Saturday Morning Breakfast Cereal
Sorry Zach, but I enjoyed this joke twice: as an philosopher and as a statistician!
Showing posts with label probability. Show all posts
Showing posts with label probability. Show all posts
Sunday, June 05, 2016
Monty Hall Trolley Problem
Saturday Morning Breakfast Cereal
Sorry Zach, but I enjoyed this joke twice: as an philosopher and as a statistician!
Saturday, June 27, 2015
On evidence
What is evidence?
There are several definitions. One appealing definition is an event is evidence for another event if observing the first event increases the prior probability of the second, unobservable, event: $B$ is evidence for $A$ if $P(A|B)>P(A)$. However, while this is not a terrible definition of evidence, it has some deficiencies.
The first deficiency is that in common language, saying that $B$ is evidence for $A$ is to imply that $B$ is a good reason to believe $A$. However, if $P(A) = 1×10^{-6}$, and $P(A|B) = 2×10^{-6}$, then observing $B$ has doubled the probability of $A$, but we still don't have a good reason to believe $A$. If we're careful, we can avoid this equivocation, but we do have to be careful. However, translating from math to common language is always fraught with peril, so by itself, the possibility of equivocation is not dispositive. Still, a definition of evidence that is less easily equivocated would be better.
A more mathematical deficiency is that using evidence in the above manner absolutely requires that $A$ be constructed before observing $B$. The problem is that all individual events have very low probability; if we assume a continuous distribution, then all individual events have zero probability. If I observe $B$, and then construct $A$ in terms of $B$, it's always possible to construct $A$ such that $P(A|B)≫P(A)$. I have to at least be able to construct $A$ without knowing $B$ beforehand; ideally I want to construct $A$ before actually knowing $B$.
This problem is especially relevant when we are looking at evidence for intention or planning. Suppose you hand me a deck of cards, and I want to determine if you shuffled them or stacked the deck. I observe the order of the deck, $B$, and note that the probability of getting that specific order is $1/52!=1.24×10^{-68}$. I therefore construct $A$ to be that you intentionally arranged the cards in order $B$. Therefore $P(B|A)=1$ and $P(A|B)={P(B|A)P(A)}/{P(B)}={P(A)}/{P(B)}≫P(A)$: Whatever the prior probability of $A$ is, I've raised the posterior probability by 68 orders of magnitude. Because I am constructing $A$ knowing $B$, any observation $B$ at all "raises" the probability of $A$ by as many orders of magnitude as there are in the sample space of $B$.
A counter-objection is that in science, we construct theories based on our prior observations all the time. We observe, for example, that on the surface of the Earth, things fall when we drop them, so we construct a theory of gravity. We would never have thought about gravity had we not already observed things falling. The rebuttal is that we test (or at least try to test) scientific theories with new observations. We predict a new, as yet unobserved, event, and then observe it.
Although it's true that $P(A|B)≫P(A)$, this calculation might be completely meaningless. Because we don't know the the prior probability of $A$, just that $P(A|B)≫P(A)$ does not tell us whether $P(A)>0.5$ (i.e. the simplest mathematical expression that we have good reasons to believe $A$). But the case is actually worse: because we have narrowed the definition of $A$, we have actually reduced its prior probability, and we may have reduced it relative to the broader definition, the definition that does not include $B$, which I'll label as $A'$, by more than the probability of $B$ improves the probability of $A$. In other words, it may the be the case that even given $P(A|B)≫P(A)$ it could still be true that $P(A|B)<P(A')$.
For example, let's assume that there are a quadrillion ($10^{15}$) ways to order a deck of cards such that the ordering is "apparent": all cards in suit and rank order (A♠-2♠,A♥-2♥,...2♣), rank and suit order (A♠-A♣,K♠-K♣,...2♣), all ranks together in random order of suit, all suits together in random order of rank, all even and odd cards together, ordered such that if we played 5 card draw, you would get four of a kind or a straight flush and I would have a full house, etc. Let us assume that if you intentionally order the deck, you will order it in one of the quadrillion apparent ways. Furthermore, let us assume that the prior probability you will choose to order the deck vs. shuffling it randomly is 0.5: you flip a coin: heads, you order the deck; tails, you shuffle it.
$A': \text " you flipped heads, and ordered the deck in an 'apparently' intentional order"$
$A: \text " you flipped heads and intentionally ordered the deck in order " B$
$B: \text " the deck appears in a specific order, which might or might not be 'apparently' ordered"$
$C: \text " the deck appears in one of the 'apparently' intentional orders"$
$P(A')=0.5$
$P(B)=1.24×10^{-68}$
$P(C)=10^{15}/{52!}=1.24x10^{-53}$
$P(C|A')=1$
$P(B|A)=1$
Therefore,
$$P(A'|C)={P(C|A')P(A')}/{P(C|A')P(A')+P(C|~A')P(~A')}≈0.99...9$$
Thus, if I see the deck in an "apparent" order, I'm pretty sure you flipped heads and intentionally ordered the deck.
However, constructing $A$ in terms of $B$, we get:
$$P(A|B)={P(B|A)P(A)}/{P(B|A)P(A)+P(B|~A)P(~A)}={P(A)}/{P(A)+1.24×10^{-68}P(~A)}$$
To calculate this probability, we have to calculate some prior probability for $A$. However, we have narrowed the definition of $A$ with respect to $A'$: $A'$ says that you order the deck in an "apparent" ordering; $A$ says that you ordered the deck in order $B$, which can include apparent or non-apparent orderings, i.e. any possible ordering. Therefore, it's arguable that the prior probability of $A$ is $0.5 * 1.24×10^{-68}$ (you flip a coin; heads, you intentionally order the deck in a specific possible order that might or might not be "apparent"; tails, you shuffle the deck into a specific possible ordering that might or might not be "apparent". Therefore, $P(A|B)=0.5$ exactly: even though $P(A|B)=1≫P(A)=1.24×10^{-68}$, observing the ordering has given me zero information about whether you intentionally ordered the deck. This conclusion holds even if the deck is "apparently" ordered: the probability that had you flipped heads, you would have picked an apparent ordering is exactly equal to the probability that an apparent ordering would appear by chance.
Worse yet, if we include even a single case where if you flip heads, you might do something other than give me a deck in some order (e.g. there is a $1:1.24×10^{68}+1$ chance you might have just kept the deck if you flipped heads), then $P(A|B)<0.5$ (only slightly, but smaller nonetheless): just receiving a deck in some order means that the posterior probability that you flipped heads is (slightly) lower than the prior probability.
Therefore, simply increasing the posterior probability relative to the prior probability is not always meaningful. (It can, of course, sometimes be meaningful, but at least the principle of prior prediction, or something with the same effect, has to hold, so we can calculate the prior probability without reference to the actual observation itself.)
Thus, we can modify our definition: an event is evidence relative to a pair of mutually exclusive hypotheses if observing the event distinguishes between those hypotheses, i.e. Given $P(A|B)=P(B|A)=0$, ${P(A|C)}/{P(B|C)}>{P(A)}/{P(B)}$. It is still the case that observing $C$ might not be a good enough reason to believe $A$ or $B$, but if we calculate the priors correctly, then constructing $A$ or $B$ in terms of an already observed $C$ entails that ${P(A|C)}/{P(B|C)}≤{P(A)}/{P(B)}$.
For example, given that we have observed that life exists, we want to distinguish between the mutually exclusive hypotheses that life exists by chance, and life exists by design. If we include the prior probabilities correctly, then given
$$A: \text " the universe exists by design"$$
$$B: \text " the universe exists by chance"$$
$$C: \text " life exists"$$
Then
$${P(A|C)}/{P(B|C)}={P(A)}/{P(B)}$$
And the existence of life is just not evidence for these pairs of hypotheses. Note that even if we adjust only hypothesis $A$ to include " the universe exists by a designer who designed life" *$A'$), then even though it is true that $P(A'|C)≫P(A')$, because we have to lower the prior probability of $A'$, it is still the case that ${P(A'|C)}/{P(B|C)}≤{P(A')}/{P(B)}$.
There are several definitions. One appealing definition is an event is evidence for another event if observing the first event increases the prior probability of the second, unobservable, event: $B$ is evidence for $A$ if $P(A|B)>P(A)$. However, while this is not a terrible definition of evidence, it has some deficiencies.
The first deficiency is that in common language, saying that $B$ is evidence for $A$ is to imply that $B$ is a good reason to believe $A$. However, if $P(A) = 1×10^{-6}$, and $P(A|B) = 2×10^{-6}$, then observing $B$ has doubled the probability of $A$, but we still don't have a good reason to believe $A$. If we're careful, we can avoid this equivocation, but we do have to be careful. However, translating from math to common language is always fraught with peril, so by itself, the possibility of equivocation is not dispositive. Still, a definition of evidence that is less easily equivocated would be better.
A more mathematical deficiency is that using evidence in the above manner absolutely requires that $A$ be constructed before observing $B$. The problem is that all individual events have very low probability; if we assume a continuous distribution, then all individual events have zero probability. If I observe $B$, and then construct $A$ in terms of $B$, it's always possible to construct $A$ such that $P(A|B)≫P(A)$. I have to at least be able to construct $A$ without knowing $B$ beforehand; ideally I want to construct $A$ before actually knowing $B$.
This problem is especially relevant when we are looking at evidence for intention or planning. Suppose you hand me a deck of cards, and I want to determine if you shuffled them or stacked the deck. I observe the order of the deck, $B$, and note that the probability of getting that specific order is $1/52!=1.24×10^{-68}$. I therefore construct $A$ to be that you intentionally arranged the cards in order $B$. Therefore $P(B|A)=1$ and $P(A|B)={P(B|A)P(A)}/{P(B)}={P(A)}/{P(B)}≫P(A)$: Whatever the prior probability of $A$ is, I've raised the posterior probability by 68 orders of magnitude. Because I am constructing $A$ knowing $B$, any observation $B$ at all "raises" the probability of $A$ by as many orders of magnitude as there are in the sample space of $B$.
A counter-objection is that in science, we construct theories based on our prior observations all the time. We observe, for example, that on the surface of the Earth, things fall when we drop them, so we construct a theory of gravity. We would never have thought about gravity had we not already observed things falling. The rebuttal is that we test (or at least try to test) scientific theories with new observations. We predict a new, as yet unobserved, event, and then observe it.
Although it's true that $P(A|B)≫P(A)$, this calculation might be completely meaningless. Because we don't know the the prior probability of $A$, just that $P(A|B)≫P(A)$ does not tell us whether $P(A)>0.5$ (i.e. the simplest mathematical expression that we have good reasons to believe $A$). But the case is actually worse: because we have narrowed the definition of $A$, we have actually reduced its prior probability, and we may have reduced it relative to the broader definition, the definition that does not include $B$, which I'll label as $A'$, by more than the probability of $B$ improves the probability of $A$. In other words, it may the be the case that even given $P(A|B)≫P(A)$ it could still be true that $P(A|B)<P(A')$.
For example, let's assume that there are a quadrillion ($10^{15}$) ways to order a deck of cards such that the ordering is "apparent": all cards in suit and rank order (A♠-2♠,A♥-2♥,...2♣), rank and suit order (A♠-A♣,K♠-K♣,...2♣), all ranks together in random order of suit, all suits together in random order of rank, all even and odd cards together, ordered such that if we played 5 card draw, you would get four of a kind or a straight flush and I would have a full house, etc. Let us assume that if you intentionally order the deck, you will order it in one of the quadrillion apparent ways. Furthermore, let us assume that the prior probability you will choose to order the deck vs. shuffling it randomly is 0.5: you flip a coin: heads, you order the deck; tails, you shuffle it.
$A': \text " you flipped heads, and ordered the deck in an 'apparently' intentional order"$
$A: \text " you flipped heads and intentionally ordered the deck in order " B$
$B: \text " the deck appears in a specific order, which might or might not be 'apparently' ordered"$
$C: \text " the deck appears in one of the 'apparently' intentional orders"$
$P(A')=0.5$
$P(B)=1.24×10^{-68}$
$P(C)=10^{15}/{52!}=1.24x10^{-53}$
$P(C|A')=1$
$P(B|A)=1$
Therefore,
$$P(A'|C)={P(C|A')P(A')}/{P(C|A')P(A')+P(C|~A')P(~A')}≈0.99...9$$
Thus, if I see the deck in an "apparent" order, I'm pretty sure you flipped heads and intentionally ordered the deck.
However, constructing $A$ in terms of $B$, we get:
$$P(A|B)={P(B|A)P(A)}/{P(B|A)P(A)+P(B|~A)P(~A)}={P(A)}/{P(A)+1.24×10^{-68}P(~A)}$$
To calculate this probability, we have to calculate some prior probability for $A$. However, we have narrowed the definition of $A$ with respect to $A'$: $A'$ says that you order the deck in an "apparent" ordering; $A$ says that you ordered the deck in order $B$, which can include apparent or non-apparent orderings, i.e. any possible ordering. Therefore, it's arguable that the prior probability of $A$ is $0.5 * 1.24×10^{-68}$ (you flip a coin; heads, you intentionally order the deck in a specific possible order that might or might not be "apparent"; tails, you shuffle the deck into a specific possible ordering that might or might not be "apparent". Therefore, $P(A|B)=0.5$ exactly: even though $P(A|B)=1≫P(A)=1.24×10^{-68}$, observing the ordering has given me zero information about whether you intentionally ordered the deck. This conclusion holds even if the deck is "apparently" ordered: the probability that had you flipped heads, you would have picked an apparent ordering is exactly equal to the probability that an apparent ordering would appear by chance.
Worse yet, if we include even a single case where if you flip heads, you might do something other than give me a deck in some order (e.g. there is a $1:1.24×10^{68}+1$ chance you might have just kept the deck if you flipped heads), then $P(A|B)<0.5$ (only slightly, but smaller nonetheless): just receiving a deck in some order means that the posterior probability that you flipped heads is (slightly) lower than the prior probability.
Therefore, simply increasing the posterior probability relative to the prior probability is not always meaningful. (It can, of course, sometimes be meaningful, but at least the principle of prior prediction, or something with the same effect, has to hold, so we can calculate the prior probability without reference to the actual observation itself.)
Thus, we can modify our definition: an event is evidence relative to a pair of mutually exclusive hypotheses if observing the event distinguishes between those hypotheses, i.e. Given $P(A|B)=P(B|A)=0$, ${P(A|C)}/{P(B|C)}>{P(A)}/{P(B)}$. It is still the case that observing $C$ might not be a good enough reason to believe $A$ or $B$, but if we calculate the priors correctly, then constructing $A$ or $B$ in terms of an already observed $C$ entails that ${P(A|C)}/{P(B|C)}≤{P(A)}/{P(B)}$.
For example, given that we have observed that life exists, we want to distinguish between the mutually exclusive hypotheses that life exists by chance, and life exists by design. If we include the prior probabilities correctly, then given
$$A: \text " the universe exists by design"$$
$$B: \text " the universe exists by chance"$$
$$C: \text " life exists"$$
Then
$${P(A|C)}/{P(B|C)}={P(A)}/{P(B)}$$
And the existence of life is just not evidence for these pairs of hypotheses. Note that even if we adjust only hypothesis $A$ to include " the universe exists by a designer who designed life" *$A'$), then even though it is true that $P(A'|C)≫P(A')$, because we have to lower the prior probability of $A'$, it is still the case that ${P(A'|C)}/{P(B|C)}≤{P(A')}/{P(B)}$.
Friday, June 12, 2015
Even more on Fine Tuning
The central portion of the Fine Tuning Argument (FTA) is valid: Assuming that $P(L|C)=1$ by definition, then $P(C|L)={P(L|C)P(C)}/{P(L)}≥P(C)$. Assuming that $0<P(C)<1$ and $P(L)<1$ then $P(C|L)>P(C)$. Observing that life exists definitely raises the ex ante probability that a creator exists for this world. Life is, in a sense, "evidence for" a creator.
However, to accept this argument as meaningful, we have to assume that $P(C)<1$. This assumption seems to require that we bite a serious philosophical bullet. If we take a frequentist ontology, then $P(C)<1$ means that there exist possible worlds where no creator exists. This contradicts one theistic definition of God as a modally necessary being: if God exists, God exists in all possible worlds. In this case, $P(C)=1$, and $P(C|L)=1$; because we were already certain, by definition, that God existed. Under a Bayesian ontology, $P(C)$ represents our subjective confidence that God exists, and $P(C)<1$ entails that we can reasonably be in doubt that God exists, which contradicts at least Plantinga's argument that God as properly basic: $C⇒P(C)=1$. Thus, if the FTA is rationally persuasive, then God must be neither modally necessary nor properly basic. But suppose we bite these philosophical bullets.
Another portion of the FTA is also valid: Assuming that $P(L|~C)<P(L)$, then $P(~C|L)={P(L|~C)P(~C)}/{P(L)}<P(~C)$: Observing that life exists definitely lowers the ex ante probability that no creator exists for this world.
But so what? We really want to know whether or not $P(C)>P(~C)$. The question is not whether observing life makes the race closer, we want to know if observing life changes the winner. We want to know if $P(C|L)>P(~C|L)$. Because we want to know if this particular inequality is true that we must condition on $L$. The FTA does not, however, help us answer this question. Regardless of how small $P(L|~C)$ is, we do not know whether or not it is less than $P(C)$, and the FTA requires that $P(C)>P(L|~C)$. If we simply assume that $P(C)>P(L|~C)$, then we can just as simply assume that $P(C)<P(L|~C)$ (neither assumption entails a contradiction), so the FTA works only on an arbitrary choice of assumptions, which essentially begs the question. And, according to Ikeda and Jeffries, we have a plausible argument that $P(C|L&F)<P(C|L)$, because $P(~F|C&L)>0$. If we assume that $P(F|C)=1$, then $P(F|C)=P(L|C)$ and we're back where we started.
However, to accept this argument as meaningful, we have to assume that $P(C)<1$. This assumption seems to require that we bite a serious philosophical bullet. If we take a frequentist ontology, then $P(C)<1$ means that there exist possible worlds where no creator exists. This contradicts one theistic definition of God as a modally necessary being: if God exists, God exists in all possible worlds. In this case, $P(C)=1$, and $P(C|L)=1$; because we were already certain, by definition, that God existed. Under a Bayesian ontology, $P(C)$ represents our subjective confidence that God exists, and $P(C)<1$ entails that we can reasonably be in doubt that God exists, which contradicts at least Plantinga's argument that God as properly basic: $C⇒P(C)=1$. Thus, if the FTA is rationally persuasive, then God must be neither modally necessary nor properly basic. But suppose we bite these philosophical bullets.
Another portion of the FTA is also valid: Assuming that $P(L|~C)<P(L)$, then $P(~C|L)={P(L|~C)P(~C)}/{P(L)}<P(~C)$: Observing that life exists definitely lowers the ex ante probability that no creator exists for this world.
But so what? We really want to know whether or not $P(C)>P(~C)$. The question is not whether observing life makes the race closer, we want to know if observing life changes the winner. We want to know if $P(C|L)>P(~C|L)$. Because we want to know if this particular inequality is true that we must condition on $L$. The FTA does not, however, help us answer this question. Regardless of how small $P(L|~C)$ is, we do not know whether or not it is less than $P(C)$, and the FTA requires that $P(C)>P(L|~C)$. If we simply assume that $P(C)>P(L|~C)$, then we can just as simply assume that $P(C)<P(L|~C)$ (neither assumption entails a contradiction), so the FTA works only on an arbitrary choice of assumptions, which essentially begs the question. And, according to Ikeda and Jeffries, we have a plausible argument that $P(C|L&F)<P(C|L)$, because $P(~F|C&L)>0$. If we assume that $P(F|C)=1$, then $P(F|C)=P(L|C)$ and we're back where we started.
More on Fine Tuning
Luke Barnes has asked me to look at Part 1 of his defense of Fine Tuning against Ikeda and Jeffries' critique. Part 1 is, unfortunately, over-complicated, and, as best I can tell, the complications obscure rather than explain the underlying point.
We can concede the most basic Fine Tuning argument: The existence of life does indeed raise the probability that a life-designer exists and designed life. At a very basic (not fundamental) level, the existence of life is indeed evidence for a life-designer. This conclusion has absolutely nothing to do, however, with Fine Tuning. Even if we were to conclude that all logically possible physics guaranteed the existence of life, then it would still be the case that the existence of life was evidence for a life designer (or a life-friendly-physics designer). I don't even need to get into complicated probability math. We can simply look at the alternative: if no life existed, then that would absolutely falsify the existence of a life-designer, in just the same sense that the fact that we observe that no hoverboards exists absolutely falsifies the existence of someone who has (successfully) designed a hoverboard.
The problem with defining evidence in this sense is that it's far too broad. I used to play poker. Sometimes I would win, sometimes I would lose. Just the fact that I lost on some particular day is evidence in the above sense that someone was cheating that day. Again, in a similar sense, had I won that day, I would have known with certainty that no one was cheating.* Possibly cheating is greater than definitely not cheating. Similarly, the fact that each star in each galaxy is arranged in the particular positions we observe is evidence that it was done intentionally: had we observed a different arrangement, that would definitely falsify the hypothesis that they were deliberately arranged in the positions we do observe.
*Or if they were cheating, they were cheating to lose, against which I have no objection.
However, the fact that I lost some days and won some days, and that all the stars in all the galaxies are in the positions that we observe are also evidence for the fact that these events happened by chance. Again, observing something different would falsify the hypothesis that the specific event occurred by chance. If a specific event had not happened, then the hypothesis that the event actually happened by chance would be certainly false.
So observing the existence of life is evidence for the hypothesis that life was created, and it's also evidence for the hypothesis that life happened by chance. Wait, what? How can something be evidence for contradictory hypotheses?! Well, the hypotheses are not contradictory. They're mutually exclusive, but neither "life was designed" nor "life exists by chance" is the complement of the other. Logical consistency demands only that evidence for some hypothesis must be evidence against its complement , not against some other event that is merely mutually exclusive.
Thus, that life exists proves that the complement of the union ("life was designed" or "life happened by chance") is definitely false, which implicitly proves that both "life was designed" and "life happened by chance" have a higher ex post probability than they had ex ante. However, we still have to distinguish between the two probabilities. Hence we start doing things like conditioning on the existence of life, and we get Ikeda and Jeffries.
We can concede the most basic Fine Tuning argument: The existence of life does indeed raise the probability that a life-designer exists and designed life. At a very basic (not fundamental) level, the existence of life is indeed evidence for a life-designer. This conclusion has absolutely nothing to do, however, with Fine Tuning. Even if we were to conclude that all logically possible physics guaranteed the existence of life, then it would still be the case that the existence of life was evidence for a life designer (or a life-friendly-physics designer). I don't even need to get into complicated probability math. We can simply look at the alternative: if no life existed, then that would absolutely falsify the existence of a life-designer, in just the same sense that the fact that we observe that no hoverboards exists absolutely falsifies the existence of someone who has (successfully) designed a hoverboard.
The problem with defining evidence in this sense is that it's far too broad. I used to play poker. Sometimes I would win, sometimes I would lose. Just the fact that I lost on some particular day is evidence in the above sense that someone was cheating that day. Again, in a similar sense, had I won that day, I would have known with certainty that no one was cheating.* Possibly cheating is greater than definitely not cheating. Similarly, the fact that each star in each galaxy is arranged in the particular positions we observe is evidence that it was done intentionally: had we observed a different arrangement, that would definitely falsify the hypothesis that they were deliberately arranged in the positions we do observe.
*Or if they were cheating, they were cheating to lose, against which I have no objection.
However, the fact that I lost some days and won some days, and that all the stars in all the galaxies are in the positions that we observe are also evidence for the fact that these events happened by chance. Again, observing something different would falsify the hypothesis that the specific event occurred by chance. If a specific event had not happened, then the hypothesis that the event actually happened by chance would be certainly false.
So observing the existence of life is evidence for the hypothesis that life was created, and it's also evidence for the hypothesis that life happened by chance. Wait, what? How can something be evidence for contradictory hypotheses?! Well, the hypotheses are not contradictory. They're mutually exclusive, but neither "life was designed" nor "life exists by chance" is the complement of the other. Logical consistency demands only that evidence for some hypothesis must be evidence against its complement , not against some other event that is merely mutually exclusive.
Thus, that life exists proves that the complement of the union ("life was designed" or "life happened by chance") is definitely false, which implicitly proves that both "life was designed" and "life happened by chance" have a higher ex post probability than they had ex ante. However, we still have to distinguish between the two probabilities. Hence we start doing things like conditioning on the existence of life, and we get Ikeda and Jeffries.
Sunday, June 07, 2015
Fine tuning and probability
Luke Barnes criticizes this version of Ikeda and Jefferys' paper, "The Anthropic Principle Does Not Support Supernaturalism." (I usually link to this version; I assume the content is identical.) Barnes' post is five years old, but it's still be referenced by some people, so I guess it's still alive enough to be worth rebutting. In his criticism, Barnes runs afoul of some of probability's conceptual problems.
In general, probability is very philosophically problematic. Definitely with the frequentist interpretation, but also with the Bayesian interpretation, using probability requires us to adopt an ontological commitment to the existence of unknowable things: frequentism requires us to talk about experiments not performed, Bayesianism requires us to talk precisely about what we don't know. It is philosophically respectable to reject probability entirely: the past is fixed and certain, the future is determined by the present, and it is incoherent to measure our ignorance. However, if you're going to use probability to make inferences, you have to bite all the philosophical bullets that probability requires.
To briefly recap, the Fine Tuning Argument (FTA) says that, given relatively unproblematic assumptions about the laws of physics at the most general level, the probability is very low that the (apparently) arbitrary constants of the laws of physics, i.e. those constants whose values cannot be determined theoretically and must be determined observationally, are such that life can exist. We human beings apparently got ridiculously lucky when the universe formed almost perfectly set up for life to evolve. The Weak Anthropic Principle (WAP) rebuts the FTA: if the universe were not "fine tuned" for life, we would not be around to observe otherwise; of course we observe that the universe is "fine tuned." Barnes counter-argues that if we are required to condition on the result (life exists), then probabilistic arguments are generally useless. Barnes is correct: if we are required in principle to condition a probabilistic argument on the result, then probabilistic arguments are useless. However, he is not correct that Ikeda and Jeffries argue that we are required in principle to condition on the result.
We have to make additional assumptions to make probability work. First, we have to be conceptually able to "observe" both success or failures. If we cannot in principle observe failures, then we cannot make any inference at all about the probability of the successes we observe. This is a trick that "psychics" use: they show the hits but hide the misses. Second, we have to be conceptually able to describe success before we actually observe it. In a large probability space, every actual event has a very low probability. Shuffle a deck of cards and then look at it: before you shuffled it, the probability that the deck would end up in exactly that order is $1/{52!}≈1.24×10^{-68}$.
The difference between the FTA and Barnes' example is that in the FTA, we cannot in principle, even conceptually, observe a universe that does not have life in it. In contrast, I personally could in principle have observed Magneto and his grandson being killed by the shrapnel. Furthermore, even though the grandson could not have actually observed that outcome, he could conceptually have "observed" the alternative outcome, e.g. by observing that other people actually died.
Ikeda and Jeffries invert the FTA. Instead of asking the probability of naturalism, they ask the probability of God existing. We first assume first that if God does not exist, then life will exist only in a life-friendly universe. Life existing entails life friendliness $P(F|~G&L)=1$. We don't have to condition on L, but if we do, then if $G$ is false then $F$ must be true.
We then consider two alternative hypotheses. The first hypothesis is that if God1 exists, then He would have created a life-friendly universe, If that is the hypothesized God, then observing life-friendly universe with life in it (our only actual observation) does not distinguish between the existence and non-existence of God. The second hypothesis is that if God2 exists, then He would have created life, and might have created life in a non-life-friendly universe. In this case, Ikeda and Jeffries correctly argue that observing life in a life-friendly universe lowers (slightly) the probability that God2 exists (in the same sense that observing a non-black non-crow raises the probability slightly that all crows are black).
We don't have to condition on life existing. However, we're worse off. We could not conceptually have predicted before observing any universe that life would exist in it. The only reason we have to assume that if God exists, He would have created life is just that we are in fact alive. We are "predicting" the random order of the deck after we've seen it. (Positing indirect attributes of God that entail His creating life just move the problem around.) Just the hypothesis that if God exists, he would have created anything we actually see begs the question: it assumes what we are setting out to "prove." Plantinga would call that an adequate argument, but Plantinga is a doofus who doesn't understand modal logic or probability.
In short, the Fine Tuning Argument for the existence of God is dead. It either provides evidence against the existence of God, or it fails to satisfy the requirements to even make a probabilistic argument in the first place.
In general, probability is very philosophically problematic. Definitely with the frequentist interpretation, but also with the Bayesian interpretation, using probability requires us to adopt an ontological commitment to the existence of unknowable things: frequentism requires us to talk about experiments not performed, Bayesianism requires us to talk precisely about what we don't know. It is philosophically respectable to reject probability entirely: the past is fixed and certain, the future is determined by the present, and it is incoherent to measure our ignorance. However, if you're going to use probability to make inferences, you have to bite all the philosophical bullets that probability requires.
To briefly recap, the Fine Tuning Argument (FTA) says that, given relatively unproblematic assumptions about the laws of physics at the most general level, the probability is very low that the (apparently) arbitrary constants of the laws of physics, i.e. those constants whose values cannot be determined theoretically and must be determined observationally, are such that life can exist. We human beings apparently got ridiculously lucky when the universe formed almost perfectly set up for life to evolve. The Weak Anthropic Principle (WAP) rebuts the FTA: if the universe were not "fine tuned" for life, we would not be around to observe otherwise; of course we observe that the universe is "fine tuned." Barnes counter-argues that if we are required to condition on the result (life exists), then probabilistic arguments are generally useless. Barnes is correct: if we are required in principle to condition a probabilistic argument on the result, then probabilistic arguments are useless. However, he is not correct that Ikeda and Jeffries argue that we are required in principle to condition on the result.
We have to make additional assumptions to make probability work. First, we have to be conceptually able to "observe" both success or failures. If we cannot in principle observe failures, then we cannot make any inference at all about the probability of the successes we observe. This is a trick that "psychics" use: they show the hits but hide the misses. Second, we have to be conceptually able to describe success before we actually observe it. In a large probability space, every actual event has a very low probability. Shuffle a deck of cards and then look at it: before you shuffled it, the probability that the deck would end up in exactly that order is $1/{52!}≈1.24×10^{-68}$.
The difference between the FTA and Barnes' example is that in the FTA, we cannot in principle, even conceptually, observe a universe that does not have life in it. In contrast, I personally could in principle have observed Magneto and his grandson being killed by the shrapnel. Furthermore, even though the grandson could not have actually observed that outcome, he could conceptually have "observed" the alternative outcome, e.g. by observing that other people actually died.
Ikeda and Jeffries invert the FTA. Instead of asking the probability of naturalism, they ask the probability of God existing. We first assume first that if God does not exist, then life will exist only in a life-friendly universe. Life existing entails life friendliness $P(F|~G&L)=1$. We don't have to condition on L, but if we do, then if $G$ is false then $F$ must be true.
We then consider two alternative hypotheses. The first hypothesis is that if God1 exists, then He would have created a life-friendly universe, If that is the hypothesized God, then observing life-friendly universe with life in it (our only actual observation) does not distinguish between the existence and non-existence of God. The second hypothesis is that if God2 exists, then He would have created life, and might have created life in a non-life-friendly universe. In this case, Ikeda and Jeffries correctly argue that observing life in a life-friendly universe lowers (slightly) the probability that God2 exists (in the same sense that observing a non-black non-crow raises the probability slightly that all crows are black).
We don't have to condition on life existing. However, we're worse off. We could not conceptually have predicted before observing any universe that life would exist in it. The only reason we have to assume that if God exists, He would have created life is just that we are in fact alive. We are "predicting" the random order of the deck after we've seen it. (Positing indirect attributes of God that entail His creating life just move the problem around.) Just the hypothesis that if God exists, he would have created anything we actually see begs the question: it assumes what we are setting out to "prove." Plantinga would call that an adequate argument, but Plantinga is a doofus who doesn't understand modal logic or probability.
In short, the Fine Tuning Argument for the existence of God is dead. It either provides evidence against the existence of God, or it fails to satisfy the requirements to even make a probabilistic argument in the first place.
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