Multiple Universes and Self-Locating Evidence

The qualitative evidence gained by reading nearly any book will overwhelmingly support the multiverse.

“In fact, the qualitative evidence gained by reading nearly any book will overwhelmingly support the multiverse.”

• AssumptionThe authors state this as an assumption: In the simplified setting, the intrinsic qualitative profiles of each universe are independent and identically distributed.

The prior probability of 'α is Universe One' is less than one, even though this is a metaphysically necessary truth.

“But the prior probability of”

Given the number of inhabited universes, our self-locating evidence tells us nothing about how many uninhabited universes there are (lemma 5 in appendix A).

“Second, we have a lemma that says that, given the number of inhabited universes, our self-locating evidence tells us nothing about how many uninhabited universes there are (lemma 5 in appendix A).”

• AssumptionThe authors state this as an assumption: Qualitative priors have a simple form that builds in the same kind of independence assumptions that Hacking and White supposed.
• AssumptionThe authors state this as an assumption: The relation of evidential equivalence between centers is local: intrinsic duplicate universes are evidentially equivalent.
• AssumptionThe authors state this as an assumption: A posterior is evidential: no world is confirmed relative to any evidentially equivalent world.

In the toy model without fine-tuning, Self Indication confirms the multiverse by a factor of 4, bringing its probability from 1/3 to 2/3.

“So the multiverse is relatively confirmed by a factor of 4, bringing its probability from 1/3 to 2/3.”

Conditional on exactly one universe being inhabited, neither U1 nor U2 is confirmed relative to the other.

“So, conditional on exactly one universe being inhabited, neither U 1 nor U 2 is confirmed relative to the other.”

For a separable prior, if p < 1/2, then it is more probable that exactly one universe is inhabited if there are two universes than if there is just one.

“First, for a separable prior Pr(-), a little calculation shows that if p < 1/2, then it is more probable that exactly one universe is inhabited if there are two universes than if there is just one.”

• AssumptionThe authors state this as an assumption: Qualitative priors have a simple form that builds in the same kind of independence assumptions that Hacking and White supposed.
• AssumptionThe authors state this as an assumption: Each universe's intrinsic profile is independent and identically distributed.

In the toy model without fine-tuning, Compartmentalized Conditionalization confirms the multiverse by a factor of approximately 2.7, bringing its probability from 1/3 to about 0.57.

“So the multiverse is relatively confirmed by a factor of ≈ 2.7, bringing the probability from 1/3 to about 0.57.”

In the fine-tuning toy model, Self Sampling confirms the multiverse by a factor of approximately 3.4, taking its probability from 1/3 to approximately 0.63.

“So the multiverse is confirmed by a factor of ≈ 3.4, taking its probability from 1/3 to ≈ 0.63.”

Theorem 1: For any separable qualitative prior and any evidential posterior, Locality and Agents imply that, for any possible multiverse size 1 < n < 1/p, where p is the fine-tuning parameter, the proposition there are n universes is confirmed relative to there is just one universe.

“Theorem 1.For any separable qualitative prior and any evidential posterior, Locality and Agents imply that, for any possible multiverse size 1 < n < 1/p, where p is the fine-tuning parameter, the proposition there are n universes is confirmed relative to there is just one universe.”

• AssumptionThe authors state this as an assumption: There is a prior probability distribution over qualitative hypotheses.
• AssumptionThe authors state this as an assumption: Qualitative priors have a simple form that builds in the same kind of independence assumptions that Hacking and White supposed.
• AssumptionThe authors state this as an assumption: Each universe's intrinsic profile is independent and identically distributed.
• AssumptionThe authors state this as an assumption: The relation of evidential equivalence between centers is local: intrinsic duplicate universes are evidentially equivalent.
• AssumptionThe authors state this as an assumption: The posterior probability that there are no agents is zero.
• AssumptionThe authors state this as an assumption: A posterior is evidential: no world is confirmed relative to any evidentially equivalent world.

In the fine-tuning toy model, Self Indication confirms the multiverse by a factor of 4, bringing its probability from 1/3 to 2/3.

“So the multiverse hypothesis is relatively confirmed by a factor of 4, bringing its probability from 1/3 to 2/3.”

In our toy model, the confirmation factor was n (the number of universes in the multiverse) regardless of the fine-tuning parameter.

“In our toy model we saw that the confirmation factor was n (the number of universes in the multiverse) regardless of the fine-tuning parameter.”

In the fine-tuning toy model, Compartmentalized Conditionalization confirms the multiverse by a factor of approximately 3.8, taking its probability from 1/3 to a bit under 0.66.

“So the multiverse hypothesis is relatively confirmed by a factor of ≈ 3.8, which takes its probability from 1/3 to a bit under 0.66.”

Lemma 5: Let Pr be a separable qualitative prior on a local multiverse structure, and let Pr* be an evidential posterior. For any possible multiverse sizes m and n, Un is not confirmed relative to Um conditional on Ik.

“Lemma 5. Let Pr(-) be a separable qualitative prior on a local multiverse structure, and let Pr * (-) be an evidential posterior.For any possible multiverse sizes m and n, U n is not confirmed relative to U m conditional on I k .”

• AssumptionThe authors state this as an assumption: Qualitative priors have a simple form that builds in the same kind of independence assumptions that Hacking and White supposed.
• AssumptionThe authors state this as an assumption: The relation of evidential equivalence between centers is local: intrinsic duplicate universes are evidentially equivalent.
• AssumptionThe authors state this as an assumption: A posterior is evidential: no world is confirmed relative to any evidentially equivalent world.

For separable priors, the probability Pr(¬I0|Un) of at least one universe being inhabited if there are n universes total is 1-(1-p)^n, where p is the fine-tuning parameter.

“For separable priors, the probability Pr(¬I 0 | U n ) of at least one universe being inhabited if there are n universes total is 1 -(1p) n , where p is the fine-tuning parameter.”

Lemma 1(a): If none of the universes u1,...,uk are duplicates of any of the universes uk+1,...,un, then Pr([u1...un] | Un) = n choose k Pr([u1...uk] | Uk) Pr([uk+1...un] | Un-k).

“Pr([u 1 • • • u n ] | U n ) n k Pr([u 1 • • • u k ] | U k ) Pr([u k+1 • • • u n ] | U n-k )”

• AssumptionThe authors state this as an assumption: Qualitative priors have a simple form that builds in the same kind of independence assumptions that Hacking and White supposed.
• AssumptionThe authors state this as an assumption: Each universe's intrinsic profile is independent and identically distributed.

Theorem 2: For any separable prior and any ordinary evidential posterior, Locality and Agents imply that for any possible multiverse size n > 1, there are n universes is confirmed relative to there is just one universe.

“Theorem 2. For any separable prior and any ordinary evidential posterior, Locality and Agents imply that for any possible multiverse size n > 1, there are n universes is confirmed relative to there is just one universe.”

• AssumptionThe authors state this as an assumption: There is a prior probability distribution over qualitative hypotheses.
• AssumptionThe authors state this as an assumption: Qualitative priors have a simple form that builds in the same kind of independence assumptions that Hacking and White supposed.
• AssumptionThe authors state this as an assumption: Each universe's intrinsic profile is independent and identically distributed.
• AssumptionThe authors state this as an assumption: The relation of evidential equivalence between centers is local: intrinsic duplicate universes are evidentially equivalent.
• AssumptionThe authors state this as an assumption: The posterior probability that there are no agents is zero.
• AssumptionThe authors state this as an assumption: A posterior is evidential: no world is confirmed relative to any evidentially equivalent world.
• AssumptionThe authors state this as an assumption: A posterior is ordinary: for each n > 0, conditional on every universe containing agents, the existence of just one universe is not confirmed relative to the existence of n universes.

Lemma 1(b): For 0 ≤ k ≤ n, Pr(Ik | Un) = n choose k p^k (1-p)^{n-k}.

“For 0 ≤ k ≤ n, Pr(I k | U n ) n k p k (1 -p) n-k”

• AssumptionThe authors state this as an assumption: Qualitative priors have a simple form that builds in the same kind of independence assumptions that Hacking and White supposed.
• AssumptionThe authors state this as an assumption: Each universe's intrinsic profile is independent and identically distributed.

It follows that there are n universes and at least one universe is inhabited is not disconfirmed relative to there is one inhabited universe.

“It follows that there are n universes and at least one universe is inhabited is not disconfirmed relative to there is one inhabited universe.”

Lemma 1(c): If 1 < n, it is more likely that there is at least one inhabited universe if there are n universes than if there is just one: Pr(¬I0 | Un) > Pr(¬I0 | U1).

“If 1 < n , it is more likely that there is at least one inhabited universe if there are n universes than if there is just one: Pr(¬I 0 | U n ) > Pr(¬I 0 | U 1 )”

• AssumptionThe authors state this as an assumption: Qualitative priors have a simple form that builds in the same kind of independence assumptions that Hacking and White supposed.
• AssumptionThe authors state this as an assumption: Each universe's intrinsic profile is independent and identically distributed.

The prior probability that at least one universe is inhabited is higher if there are many universes than if there is just one.

“But the prior probability that at least one universe is inhabited is higher if there are many universes than if there is just one.”

• AssumptionThe authors state this as an assumption: Qualitative priors have a simple form that builds in the same kind of independence assumptions that Hacking and White supposed.
• AssumptionThe authors state this as an assumption: Each universe's intrinsic profile is independent and identically distributed.

Lemma 1(d): If 1 < n < 1/p, it is more likely that there is exactly one inhabited universe if there are n universes than if there is just one: Pr(I1 | Un) > Pr(I1 | U1).

“If 1 < n < 1/p , it is more likely that there is exactly one inhabited universe if there are n universes than if there is just one: Pr(I 1 | U n ) > Pr(I 1 | U 1 )”

• AssumptionThe authors state this as an assumption: Qualitative priors have a simple form that builds in the same kind of independence assumptions that Hacking and White supposed.
• AssumptionThe authors state this as an assumption: Each universe's intrinsic profile is independent and identically distributed.

Our relevant evidence includes the more specific fact that there is fine-tuned life in this universe, our universe, not just the general fact that there is fine-tuned life in some universe.

“Our relevant evidence is not just the general evidence that there is fine-tuned life in some universe or other: rather, we know the more specific fact that there is fine-tuned life in this universe, our universe.”

Leading approaches to self-locating evidence agree that the fact that our own universe contains fine-tuned life confirms the existence of a multiverse, at least in a suitably idealized setting.

“As it turns out, leading approaches to self-locating evidence agree that the fact that our own universe contains fine-tuned life indeed confirms the existence of a multiverse (at least in a suitably idealized setting).”

• AssumptionThe authors state this as an assumption: Fine-tuning can be interpreted as making explicit probabilistic claims.

Any non-trivial local qualitative proposition q counts in favor of a multiverse.

“Any non-trivial local qualitative proposition q counts in favor of a multiverse.”

• AssumptionThe authors state this as an assumption: In the simplified setting, the intrinsic qualitative profiles of each universe are independent and identically distributed.

Fine-tuning evidence works like local qualitative evidence in supporting a multiverse.

“Fine-tuning evidence works this way, too.”

• AssumptionThe authors state this as an assumption: In the simplified setting, the intrinsic qualitative profiles of each universe are independent and identically distributed.

For any local proposition with probability conditional on a single universe strictly between zero and one, its probability conditional on many universes is greater.

“Whatever probability such a proposition may have conditional on there being a single universe (strictly between zero and one), the probability must be greater conditional on there being many universes.”

• AssumptionThe authors state this as an assumption: In the simplified setting, the intrinsic qualitative profiles of each universe are independent and identically distributed.

Even without fine-tuning, we already had some evidence with the same feature: local qualitative facts that have very low prior probability given a single universe simply due to including a lot of detail.

“But even without fine-tuning, we already had some evidence with this same feature: local qualitative facts that have very low prior probability, given a single universe, simply due to including a lot of detail.”

• AssumptionThe authors state this as an assumption: In the simplified setting, the intrinsic qualitative profiles of each universe are independent and identically distributed.

Any local proposition that is true in a single universe is also true in any multiverse that includes a copy of that single universe.

“Any local proposition that is true in a single universe is also true in any multiverse that includes a copy of that single universe.”

It does not follow from the fact that 'Universe One has life' has prior probability 1/n that 'α has life' has prior probability 1/n.

“So it does not follow from the fact that”

The existence of life and complexity is very unlikely given a single universe.

“It tells us that the existence of life and complexity is very unlikely given a single universe-much, much less likely than one might have naïvely expected.”

• AssumptionThe authors state this as an assumption: In the simplified setting, the intrinsic qualitative profiles of each universe are independent and identically distributed.

Any updating rule that satisfies a few reasonable conditions in this setting will have the feature that fine-tuned life confirms a multiverse.

“we present two theorems showing that in this setting, any updating rule that satisfies a few reasonable conditions will have the same feature.”

• AssumptionThe authors state this as an assumption: Fine-tuning can be interpreted as making explicit probabilistic claims.

The general qualitative proposition that there is fine-tuned life is straightforwardly evidence for the multiverse.

“As all parties concede, the general qualitative proposition there is finetuned life is straightforwardly evidence for the multiverse.”

• cites Hacking 1987 sourceNot checked — no full text of the source
• cites White 2000 sourceNot checked — no full text of the source

• AssumptionThe authors state this as an assumption: Fine-tuning can be interpreted as making explicit probabilistic claims.

Specific evidence can screen off the import of general evidence.

“Specific evidence can screen off the import of general evidence.”

• cites White 2000 sourceNot checked — no full text of the source

Our specific evidence that this universe contains fine-tuned life does confirm a multiverse.

“We will make the case that that our specific evidence does confirm a multiverse.”

• AssumptionThe authors state this as an assumption: Fine-tuning can be interpreted as making explicit probabilistic claims.

If fine-tuning does not provide support for the multiverse, it is not for the reason that Hacking and White defended.

“But at least we can see that if fine-tuning does not provide support for the multiverse, it is not for the reason that Hacking and White defended.”

The single-center restriction of Titelbaum's Certainty Loss Framework avoids the apparent counterexamples of the kind just considered.

“This restriction avoids the apparent counterexamples of the kind we just considered.”

The support that Self Indication gives to the multiverse is not really a fine-tuning argument.

“So he's right: the support that Self Indication gives to the multiverse is not really a fine-tuning argument.”

In the simple setting, the specificity of one's mundane evidence makes no difference to how strongly a multiverse is confirmed under Self Indication and Self Sampling.

“Notice that the formulas for Self Indication and Self Sampling in table 1 do not depend on q: this means that in our simple setting, the specificity of one's mundane evidence makes no difference to how strongly a multiverse is confirmed.”

Under Self Sampling, the relative probability of each world w containing any centers is boosted according to the proportion of live centers among all centers in w.

“The effect is that the relative probability of each world w containing any centers is boosted according to the proportion of live centers among all centers in w.”

Locality is much weaker than internalism.

“But Locality is much weaker than internalism.”

These factors cancel out, leaving the confirmation the same for any fine-tuning parameter.

“These factors cancel out, leaving the confirmation the same for any fine-tuning parameter.”

There being zero observers makes it less likely that you would observe anything.

“after all, there being zero observers surely makes it less likely that you would observe anything.”

It is much more difficult to assess prior probabilities for propositions like our universe has fine-tuned life than to assess prior probabilities for qualitative hypotheses about the number of universes and their intrinsic profiles.

“For a start, while we have helped ourselves to prior probabilities for qualitative hypotheses about the number of universes and their intrinsic profiles, it is much more difficult to assess prior probabilities for propositions like our universe has fine-tuned life.”

In the three-person coin-and-light example, seeing a red light is evidence that the coin came up tails, with posterior odds of 2:1.

“It seems clear that seeing a red light is evidence that the coin came up tails.(Indeed, it seems clear how strong this evidence is: one is twice as likely to see a red light if the coin came up tails, so the posterior odds should be 2 : 1.)”

• cites Weintraub 2004 sourceNot checked — no full text of the source

Two different centers in the same quantum multiverse can each be consistent with one's evidence, but should not be assigned the same probability because they are on branches with different quantum amplitudes.

“Two different centers in the same quantum multiverse can each be consistent with one's evidence, but should not be assigned the same probability because they are on branches with different quantum amplitudes.”

Compartmentalized Conditionalization relatively confirms each hypothesis H in proportion to the conditional probability that there are any live centers, given H.

“Compartmentalized Conditionalization relatively confirms each hypothesis H in proportion to the conditional probability that there are any live centers, given H.”

The correct principle is that evidence I am F supports hypothesis H only if it is more likely given H that I would be F (whether or not I observe this).

“The correct principle is that the evidence I am F supports hypothesis H only if it is more likely given H that I would be F (whether or not I observe this).”

For qualitative propositions, Compartmentalized Conditionalization with evidence that one is F is equivalent to standard conditionalization on the qualitative proposition that someone is F.

“When it comes to qualitative propositions, Compartmentalized Conditionalization for the evidence being F amounts to standard conditionalization on the qualitative proposition someone is F.”

Self Indication does not solve the problem of mundane evidence confirming extravagant hypotheses.

“This does not solve the problem of mundane evidence confirming extravagant hypotheses.”

All three of the rules we have considered support the first conclusion (being in a universe that contains fine-tuned life is strong evidence for the multiverse).

“All three of the rules we have considered support the first conclusion (as will the broader theoretical considerations we offer in section 7).”

For the kind of simple multiverse model we have all taken for granted (with its independence assumptions), that evidence does support a multiverse, given any reasonable spelled-out theory of self-locating evidence that we know of.

“For the kind of simple multiverse model we have all taken for granted (with its independence assumptions), that evidence does support a multiverse, given any reasonable spelled-out theory of self-locating evidence that we know of.”

Non-trivial propositions of the form someone is F, where F is any local property, always confirm large random multiverses over single universes.

“non-trivial propositions of the form someone is F (where F is any local property) will always confirm large random multiverses over single universes.”

Each rule multiplies the prior probability of each world w by a confirmation factor η(w) and then renormalizes.

“Each rule has the effect of multiplying the prior probability of each world w by some “confirmation factor” η(w), and then renormalizing.”

The smaller the fine-tuning parameter, the smaller the expected number of conscious beings given a multiverse.

“it is correct that the smaller the finetuning parameter, the smaller the expected number of conscious beings given a multiverse.”

In the doomsday device example, Self Sampling boosts the posterior probability of the doomsday hypothesis from one in a million to close to one.

“Since there are more centers in glorious-future worlds than doomsday worlds by a factor of billions, the posterior probability of the doomsday hypothesis is boosted from one in a million to close to one.”

• cites Carter 1983 sourceNot checked — no full text of the source
• cites Leslie 1990 sourceNot checked — no full text of the source
• cites Bostrom 2002Not checked — no full text of the source

Unlike the other rules considered, Self Sampling need not confirm multiverses given just mundane local evidence.

“So unlike the other rules we have considered, Self Sampling need not confirm multiverses given just mundane local evidence.”

• AssumptionThe authors state this as an assumption: Each universe has the same probabilities of producing centers of any particular kind, regardless of what other universes there may be.

White's more central point is correct: fine-tuning does not increase the strength of confirmation that the Self Indication rule gives to the multiverse.

“Still, White's more central point in this passage is correct: fine-tuning does not increase the strength of confirmation that the Self Indication rule gives to the multiverse.”

A multiverse of any finite size is confirmed by fine-tuned life-as long as it is fine-tuned enough.

“So a multiverse of any finite size is confirmed by fine-tuned life-as long as it is fine-tuned enough.”

The restricted rule of Titelbaum's Certainty Loss Framework falls silent on multiverse cases that can involve multiple agents in the same world with the same self-locating evidence.

“tiverse cases we are interested in, which can involve multiple agents in the same world with the same self-locating evidence.”

Compartmentalized Conditionalization and Self Sampling both say that how much the multiverse gets confirmed is sensitive to fine-tuning.

“Compartmentalized Conditionalization or Self Sampling both say that how much the multiverse gets confirmed is sensitive to fine-tuning.”

Self Indication does not escape the result that mundane evidence overwhelmingly confirms multiverse hypotheses.

“Unfortunately, Self Indication does not escape the result that mundane evidence overwhelmingly confirms multiverse hypotheses, either.”

The three rules discussed give purchase on claims about confirmation by self-locating evidence that does not have to go through claims about prior probabilities of self-locating evidence.

“In fact, the three rules we have discussed give us purchase on claims about confirmation by self-locating evidence that does not have to go through claims about prior probabilities of self-locating evidence.”

Compartmentalized Conditionalization is the only one of the three rules for which the specificity of mundane evidence gives the multiverse a boost.

“So Compartmentalized Conditionalization is the only one of the three rules for which the specificity of mundane evidence gives the multiverse a boost.”

Each of the self-locating rules discussed in section 4 can be reinterpreted as recipes for generating self-locating priors from qualitative priors.

“Furthermore, each of the self-locating rules we discussed in section 4 can be reinterpreted as recipes for generating self-locating priors from qualitative priors.”

Self Sampling has a compensating mechanism: if a world contains centers that do not fit one's evidence, these take away from the probability of the centers that do match one's evidence.

“Self Sampling has a compensating mechanism, already on display in the simple example: if a world contains centers that don't fit your evidence, then these take away from the probability of the centers that do match your evidence.”

None of these particular recipes generate priors that vindicate White's independence claim that whether our universe contains life is independent of how many universes there are.

“But as it turns out, none of these particular recipes generate priors that vindicate White's independence claim-that whether our universe contains life is independent of how many universes there are.”

Being in a universe that contains fine-tuned life is strong evidence for the multiverse.

“One is that being in a universe that contains fine-tuned life is strong evidence for the multiverse.”

Self Sampling relatively confirms each hypothesis H according to the expected proportion of live centers among all centers, where this proportion is considered zero in worlds containing no centers at all.

“And Self Sampling relatively confirms each hypothesis H according to the expected proportion of live centers among all centers, where this proportion is considered to be zero in worlds containing no centers at all.”

Theorem 1 tells us that for any n, there is some fine-tuning parameter p for which a size-n multiverse is confirmed.

“Theorem 1 tells us that for any n, there is some fine-tuning parameter p for which a size-n multiverse is confirmed.”

Setting aside the possibility of non-local evidence and holding fixed the independence assumptions in the background, the only hope for a single universe is non-qualitative evidence.

“Setting aside the possibility of non-local evidence (and holding fixed the independence assumptions in the background), the only hope for a single universe is non-qualitative evidence.”

Self Sampling's recommendations are sensitive to what exactly counts as a center, which is the reference class problem.

“A second problem for Self-Sampling is that it is not entirely clear what it says: its recommendations are sensitive to what exactly counts as a center (in a way that those of the other two rules are not).”

• cites Bostrom 2002Not checked — no full text of the source
• cites Arntzenius and Dorr, ManuscriptNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Self Indication relatively confirms each hypothesis H in proportion to the expected number of live centers conditional on H.

“Self Indication relatively confirms each hypothesis H in proportion to the expected number of live centers conditional on H.”

The way of thinking outlined in appendix B basically vindicates White's Observation Principle.

“The way of thinking about things outlined there basically vindicates what White calls the “Observation Principle”: “An observation I make gives me evidence for hypothesis H only if it is more likely given H that I would make that observation” (White 2003, p. 244).”

• cites White 2003Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Unlike the other two rules, Self Sampling does not have the effect that mundane local evidence confirms vast worlds.

“Unlike the other two rules, Self Sampling does not have the effect that mundane local evidence confirms vast worlds.”

Compartmentalized Conditionalization is hopeless for avoiding confirmation of multiverses.

“This same feature, though, makes Compartmentalized Conditionalization hopeless for avoiding confirmation of multiverses.”

In the three-person coin-and-light example, the proposition that at least one person sees a red light is evidence for neither hypothesis.

“But the proposition at least one person sees a red light is evidence for neither hypothesis.”

• cites Weintraub 2004 sourceNot checked — no full text of the source

Self Indication says that the multiverse gets confirmed to exactly the same degree whether or not life is fine-tuned.

“Self Indication says that the multiverse gets confirmed to exactly the same degree whether or not life is fine-tuned.”

In cases where the single-center restriction holds, Compartmentalized Conditionalization and Self-Indication give precisely the same results.

“(In cases where the single-center restriction does hold, Compartmentalized Conditionalization and Self-Indication give precisely the same results.)”

The smaller the fine-tuning parameter, the smaller the expected number of conscious beings given a single universe.

“But also, in the same way, the smaller the fine-tuning parameter, the smaller the expected number of conscious beings given a single universe.”

Given a separable, evidentially natural self-locating prior on a local multiverse frame and a center i such that Pr(E(i)) > 0 and Pr_i(-) is ordinary, for any possible multiverse size n > 1, n universes are confirmed relative to one.

“Theorem 4. Let Pr(-) be a separable, evidentially natural self-locating prior on a local multiverse frame.Let i be a center such that Pr(E(i)) > 0 and Pr i (-) is ordinary (definition 7).For any possible multiverse size n > 1, n universes are confirmed relative to one.”

• AssumptionThe authors state this as an assumption: In each world there is one additional null center, which represents the possibility of not being an agent.
• AssumptionThe authors state this as an assumption: Any duplication map from one universe to another maps each agent in the first universe to an agent in the other; whether something counts as an agent does not depend on anything beyond its universe.
• AssumptionThe authors state this as an assumption: For simplicity, each non-null center i has a unique evidential type: there is a unique set of centers E(j) that includes i.

Starting with any qualitative prior, generating a self-locating prior using the recipe and conditionalizing on self-locating evidence for a center i yields the same result as the original rule gives for i.

“These indexical priors agree with the original rules, in the following sense.Start with any qualitative prior Pr q (-).If you generate a self-locating prior using the rule's recipe above, then you conditionalize the prior on the selflocating evidence for a center i (where E(i) has positive prior probability), you get the same result as the original rule gives for i.We omit the proofs.”

If Hacking and White were right, then our evidence should not count against a multiverse, either-even in the absence of fine-tuning.

“If they were right, then our evidence should not count against a multiverse, either-even in the absence of fine-tuning.”

Any two worlds in I_k Q are evidentially equivalent: each of the k inhabited universes in one world can be mapped to a duplicate universe in the other, and locality says that duplicate universes are evidentially equivalent.

“Any two worlds in I k Q are evidentially equivalent: each of the k inhabited universes in one world can be mapped to a duplicate universe in the other, and locality says that duplicate universes are evidentially equivalent.”

The set Q of distinct local profiles Q[u1...uk] where (u1,...,uk) is a sequence of inhabited universes and Pr(Q)>0 are mutually exclusive.

“Let Q be the set of distinct local profiles Q [u 1 • • • u k ] where (u 1 , . . ., u k ) is a sequence of inhabited universes and Pr(Q) > 0. These propositions are mutually exclusive.”

This again tells against Hacking and White's diagnosis of the fine-tuning argument.

“This again tells against Hacking and White's diagnosis of the fine-tuning argument.”

Our general results show that it will be difficult to come up with any reasonable theory that does vindicate the idea that this more specific evidence does not confirm the multiverse.

“What's more, our general results show that it will be difficult to come up with any reasonable theory that does vindicate it.”

The existence of fine-tuned life, taken on its own, is straightforwardly evidence for a multiverse.

“The existence of fine-tuned life, taken on its own, is straightforwardly evidence for a multiverse.”

Given a separable, evidentially natural self-locating prior on a local multiverse frame, for any possible multiverse size n such that 1 < n < 1/p, where p is the fine-tuning parameter, for any center i such that Pr(E(i)) > 0, n universes are confirmed relative to one: Pr_i(U_n)/Pr_i(U_1) > Pr(U_n)/Pr(U_1) or else Pr(U_1)=0.

“Theorem 3. Let Pr(-) be a separable, evidentially natural self-locating prior on a local multiverse frame.For any possible multiverse size n such that 1 < n < 1/p (where p is the fine-tuning parameter), for any center i such that Pr(E(i)) > 0, n universes are confirmed relative to one: Pr i (U n ) Pr i (U 1 ) > Pr(U n ) Pr(U 1 ) or else Pr(U 1 ) 0.”

• AssumptionThe authors state this as an assumption: In each world there is one additional null center, which represents the possibility of not being an agent.
• AssumptionThe authors state this as an assumption: Any duplication map from one universe to another maps each agent in the first universe to an agent in the other; whether something counts as an agent does not depend on anything beyond its universe.
• AssumptionThe authors state this as an assumption: For simplicity, each non-null center i has a unique evidential type: there is a unique set of centers E(j) that includes i.

For large n, the confirmation factor in Corollary 1 approaches 1/p.

“For large n, this confirmation factor approaches 1/p.”

Given a separable qualitative prior on a local multiverse structure and an evidential posterior that satisfies Agents, for any possible multiverse size n with 1 < n < 1/p, where p is the fine-tuning parameter, U_n is confirmed relative to U_1.

“Theorem 1.Let Pr(-) be a separable qualitative prior on a local multiverse structure, and let Pr * (-) be an evidential posterior that satsfies Agents.For any possible multiverse size n, if 1 < n < 1/p, where p is the fine-tuning parameter, then U n is confirmed relative to U 1 .”

Given a separable qualitative prior on a local multiverse structure and an ordinary evidential posterior that satisfies Agents, for any possible multiverse size n > 1, U_n is confirmed relative to U_1.

“Theorem 2. Let Pr(-) be a separable qualitative prior on a local multiverse structure.Let Pr * (-) be an ordinary evidential posterior that satisfies Agents.For any possible multiverse size n > 1, U n is confirmed relative to U 1 .”

For separable priors and ordinary evidential posteriors, self-locating evidence confirms a multiverse at least as strongly as the qualitative evidence 'There is life' does.

“In other words, for separable priors and ordinary evidential posteriors, self-locating evidence confirms a multiverse at least as strongly as the qualitative evidence”

For any evidentially natural self-locating prior Pr(-), for any center i such that Pr(E(i)) > 0, the posterior Pr_i(-) is evidential (definition 6).

“Lemma 6.For any evidentially natural self-locating prior Pr(-), for any center i such that Pr(E(i)) > 0, the posterior Pr i (-) is evidential (definition 6).”

The necessary and sufficient condition for Compartmentalized Conditionalization to be representable by a self-locating prior is that t(i) is well-defined: all evidential types that contain a common center i must have the same number of elements in i's world.

“The necessary and sufficient condition for Compartmentalized Conditionalization to be representable by a self-locating prior is that t(i) is well-defined: that is, all evidential types that contain a common center i must have the same number of elements in i's world.”

Furthermore, the proof of the theorem shows us that for large n and small p, the only way to avoid confirming a multiverse is if, in the absence of fine-tuning, our evidence strongly disconfirms a multiverse.

“Furthermore, the proof of the theorem shows us that for large n and small p, the only way to avoid confirming a multiverse is if, in the absence of fine-tuning, our evidence strongly disconfirms a multiverse.”

Some standard rules for self-locating probabilities do not vindicate the idea that this more specific evidence does not confirm the multiverse.

“Some standard rules for self-locating probabilities do not vindicate this idea.”

Given a separable prior and an ordinary evidential posterior that satisfies Agents, for any possible multiverse size n, either Pr*(U_1)=0 or else Pr*(U_n)/Pr*(U_1) ≥ [Pr(U_n)/Pr(U_1)] * [1-(1-p)^n]/p, where p is the fine-tuning parameter.

“Corollary 1.Let Pr(-) be a separable prior, and let Pr * (-) be an ordinary evidential posterior that satisfies Agents.For any possible multiverse size n, either Pr * (U 1 ) 0 or else Pr * (U n ) Pr * (U 1 ) ≥ Pr(U n ) Pr(U 1 ) 1 -(1 -p) n p”

Neither U_n nor U_m is relatively confirmed conditional on I_k Q.

“Thus lemma 4 tells us that neither of U n or U m is relatively confirmed conditional on I k Q.”