mean by exactly 50% without variation? Does this mean the sequence is s,~s,s,~s.... ? Or is it like flipping coin as a random variable?

Bernoulli trial comes to mind.

As to comparing methods. In reliability engineering there are two common statistical reliability rests, exponential and sequential. Without getting into details if you take experimental data and apply both analysis you will get different results. Which method you use depends on the hypothesis of what you are trying to prove.

With Bayesian or any statistical techniques you have to start with an hypothesis you want to prove. If I were to use Bayesian I would start with an hypothesis, given the known reliability of a system is the new reliability less than or equal to some number with the addition of new components, within a statistical confidence interval.

Point being different statistical methods applied to the same data or problem does not necessarily yield the same results.

One of my first projects was setting up a field statistical reliability program, and a manufacturing statistical process control system

 

[steve_bank]

Restating the Jesus problem I posed on the other tread. My intent was to show that Bayesian does not seem to apply. It is simple probailities.

Given Jesus has a 50% probability of existng, and if he existed there were three choices as to who he was.

There is a room with three doors labeled 1,2 and 3.

You walk into the room and flip a coin. If tails you turn around and leave.
If heads you randomly pick a ball and go through the door with the number on the ball.

What percentage of the time do you go through one of the doors?

P1 =1/2,p2 = 1/3
The probability of going through any of the doors is 50% of the time you enter the room. The probability of going through either 1,2,or 3 is .1/6 0r 5/3. P1 * p2. Probabilities must always add up.


I don't see what Swami eeans by sea rated.

For Pa and Pb s1 + s2 must equal .5. The table does not make sense for s1 90% and s2 = 10% with s at 50% or .5 probability. Do you mean Ps1= .9*.5 = .45 and Ps2 = .1*.5 = .05?

 

[Swammerdami]

I don't know how many thousands of trips I've made to that famous casino in Monte Carlo^* but just for you Steve I'll make another visit. As you can see I've done the dice rolls 100 Million times -- these computers are a bit faster than when you and I were young and foolish, Steve -- I hope this is enough to avoid an accusation that my result is just "anecdotal."

I normally avoid games of chance for something so trivial, but with the mistaken views sometimes seen, I'm being careful. I've also coded this VERY prosaically so easy to understand.

* - My visits to Monte Carlo are purely metaphorical. Although I've been in physical casinos in 12 countries, the closest I've ever been to the French Riviera were my delightful matinees at Madame Gigi's Pleasure Salon in Marseille.

#include        <stdlib.h>
#include        <stdio.h>

int     State, Clue_A, Clue_B;
int     Iter, Pertin;
int     Phits[3], Thits[3], TAhits[3][2], TBhits[3][2];

#define NUMITER         100000000
#define RCHOICE(p)      (random() < (p) * RAND_MAX)

int doit(void)
{
        State = RCHOICE(0.5) ? 0 : RCHOICE(0.5) ? 2 : 1;
        switch (State) {
        case 0:
                Clue_A = RCHOICE(0.5);
                Clue_B = RCHOICE(0.5);
                break;
        case 2:
                Clue_A = RCHOICE(0.1);
                Clue_B = RCHOICE(0.9);
                break;
        case 1:
                Clue_A = RCHOICE(0.9);
                Clue_B = RCHOICE(0.1);
                break;
        }
        return Clue_A && Clue_B;
}

int main(int argc, char **argv)
{
        for (Iter = 0; Iter < NUMITER; Iter++) {
                if (doit()) {
                        Pertin += 1;
                        Phits[State] += 1;
                }
                Thits[State] += 1;
                TAhits[State][Clue_A] += 1;
                TBhits[State][Clue_B] += 1;
        }
        printf("Priors: %f %f %f\n",
                        Thits[0] / (double)Iter,
                        Thits[1] / (double)Iter,
                        Thits[2] / (double)Iter);
        printf("CP(A|.): %f %f %f  ... %f\n",
                        TAhits[0][1] / (double)Thits[0],
                        TAhits[1][1] / (double)Thits[1],
                        TAhits[2][1] / (double)Thits[2],
                        (TAhits[1][1] + TAhits[2][1])
                                / (double)(Thits[1] + Thits[2]));
        printf("CP(B|.): %f %f %f  ... %f\n",
                        TBhits[0][1] / (double)Thits[0],
                        TBhits[1][1] / (double)Thits[1],
                        TBhits[2][1] / (double)Thits[2],
                        (TBhits[1][1] + TBhits[2][1])
                                / (double)(Thits[1] + Thits[2]));
        printf("Posteriors: %f %f %f\n",
                        Phits[0] / (double)Pertin,
                        Phits[1] / (double)Pertin,
                        Phits[2] / (double)Pertin);

        exit(0);
}

 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 

Execution Results:

Priors: 0.500363 0.249830 0.249807
CP(A|.): 0.499987 0.899896 0.099905  ... 0.499918
CP(B|.): 0.499286 0.100466 0.899655  ... 0.500042
Posteriors: 0.734962 0.132916 0.132122

The results agree with the numbers I gave in OP. Given my increasingly senescent brain, that comes as something of a relief!

 

[Swammerdami]

A little clarification?

What do you mean by exactly 50% without variation? Does this mean the sequence is s,~s,s,~s.... ? Or is it like flipping coin as a random variable?

In the other thread, the peculiar claim was made that the error I demonstrated in OP would be avoided by Carrier's use of probability "ranges." In the table and the simulation I don't explore WHY the numbers are what they are; they're just results from a "black box."

There are no ranges. P(A|S) as found in the simulation is 49.9959%. A longer stay at the casino would have produced a number even closer to the exact 50.00000%. No range. (Though even if there were, Carrier's results would still be distinctly wrong.)

Again, I do NOT seek to impugn Carrier. I'll guess even his meager grasp of probability theory would be adequate to understand the point. I'm just using "Carrier" as shorthand for a view expressed in the other thread.

~ ~ ~ ~ ~ ~ ~ ~

With the table and simulation code on display, I don't expect any serious objection to the simple arithmetic.

There are lots of interesting probability-based puzzles. Perhaps we can use the thread for some of those!

 

[Merle]

In another thread, there seemed to be disagreement about how to calculate simple probabilities when there are three or more outcomes to consider.

First, since we are discussing this in terms of what was said on another thread, those who are interested in seeing what was said on that thread might want to visit https://iidb.org/threads/estimating-the-probability-of-an-historic-jesus.26980/ . (At least I assume that is the thread we are talking about.)

I can't speak for Carrier, but I will give you the way I think Carrier would have responded if the data above were truly something that he found when he was estimating the range of the probability of Jesus being historical. ( in his book, https://www.amazon.com/Historicity-Jesus-Might-Reason-Doubt/dp/1909697494 )

The Bayesian formula would be:

P(S)/P(not.S) = P(H|b)/P(not.H|b) * P(A|S)/P(A|not.S)* P(B|S)/P(B|not.S)

where the notation P(x|y) means the probability of x being true if it is given that y is true.

Carrier uses upper and lower limit analysis. So in this case, I understand Carrier would calculate:

Upper limit P(S)/P(not.S) = P(H|b)/P(not.H|b) * P(A|S)/P(A|not.S)* P(B|S)/P(B|not.S) = 50/50 * .9/.1 * .9/.1 = 40.5
Lower limit P(S)/P(not.S) = P(H|b)/P(not.H|b) * P(A|S)/P(A|not.S)* P(B|S)/P(B|not.S) = 50/50 * .1/.9 * .1/.9 =0.006173

Converting this to P(S) we calculate the probability of S as:
Upper limit P(S) = 40.5/40.5+1= 97.6%
Lower limit P(S) = 0.006173/0.006173+1= 0.6%

We conclude that the probability of S is somewhere between 0.6% and 97.6%, which is practically meaningless.

Of course, in this instance, one can do a much better job of estimating if one first divides the analysis into two separate calculations, one calculating P(S1) and one calculating P(S2). And yes, when one sees something like this, then the obvious solution is to divide the analysis into two separate calculations.

But also, as I emphasized in the other thread, the fact that one can find an example that one's analysis can work much better if subdivided into two sub-calculations, that does not prove that all calculations work this way. If a simple calculation for the range for P(S) gives a precision that is good enough for our purposes, we can stop there.

This discussion came up in regards to historical Jesus research, in which it was claimed that Carrier's analysis was insufficient because he did not divide it into subgroups analyzing each scenario separately. I have acknowledged that this may have helped him reach greater precision, but who cares? Carrier's intent was not to calculate the exact odds that Jesus existed. Rather, as the title of his book declares, regarding the historicity of Jesus, "we might have reason for doubt." He did not need a precise calculation of the probability to 3 decimal places to conclude "we might have reason for doubt."

 

[Swammerdami]

In another thread, there seemed to be disagreement about how to calculate simple probabilities when there are three or more outcomes to consider.

First, since we are discussing this in terms of what was said on another thread, those who are interested in seeing what was said on that thread might want to visit https://iidb.org/threads/estimating-the-probability-of-an-historic-jesus.26980/ . (At least I assume that is the thread we are talking about.)


Carrier uses upper and lower limit analysis. So in this case, I understand Carrier would calculate:

Upper limit P(S)/P(not.S) = P(H|b)/P(not.H|b) * P(A|S)/P(A|not.S)* P(B|S)/P(B|not.S) = 50/50 * .9/.1 * .9/.1 = 40.5
Lower limit P(S)/P(not.S) = P(H|b)/P(not.H|b) * P(A|S)/P(A|not.S)* P(B|S)/P(B|not.S) = 50/50 * .1/.9 * .1/.9 =0.006173

Where does Carrier even get the .9's and .1's from? He's not even distinguishing S1 and S2 which is the only place where they arise in this toy problem.

We conclude that the probability of S is somewhere between 0.6% and 97.6%, which is practically meaningless.

Ya think?

But also, as I emphasized in the other thread, the fact that one can find an example that one's analysis can work much better if subdivided into two sub-calculations, that does not prove that all calculations work this way. If a simple calculation for the range for P(S) gives a precision that is good enough for our purposes, we can stop there.

Two points:

(1) Apparently you think the EXTREMELY trivial example in OP with the humongous total of TWO (2) clues -- or the closely related "getaway" problem in the other thread -- is a trickier more difficult problem than assessing the Historicity of Jesus. Oooooo-Kayyyy. Would you be horrified to learn that scholars contemplating the historicity problem might try to cope with THREE clues?? Or even more?????

(2) You didn't "emphasize" that in the other thread at all. You dug in your heels, adamantly insisting that Carrier and you were right to treat historicity and non-historicity as each single scenario.

This discussion came up in regards to historical Jesus research, in which it was claimed that Carrier's analysis was insufficient because he did not divide it into subgroups analyzing each scenario separately

Wrong. The discussion began in OP where I was laying general background for probability estimates, referring to Historicity just for definiteness. The discussion never got to the point where detailed scenarios were laid out, let alone a plurality.

Let us start with a standard mathematical formulation
P_historic = p(A₁ | E) + p(A₃ | E) + p(A₅ | E) + p(A₇ | E) + ...
P_non-historic = p(A₂ | E) + p(A₄ | E) + p(A₆ | E) + p(A₈ | E) + ...

I use the notation in the standard way: p(A_k | E) denotes the probability that scenario A_k is reality given the totality of evidence E. I arbitrarily assign odd index numbers to scenarios where Jesus is historical and even numbers for non-historical scenarios. The set { A_k } is a partition of possible realities: Exactly one of these scenarios matches the actual reality. The quotient P_hist / (P_hist + P_nonh) will give us the answer: the probability (or rather our best estimate thereof) that Jesus of Nazareth actually existed as an historical person.

This is all much easier said than done! ...

To which you objected:

What is the probability that Jesus of Nazareth actually existed as an historical person? (We shall assume that most of the Gospel stories are fictions, and that no miracles were performed. We simply wonder if there was a real person called Jesus of Nazareth who inspired the early Christians.)

Carrier defines this as the historicity hypothesis. This is the view he is testing. This is the view that, as he says in his title, we might have reason to doubt.

So we need not test A₁, A₃, etc. We are asking only one question: How likely is it that a historical Jesus as defined above existed?

 

[Swammerdami]

Please people; let's try to keep conversation focused on the craft of probability estimation, as correctly applied.

We have a VERY simple example in view where correct analysis (using three outcomes instead of two) correctly deduces 73.5% as the chance for Not.S. Whether Carrier's faulty methods would have produced the incorrect p = 50% or the useless 1% < p < 99% is not a useful topic for debate.

 

[Bomb#20]

Outcomes	Not-S	S	S1	S2
Prior probabilities	50%	50%	25%	25%
P(A|.)	50%	50%	90%	10%
P(B|.)	50%	50%	10%	90%
A&B true --> Posterior probabilities (Carrier)	50%	50%	-	-
A&B true --> Posterior probabilities (Swammi)	73.5%	(26.5%)	13.2%	13.2%

It looks like Carrier got his conclusion by taking for granted that clues A and B are uncorrelated. In your scenario they're strongly (negatively) correlated when S is true. But your calculation of the posterior probability as 73.5% appears to similarly take for granted that A and B are uncorrelated when S is false. You say A and B are "assumed to be independent"; but it's not clear what "assumed to be independent" means in a problem where they're correlated when S is true. Perhaps you and Carrier mean different things by "independent".

Suppose the Not-S column were similarly broken down into Not-S₃ and Not-S₄ subcolumns, with different probabilities for A and B in each. It seems to me depending on what numbers you put in the subcolumns you could get whole a range of posterior probabilities.

 

[Swammerdami]

Outcomes	Not-S	S	S1	S2
Prior probabilities	50%	50%	25%	25%
P(A|.)	50%	50%	90%	10%
P(B|.)	50%	50%	10%	90%
A&B true --> Posterior probabilities (Carrier)	50%	50%	-	-
A&B true --> Posterior probabilities (Swammi)	73.5%	(26.5%)	13.2%	13.2%

It looks like Carrier got his conclusion by taking for granted that clues A and B are uncorrelated. In your scenario they're strongly (negatively) correlated when S is true. But your calculation of the posterior probability as 73.5% appears to similarly take for granted that A and B are uncorrelated when S is false. You say A and B are "assumed to be independent"; but it's not clear what "assumed to be independent" means in a problem where they're correlated when S is true. Perhaps you and Carrier mean different things by "independent".

Of course A is going to be strongly correlated with the scenario you're considering -- it wouldn't be a useful clue otherwise. You can see from the simulation source code how I assume those variables behave -- they are as uncorrelated as can be and still comply with their stated probabilities.

Does Carrier even write about separate scenarios or speak of clue "independence"? You ARE quite right that correlations among real-world clues is a major stumbling-block requiring great care. But the above table is a toy illustration.

Suppose the Not-S column were similarly broken down into Not-S₃ and Not-S₄ subcolumns, with different probabilities for A and B in each. It seems to me depending on what numbers you put in the subcolumns you could get whole a range of posterior probabilities.

Well, yes! Difficult analyses can get difficult. I started this thread mainly to seek acknowledgment of that fact. Can I put you in the Agree column?

Note that I did NOT pursue separate scenarios for the Jesus problem. I just mentioned the possibility of multiple scenarios in the preliminary remarks of OP. This attracted vehement opposition, perhaps because Carrier's thinking had never gotten so far.

I HAVE mentioned several times, as an example of the need to analyze multiple scenarios of mythicism, the Chrestus ≠ Christ controversy. This has a big effect on the chronology. If mythicist experts can answer only "Maybe" the TWO cases will have to be pursued.. (In 3 or 4 threads on the topic, I have NEVER gotten a well-defined scenario, just successions of Maybes.)

 

[Bomb#20]

Well, yes! Difficult analyses can get difficult. I started this thread mainly to seek acknowledgment of that fact. Can I put you in the Agree column?

Heh. I think the problem as described was insufficiently precise to allow a definitive conclusion as to whether Carrier's result was incorrect.

Note that I did NOT pursue separate scenarios for the Jesus problem. I just mentioned the possibility of multiple scenarios in the preliminary remarks of OP. This attracted vehement opposition, perhaps because Carrier's thinking had never gotten so far.

In my experience Carrier is usually an idiot; but I haven't read his input on this topic.

I HAVE mentioned several times, as an example of the need to analyze multiple scenarios of mythicism, the Chrestus ≠ Christ controversy. This has a big effect on the chronology. If mythicist experts can answer only "Maybe" the TWO cases will have to be pursued.. (In 3 or 4 threads on the topic, I have NEVER gotten a well-defined scenario, just successions of Maybes.)

I quit paying attention to mythicism arguments years ago, out of exasperation that nobody ever seemed to define criteria for someone to qualify as the historical Jesus.

 

[steve_bank]

I agree it poorly defined.

What does s1 and s2 not separated mean? In statistical terms do you mean correlated vs uncorrelated(random) variables?

Pa does bot make sense in the table. if s is a random variable with two staes and s1 s2 are continget on s then s1 + s2 has to equal 50%.

Please explain Swami.

Wrte out the probailty equations fr Pa and Pb in the tables.

 

[Merle]

I quit paying attention to mythicism arguments years ago, out of exasperation that nobody ever seemed to define criteria for someone to qualify as the historical Jesus.

Richard Carrier is very clear on what would qualify as a historical Jesus:

1. An actual man at some point named Jesus acquired followers in life who continued as an identifiable movement after his death.
2. This is the same Jesus who was claimed by some of his followers to have been executed by the Jewish or Roman authorities.
3. This is the same Jesus some of whose followers soon began worshiping as a living god (or demigod).

If a person meets all three, then Carrier would say that is a historical Jesus exists. Is that not clear enough for you?

 

[Merle]

Please people; let's try to keep conversation focused on the craft of probability estimation, as correctly applied.

We have a VERY simple example in view where correct analysis (using three outcomes instead of two) correctly deduces 73.5% as the chance for Not.S.

We all agree that sometimes an analysis will be more accurate if you break it up into two separate scenarios.

I claim that there are some instances where breaking an analysis into two scenarios does not help you much.

If every time we see an analysis we need to break it into two scenarios, then do we need to break those into two more, and then two more again, until we have an infinite number of scenarios?

 

[Swammerdami]

We all agree that sometimes an analysis will be more accurate if you break it up into two separate scenarios.

I claim that there are some instances where breaking an analysis into two scenarios does not help you much.

Stated differently, you claim that the Mythicist and Historicist Jesus Hypotheses are even simpler than, and single-scenario compared with, the ostentatiously simplistic two-clue puzzle in OP.

Yeah. We get that that's your opinion.

If every time we see an analysis we need to break it into two scenarios, then do we need to break those into two more, and then two more again, until we have an infinite number of scenarios?

Wikipedia has a page devoted to Named Logic Fallacies. I don't know if the list includes "If we allow someone to divide a pie into two pieces, then he's going to divide it into infinitely many pieces." Some fallacies are just too stupid to bother naming!

 

[Swammerdami]

I agree it poorly defined.

What does s1 and s2 not separated mean? In statistical terms do you mean correlated vs uncorrelated(random) variables?

Pa does bot make sense in the table. if s is a random variable with two staes and s1 s2 are continget on s then s1 + s2 has to equal 50%.

Please explain Swami.

Wrte out the probailty equations fr Pa and Pb in the tables.

I'm not sure what your questions are..
P(A|.) is the label for one row; substitute the column label for the dot to get P(A|S), P(A|not.S), P(A|S1), P(A}S2).

Thus P(A|S1) = 90% means that in Scenario S1, A is true 90% of the time. (Never mind which is cause and effect, if any. Treat it as a pure math exercise with the probabilities given.)

Note that the source code provided ensures exactly that and may answer other questions as well. See something like

case 1: // S_1​

Clue_A = (random() < 0.90 * RAND_MAX);​

(I'd intended that source code to be easy-to-read, but looking now I see it assumes familiarity with C and its random() library.)

In the other thread's example. S is the proposition that Sam committed a certain robbery. One detective divides S into two scenarios (Sam hired George as his getaway driver OR Sam drove himself). Other detectives do not consider the getaway at all, due to oversight or perhaps afraid it might confuse them further.

As you can see from the table, the clues are useless for the detectives who do NOT subdivide into two scenarios. Clue B for example might be that George has a good alibi on the day of the robbery.

 

[Merle]

Stated differently, you claim that the Mythicist and Historicist Jesus Hypotheses are even simpler than, and single-scenario compared with, the ostentatiously simplistic two-clue puzzle in OP.

Yeah. We get that that's your opinion.

Wrong. I never said that the Historicist Jesus Hypothesis is simpler than your illustration. You just made that up. Please do not make up things about what other people say.

Again, your analysis is made with a feature that require it to be analyzed as two scenarios.

The fact that Carrier's analysis does not have this feature does not mean that his analysis does not have many other, different complexities that you illustration does not have.

Wikipedia has a page devoted to Named Logic Fallacies. I don't know if the list includes "If we allow someone to divide a pie into two pieces, then he's going to divide it into infinitely many pieces." Some fallacies are just too stupid to bother naming!

Uh, is it or is it not true that all scenarios need to be broken into two scenarios? Please answer.

As you know, I have contended emphatically that only some scenarios need to be split into two. Do you or do you not agree with me?

If you agree with me, then the fact that you found one scenario that has a feature that requires it to be split does not prove that Carrier's analysis needs to be split.

If you don't agree that only some scnarios need to be split, then yes, you yourself our saying that all scenarios must be split, and yes, that splitting would , by definition, go on ad infinitum.

So which way is it?

 

[Merle]

Where does Carrier even get the .9's and .1's from?

Huh? The .9s and .1 come from your opening post! And since Carrier most likely did not read your opening post, he didn't get those .9s and .1s.

You didn't "emphasize" that in the other thread at all. You dug in your heels, adamantly insisting that Carrier and you were right to treat historicity and non-historicity as each single scenario.

Huh? I was telling you that his analysis was sufficient for his purpose to show that we might have reason to doubt historicity. I readily admitted that there might be ways to make the analysis even better, such as by breaking it up into different scenarios. That does not change the fact that his analysis was adequate for his purposes.

If you think his analysis was wrong, then please state specifically what was wrong.

 

[Swammerdami]

Stated differently, you claim that the Mythicist and Historicist Jesus Hypotheses are even simpler than, and single-scenario compared with, the ostentatiously simplistic two-clue puzzle in OP.

Yeah. We get that that's your opinion.

Wrong. I never said that the Historicist Jesus Hypothesis is simpler than your illustration. You just made that up. Please do not make up things about what other people say.

Sorry. I was "reading between the lines." You understand that the S scenario in the OP problem really does need to be split in two. Yet you imply that you and/or Carrier do not need to treat multiple scenarios for the Jesus problem.

In the context of the discussion here, "does not require multiple scenarios" implies "simpler than [Swammi's] illustration." But you don't feel that way. Okay.

But on the subject of "making up things about what other people say" let's look at this:

As you know, I have contended emphatically that only some scenarios need to be split into two. Do you or do you not agree with me?

If you agree with me, then the fact that you found one scenario that has a feature that requires it to be split does not prove that Carrier's analysis needs to be split.

If you don't agree that only some scnarios need to be split, then yes, you yourself our saying that all scenarios must be split, and yes, that splitting would , by definition, go on ad infinitum.

So which way is it?

Again the implication that I am obsessed with multiple scenarios and will split them willy-nilly into an infinitude of cases. Ridiculous?

I have already specifically mentioned different scenarios of the Mythic Jesus that I would need clarification on. Some examples:

(1) What is the Mythicist stand on Josephus' "brother of Jesus" mention? Does Carrier still think that was Jesus ben Damneus? I would want to address the different cases separately. (Unless all you Mythicists go along with Carrier's laughable solution.)

(2) Were Chrestus and Christ the same person or not? Argument details are VERY different for the two cases. Unless Mythicists have a consensus answer to this question, I would want to assess the scenarios separately.

These are just two examples. I'd like to see Mythicist scenario(s) fleshed out in detail before I evaluate their probabilities. Do you have a detailed consensus scenario you can point to?

The "getaway driver" case was NOT difficult or unusual. It was intended just to illustrate that even VERY TRIVIAL problems may involve multiple scenarios. You SEEM to be (agreeing with Carrier in?) saying that the Jesus problem lacks the complexity of the simple toy problem in OP. This claim strikes me as odd.

 

[Swammerdami]

Where does Carrier even get the .9's and .1's from?

Huh? The .9s and .1 come from your opening post! And since Carrier most likely did not read your opening post, he didn't get those .9s and .1s.

But you USED the .9's and .1's in your proposed Carrierite analysis. (Numbers that don't arise without considering the sub-scenarios.)