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<h2 class="hd hd-2 unit-title">Objective Probability</h2>
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<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;"> In this lecture we will talk about objective probability. I’ll start by saying a few words about whether there are any mathematical constraints on an assignment of objective probabilities; I’ll then try to address the more philosophical question of what objective probability is.</span></p>
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<h2 class="hd hd-2 unit-title">Objective Probability Functions</h2>
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<p><span style="font-family: 'book antiqua', palatino;">Because of the Objective-Subjective Connection (Lecture 6.1.1), some of the conclusions we reached in our discussion of subjective probability carry over to the notion of objective probability. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">Recall, in particular, that in Lecture 6.2.1 we considered the hypothesis that you can’t count as a perfectly rational agent unless your credence function is a <em>probability function</em>. When the Objective-Subjective Connection is in place, that hypothesis entails that an assignment of objective probabilities must also be a probability function, and therefore satisfy the Necessity and Additivity constraints of Lecture 6.2.1. </span></p>
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<legend id="f951305c1dad43d9b71625f54d834a4a_2_1-legend" class="response-fieldset-legend field-group-hd"><span style="font-family: 'book antiqua', palatino;">Suppose we live in a deterministic world? What is the objective probability that a coin toss lands Heads? (Assume that the outcome of the coin toss is determined by the initial conditions of the universe, in conjunction with the laws)</span></legend>
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<h2 class="hd hd-2 unit-title">Bayes' Law and Conditionalization</h2>
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<p><span style="font-family: 'book antiqua', palatino;">Just like one can speak of conditional <em>subjective</em> probabilities, so one can speak of conditional <em>objective</em> probabilities. One can speak, for instance, of the objective probability that a pair of coin-tosses yields two Heads, given that it yields at least one Head. In symbols: <span class="math inline">\(p(\text{two Heads} \, |\, \text{at least one Head})\)</span>. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">How are objective conditional probabilities related to their subjective counterparts? </span></p>
<p><span style="font-family: 'book antiqua', palatino;">It is natural to answer this question by setting forth a conditional version of the Objective-Subjective Connection:</span></p>
<p style="padding-left: 30px;"><span style="font-family: 'book antiqua', palatino;"><strong>The Objective-Subjective Connection (Conditional Version)</strong></span><br /><span style="font-family: 'book antiqua', palatino;">The objective probability of <span class="math inline">\(A\)</span> given <span class="math inline">\(H\)</span> at time <span class="math inline">\(t\)</span> is the subjective conditional probability that a perfectly rational agent would assign to <span class="math inline">\(A\)</span> given <span class="math inline">\(H\)</span>, if she had perfect information about events before <span class="math inline">\(t\)</span> and no information about events after <span class="math inline">\(t\)</span>.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">The conditional version of the Objective-Subjective Connection can be used to show that the objective probabilities must evolve in accordance with a version of conditionalization, on the hypothesis that perfectly rational agents update by conditionalization. It can also be used to show that the objective probabilities satisfy Bayes' Law, on the hypothesis that you can't count as a perfectly rational agent unless your conditional and unconditional <em>subjective</em> probabilities are related by Bayes' Law. <br /><br /></span></p>
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<p>Use the conditional version of the Objective-Subjective Connection to show that the objective probabilities must evolve in accordance with a version of conditionalization. More specifically, show that for any proposition \(A\) and any times \(t_0 \lt t _1\), </p>
<p>\(p_{t_{1}}(A) = p_{t_0}(A|H_{t_1})\)</p>
<p>where \(p_{t_i}\) is the objective probability function at \(t_i\) and \(H_{t_i}\) consists of perfect information about events before \(t_i\).(You may assume that that perfectly rational agents update by conditionalization.</p>
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<p>
<span style="font-family: 'book antiqua', palatino;">Bayes&#8217; Law allows us to perform all sorts of interesting computations. Suppose, for example, that you have an urn with two red balls and two black balls, and use a random procedure to draw balls from the urn. You draw once, and keep the ball in your pocket. The probability of getting red on your first draw <span class="math inline">\((R_1)\)</span> is <span class="math inline">\(1/2\)</span>. But the probability of getting red on the second draw <span class="math inline">\((R_2)\)</span> depends on whether you get red on your first draw. If your first draw is red, the urn will be left with two black balls and one red ball. So the probability of red on your second draw is <span class="math inline">\(1/3\)</span>: <span class="math inline">\(p(R_2 | R_1) = 1/3\)</span>. But if your first draw is black <span class="math inline">\((B_1)\)</span>, the urn will be left with one black ball and two red balls. So the probability of red on your second draw is <span class="math inline">\(2/3\)</span>: <span class="math inline">\(p(R_2 | B_1) = 2/3\)</span>. Accordingly: <span class="math display">\[\begin{array}{c|c}
\mbox{First Draw} &amp; \mbox{Second Draw} \\ \hline
p(R_1) = 1/2 &amp; p(R_2 | R_1) = 1/3 \\
p(B_1) = 1/2 &amp; p(R_2 | B_1) = 2/3
\end{array}\]</span> Use Bayes&#8217; Law allows to calculate the unconditional probability, <span class="math inline">\(p(R_2)\)</span>, of getting red on your second draw.</span>
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<span style="font-family: 'book antiqua', palatino;">In Lecture 4 we talked about probabilistic dependence and independence. These notions can be characterized formally, using the notion of conditional probability. Here is one way of doing so:</span>
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<span style="font-family: 'book antiqua', palatino;">If <span class="math inline">\(p(A|B) = p(A|\mbox{not-}B)\)</span>, then <span class="math inline">\(A\)</span> is <span>probabilistically independent</span> of <span class="math inline">\(B\)</span>. Otherwise, each of <span class="math inline">\(A\)</span> and <span class="math inline">\(B\)</span> is <span>probabilistically dependent</span> on the other.</span>
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<span style="font-family: 'book antiqua', palatino;">In fact, there are several equivalent ways of defining probabilistic independence. Is the following an equivalent way of defining probabilistic independence?</span>
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<span style="font-family: 'book antiqua', palatino;"><span class="math inline">\(p(A|B) = p(A|\text{not-}B)\)</span> if and only if <span class="math inline">\(p(AB) = p(A)\cdot p(B)\)</span>.</span>
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<span style="font-family: 'book antiqua', palatino;">How about the following? Is it an equivalent way of defining probabilistic independence?</span>
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<span style="font-family: 'book antiqua', palatino;"><span class="math inline">\(p(A|B) = p(A|\text{not-}B)\)</span> if and only if <span class="math inline">\(p(A|B) = p(A)\)</span>.</span>
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<span style="font-family: 'book antiqua', palatino;"><span class="math inline">\(p(A|B) = p(A|\mbox{not-}B)\)</span> if and only if <span class="math inline">\(p(B|A) = p(B|\mbox{not-}A)\)</span>.</span>
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<span style="font-family: 'book antiqua', palatino;">(In other words, <span class="math inline">\(A\)</span> is probabilistically independent of <span class="math inline">\(B\)</span> if and only if <span class="math inline">\(B\)</span> is probabilistically independent of <span class="math inline">\(A\)</span>.)</span>
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Problem 4
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<span style="font-family: 'book antiqua', palatino;">Suppose that the objective probability of a fair coin landing Heads is \(\frac{1}{2}\). If you toss a fair coin <span class="math inline">\(n\)</span> times, what is the objective probability that it will land Heads on every single toss? (Assume that different tosses are probabilistically independent of one another.)</span>
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<span style="font-family: 'book antiqua', palatino;">\(\left(\frac{1}{2}\right)^n\)</span>
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Problem 5
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<span style="font-family: 'book antiqua', palatino;">Part of what it is for a coin to have a 50% chance of landing Heads is for it to be <em>possible</em> that the coin land Heads.</span>
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<span style="font-family: 'book antiqua', palatino;">Notice, moreover, that the possibility of a coin&#8217;s landing Heads does not depend on how many previous coin tosses have landed Heads, since coin tosses are independent of one another. From this it follows that if you toss a fair coin infinitely many times, it is <em>possible</em> (though vanishingly unlikely) that the coin lands Heads every single time.</span>
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<h2 class="hd hd-2 unit-title">What is Objective Probability?</h2>
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<p><span style="font-family: 'book antiqua', palatino;">Recall that one of Seaborgium’s isotopes, <span class="math inline">\(^{265}\mbox{Sg}\)</span>, has a half-life of 8.9 seconds. That means that if you take a particle of <span class="math inline">\(^{265}\mbox{Sg}\)</span> and wait 8.9 seconds, the probability that it will decay is 50%. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">When we speak of probability in this context, we are talking about <strong>objective probability</strong>. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">To say that <span class="math inline">\(^{265}\mbox{Sg}\)</span> has a half-life of 8.9 seconds is to describe a feature of the external world, rather than an aspect of anyone’s psychology. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">But what feature of the world is that? </span></p>
<p><span style="font-family: 'book antiqua', palatino;">In this section we will try to answer that question.</span></p>
<blockquote>
<p><span style="font-family: 'book antiqua', palatino;">(A note about terminology: philosophers sometimes refer to objective probability as “objective chance”, or simply “chance”. I will occasionally follow that practice below.)</span></p>
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<h4><span style="font-family: 'book antiqua', palatino;">Frequentism</span></h4>
<p><span style="font-family: 'book antiqua', palatino;">We will start by considering an answer that is tempting, but ultimately unsatisfactory. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">According to <strong>frequentism</strong>, what it means for a particle of <span class="math inline">\(^{265}\mbox{Sg}\)</span> to have a 50% probability of decaying within the next 8.9 seconds is for the <em>frequency</em> of decay of <span class="math inline">\(^{265}\mbox{Sg}\)</span> particles to be 50%. In other words, it means that 50% of the <span class="math inline">\(^{265}\mbox{Sg}\)</span> that exist at a given time decay within 8.9 seconds. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">Unfortunately, frequentism cannot be correct. </span></p>
<p><span style="font-family: book antiqua, palatino;">To see this, think of a "coin toss'' as the result of observing a particle of \(^{265}\mbox{Sg}\) for 8.9 seconds. If the particle decays within that period, our "coin'' is said to have landed Heads; otherwise it is said to have landed Tails. Now imagine a situation in which only three particles of \(^{265}\mbox{Sg}\) ever exist and in which the ensuing "coin tosses'' yield: Heads, Heads, Tails. Before any of the "tosses'' took place, what was the chance that they would land Heads? The answer ought to be 1/2. But frequentism entails the mistaken conclusion that the answer is 2/3.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">A slight improvement on straight frequentism is what is sometimes called “hypothetical frequentism”. The basic idea is this: if we had performed a sufficient number of "coin tosses'', we would have gotten Heads 50% of the time. (I will set aside the question of how many tosses would count as ``sufficient''). </span></p>
<p><span style="font-family: 'book antiqua', palatino;">Unfortunately, hypothetical frequentism can't be right either: it is simply not true that a fair coin -- a coin with a 50% chance of landing Heads -- must land Heads 50% of the time in the long run. It is perfectly possible, for example, for such a coin to land Heads on every single toss. Such an outcome would be extremely unlikely, to be sure. But it is certainly <em>possible</em>; that just follows from the fact that the coin has a 50% chance of landing Heads on any given toss.</span></p>
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<p><span style="font-family: 'book antiqua', palatino;">The fact that a coin is fair does not guarantee that it will land Heads 50% of the time. But a weaker statement is true. It is true that if the coin were tossed a sufficiently large number of times, then it would <em>with very high probability</em> land Heads approximately 50% of the time.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">A rigorous version of this principle is known as the <strong>Law of Large Numbers</strong>, and is a theorem of probability theory. It is because of this law that casino owners can feel confident that they will, in the long run, make a profit. A particular table, or a particular slot machine, will have bad nights now and then. But as long as the casino is visited by enough players over a sufficiently lengthy period of time, it is extremely probable that the casino’s gains will outweigh its losses.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">The Law of Large Numbers tells us that frequentism has a grain of truth, since it entails that observed frequencies can be a good guide to probabilities. But the Law of Large Numbers also tells us what frequentism got wrong. Frequencies are not necessarily a <em>perfect</em> guide to probabilities. The Law tells us that if the objective probability of a coin landing Heads is 50%, then it is <em>very probable</em> that the coin will, in the long run, land Heads approximately 50% of the time. But that means it is <em>possible</em> that it won’t. And if frequencies aren’t a perfect <em>guide</em> to probabilities, they certainly aren’t <em>identical</em> to probabilities.</span></p>
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<span style="font-family: 'book antiqua', palatino;">Here is a more precise statement of the Law of Large Numbers,</span>
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<span style="font-family: 'book antiqua', palatino;">
<strong>Law of Large Numbers</strong>
</span>
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<span style="font-family: 'book antiqua', palatino;">Suppose that events of type <span class="math inline">\(T\)</span> have a probability of <span class="math inline">\(p\)</span> of resulting in outcome <span class="math inline">\(O\)</span>. Then, for any real numbers <span class="math inline">\(\epsilon\)</span> and <span class="math inline">\(\delta\)</span> larger than zero, there is an <span class="math inline">\(N\)</span> such that the following will be true with a probability of at least <span class="math inline">\(1-\epsilon\)</span>:
<blockquote><p><span style="font-family: 'book antiqua', palatino;">If <span class="math inline">\(M &gt; N\)</span> events of type <span class="math inline">\(T\)</span> occur, the proportion of them that result in outcome <span class="math inline">\(O\)</span> will be <span class="math inline">\(p \pm \delta\)</span> (&#8220;<span class="math inline">\(\pm\)</span>&#8221; means &#8220;plus or minus&#8221;).</span></p></blockquote></span>
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<span style="font-family: 'book antiqua', palatino;">Suppose there is a casino with many slot machines. Each time a customer uses one of the slot machines, the casino has a 49% chance of losing a dollar and a 51% chance of winning a dollar.</span>
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<span style="font-family: 'book antiqua', palatino;">Is the following true or false?</span>
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<span style="font-family: 'book antiqua', palatino;">If the casino has enough customers at the slot machines, it is at least 99.99% likely to end up with a profit of at least a million dollars.</span>
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<h2 class="hd hd-2 unit-title">Rationalism</h2>
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<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;"> We have seen that frequentism cannot be correct as an account of objective probability. But if frequentism is not an option, what <em>is</em> objective probability? </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">One option is to embrace <strong>rationalism</strong>: the view that there is nothing more to objective probability than the Objective- Subjective Connection of Sections 6.1.1 and 6.3.2. </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">This means that a claim about objective probability is a claim about what a perfectly rational subject would believe, if she had perfect information about the past (but none about the future). </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">A rationalist account of objective probability is consistent with the idea that objective probabilities are not concerned with the psychology of particular subjects, since the demands of rationality need not depend on what anyone in fact believe. </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">It is natural to think that they instead depend on relevant features of the physical world. When we say, for example that [mathjaxinline]^{265}\mbox{Sg}[/mathjaxinline] has a half-life of 8.9 seconds, we’re describing a feature of the physical world. But we’re describing it indirectly, by saying that the physical world is such as to demand a certain rational stance. Specifically: the physical world makes it the case that the rational degree of belief to have about whether a particle of [mathjaxinline]^{265}\mbox{Sg}[/mathjaxinline] will decay in the next 8.9 seconds is 0.5—assuming one has perfect information about the past, but none about the future. </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">Notice, however, that on a rationalist conception of objective probability it is not clear that the objective probabilities are always well defined. To see the worry, note that rationalism entails that the objective probabilities at [mathjaxinline]t[/mathjaxinline] can only well-defined if there is a definite fact of the matter about what the <em>credences</em> of a perfectly rational agent should be, when she has perfect information about events prior to [mathjaxinline]t[/mathjaxinline] (and no information about the future). </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">But recall that we were unable to identify an satisfactory version of the Principle of Indifference (Section 6.2.4). In the absence of such a principle it is not clear that there will always be definite fact of the matter about what a perfectly rational agent ought to believe.</span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;"> Notice, in particular, that it is not clear that there is a definite fact of the matter about what a perfectly rational agent’s “initial” credences should be—her credences in the absence of any evidence about the way the world is. And if our perfectly rational agent updates by conditionalization, then unclarity about initial credences can translate into unclarity about future credences, since the subject’s future credences will only be well-defined if her initial credences, conditional on the relevant evidence, are well-defined. So the rationalist may need to be open to the possibility that there may not always be a definite fact of the matter about what the objective probabilities are. </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">I myself am a <strong>localist</strong> about objective probability. </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">I agree with the rationalist that there is no more to the notion of objective probability than the Objective-Subjective Connection, and I think the notion of perfect rationality is only well-defined in certain special circumstances; for example, circumstances in which there is an unproblematic way of deploying a Principle of Indifference. </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">I therefore believe that it is only in such circumstances that the objective probabilities are well-defined.</span></p>
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<h2 class="hd hd-2 unit-title">Best System Accounts of Objective Probability</h2>
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<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;"> Is there any way of further constraining the notion of objective probability, so as to guarantee that the objective probabilities are well-defined across a large range of cases? In this section I’ll tell you about a “best system” account of objective probability, according to which the objective probabilities are the result of striving for an optimal balance of simplicity and strength in our overall theory of the world. </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">We want our theories to be <strong>strong</strong>; in other words: we want them to provide us with as much information as possible. We also want them to be <strong>simple</strong>; in other words: we want the information to be articulated briefly and systematically. A theory that consisted of a huge list detailing what happens at every corner of the universe at every time would be extraordinarily strong, but it wouldn’t be simple. In contrast, a theory whose only principle was “2 + 2 = 4” would be extraordinarily simple, but wouldn’t be very strong. Good theories strike a balance between simplicity and strength. They allow us to derive a lot of information from simple principles. </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">The notion of objective probability might be used to help achieve such a balance. Suppose, for example, that someone flips a coin and that we want to predict the outcome. We could try to come up with a definite prediction by using classical mechanics. In order to do so, we would need precise measurements of all forces acting on the coin, and we would need to perform a complex calculation. Although it is in principle possible to do so, it would be exceedingly difficult in practice. </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">An alternative is to use a probabilistic theory, which states that the objective probability that the coin lands Heads is 50%. Such a theory is not particularly strong. But it is simple enough to be put into practice without specialized measuring equipment or sophisticated computational capabilities. Because a probabilistic theory does not tell us conclusively how the coin will land, it is of limited utility. But it is certainly not useless. It can be used to conclude, for example, that it would be a good idea to accept a bet whereby you win two dollars if the coin lands Heads and lose one dollar if the coin lands Tails. And it might give you enough information to set up a successful casino. </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">How could one determine if a probabilistic theory is accurate? Suppose someone flips a coin 100,000 times, and gets Heads 31% of the time. This would extremely unlikely to happen if the probability of Heads were 50%. So there are good reasons to reject a probabilistic theory according to which the coin has a 50% chance of landing Heads. If, on the other hand, we observed that 50.023% of the coin landed Heads, we would have little reason to doubt the accuracy of the theory. </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">Say that a<strong> best system</strong> for a particular phenomenon is a description of the phenomenon that satisfies two conditions: (1) the description is accurate (i.e. its non-probabilistic claims are all true, and its probabilistic claims are reasonably close to the observed frequencies), and (2) the description delivers an optimal combination of simplicity and strength. (Just what “optimal” means in this context is something that needs to be spelled out.) </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">Philosopher David Lewis, who we have encountered before, famously argued that the notion of a best system is key to understanding the notion of objective probability. He thought, for example, that all it takes for the objective probability of a quantum event to be [mathjaxinline]x[/mathjaxinline] is for our best theory of quantum phenomena to tell us that the objective probability of the event is [mathjaxinline]x[/mathjaxinline]. In general, he thought that what it is for the objective probability of event [mathjaxinline]E[/mathjaxinline] to be [mathjaxinline]x[/mathjaxinline] is simply for the best system for the relevant phenomenon to tell us that the objective probability of [mathjaxinline]E[/mathjaxinline] is [mathjaxinline] x[/mathjaxinline]. </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">Is this story plausible as an account of objective probability? The answer depends, in part, on whether one thinks that there are objective standards for counting a theory as an optimal combination of simplicity and strength. Lewis himself believed that there were objective standards of simplicity. On this view, there is no conflict between a best system account of objective probability and the claim that objective probabilities are independent of our psychologies. There’s also no conflict with rationalism, since one could claim that rationality demands that one’s credences be fixed in accordance with the system that is objectively best.</span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;"> There is, however, room for disagreeing with Lewis about whether there really is such a thing as objective simplicity (or, indeed, or an objective fact of the matter about what counts as an “optimal” combination of simplicity and strength). My own view is that simplicity is in the eye of the beholder: what counts as simple for humans is may not count as simple for, e.g. Martians. If this is right, then the best system account of probability fails to deliver a notion of probability that is independent of our psychologies. What it delivers instead is a notion of probability that is partly about objective features of the world and partly about our subjective standards for representing the world in ways that we find simple. (There is a lot more to be said about the best-system account of objective probabilities. If you’d like to know more, have a look at the readings I suggest in Section 6.6.)</span></p>
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<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;"> Suppose you’re not comfortable with a localist account of objective probability, according to which the objective probabilities are not generally welldefined, but you also think that a best-systems theory is too subjective to help further constrain objective probability. </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">Is there something to be done? </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">There is always the option of being a <strong>primitivist</strong> about objective probabilities. </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">A primitivist thinks of the notion of objective probability as a theoretical primitive, alongside the notion of a gluon and that of electrical charge. She concedes that it might be possible to establish interesting links between objective probability and rationality, but thinks that objective probability goes beyond such links. </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">There is certainly nothing incoherent about primitivism. After all: explanations must end somewhere. Why couldn’t they end when we get to the objective probabilities? </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">It is worth noting, however, that primitivism would deliver a somewhat intractable picture of the world. </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">To see this, suppose a primitivist tells you that the objective probability of a certain event [mathjaxinline]E[/mathjaxinline] is 30%. To what degree should you believe that it will occur? </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">Unless you have information about the future, one would have thought that the answer is clear: you should believe that [mathjaxinline]E[/mathjaxinline] will occur to degree 0.3. </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">The problem with primitivism is that it doesn’t make clear what it is about the notion of objective probability that would warrant such a conclusion, because it doesn’t explain why rational belief ought to track objective probability. (Notice, in contrast, that the best systems theorist of Section 6.3.4.1 is in a position to claim that the conclusion is warranted because a subject who adjusts her beliefs to the objective probabilities represents the world with an optimal balance of simplicity and strength, and that so representing the world is a constraint of rationality.)</span></p>
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