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klderivation [2024/12/09 21:19] – [Assumption 2: Restrictions on the cost function] pedroortegaklderivation [2024/12/24 12:58] (current) – [Connecting to the free energy objective] pedroortega
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 ====== Why does every choice come with an entropy tax? ====== ====== Why does every choice come with an entropy tax? ======
 +
 +> I present a very general derivation that shows how every choice carries an unavoidable "entropy tax," reflecting the hidden cost of shifting from old beliefs to new choices. 
 +
 +//Cite as: Ortega, P.A. “Why does every choice come with a tax?”, Tech Note 3, DAIOS, 2024.//
  
 Imagine every choice you make —whether trivial or life-changing— comes with a hidden "tax". This isn't a financial toll but an entropy tax, an inherent cost tied to the mental effort of shifting from what you initially believe (the prior) to what you choose after deliberation (the posterior). Imagine every choice you make —whether trivial or life-changing— comes with a hidden "tax". This isn't a financial toll but an entropy tax, an inherent cost tied to the mental effort of shifting from what you initially believe (the prior) to what you choose after deliberation (the posterior).
  
-This concept lies at the heart of [[https://royalsocietypublishing.org/doi/10.1098/rspa.2012.0683|information-theoretic bounded rationality]], and the entropy tax formula looks like this:+This concept lies at the heart of [[https://royalsocietypublishing.org/doi/10.1098/rspa.2012.0683|information-theoretic bounded rationality]], and the tax formula looks like this:
 \[ \[
    \text{Choice Tax} \propto \sum_x P(x|d) \log \frac{ P(x|d) }{ P(x) },    \text{Choice Tax} \propto \sum_x P(x|d) \log \frac{ P(x|d) }{ P(x) },
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 ===== Assumption 1: Temporal progress as conditioning ===== ===== Assumption 1: Temporal progress as conditioning =====
  
-First we need to model temporal progress of any kind. We'll go with a "spacetime" representation that is standard in measure theory. This works as follows. We assume that we have a collection of all the possible realizations of a process of interest. This is our sample space $\Omega$. To make things simple, let's assume this set is finite. We also place a probability distribution $P$ over all the realizations $\omega \in \Omega$.+First we need to model temporal progress of any kind. We'll go with a "spacetime" representation that is standard in measure theory. This works as follows. We assume that we have a collection of all the possible realizations of a process of interest. This is our sample space $\Omega$. To make things simple, let's assume this set is finite (but potentially huge). We also place a probability distribution $P$ over all the realizations $\omega \in \Omega$.
  
 Now, any event –be it a choice, an observation, a thought, etc.– is a subset of $\Omega$. Whenever an event $e \subset \Omega$ occurs, we condition our sample space by $e$. This means that we restrict our focus only on the elements $\omega \in e$ inside the event, and then renormalize our probabilities:  Now, any event –be it a choice, an observation, a thought, etc.– is a subset of $\Omega$. Whenever an event $e \subset \Omega$ occurs, we condition our sample space by $e$. This means that we restrict our focus only on the elements $\omega \in e$ inside the event, and then renormalize our probabilities: 
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 ===== Assumption 2: Restrictions on the cost function ===== ===== Assumption 2: Restrictions on the cost function =====
  
-Next, we'll impose constraints on the cost function. We want our cost function to capture efforts that are structurally consistent with the underlying probability space. The following requirements are natural:+Next, we'll impose constraints on the cost function. We want our cost function to capture efforts that are structurally consistent with the underlying probability space. (Later, we'll see how to relax these assumptions without compromising these structural constraints.) The following requirements are natural:
  
 {{ ::cost-axioms.png?nolink |}} {{ ::cost-axioms.png?nolink |}}
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 ===== Cost of deliberation ===== ===== Cost of deliberation =====
  
-Now, let's calculate the cost of transforming the prior choice probabilities into posterior choice probabilities:+Now, based on our sketch above, let's calculate the cost of transforming the prior choice probabilities into posterior choice probabilities:
 \[ \[
   \begin{align}   \begin{align}
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 We've obtained two expectation terms. The second is proportional to the Kullback-Leibler divergence between of the posterior to the prior choice probabilities. What is the first expectation? We've obtained two expectation terms. The second is proportional to the Kullback-Leibler divergence between of the posterior to the prior choice probabilities. What is the first expectation?
  
-The first expectation represents the expected cost of each individual choice. This is because each term $C(x \cap d|x \cap c)$ measures the cost of transforming the relative probability of a specific choice.+The first expectation represents the expected cost of each individual choice (if each choice were to occur deterministically). This is because each term $C(x \cap d|x \cap c)$ measures the cost of transforming the relative probability of a specific choice.
  
 ===== Connecting to the free energy objective ===== ===== Connecting to the free energy objective =====
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 We can transform the above equality into a variational principle by replacing the individual choice costs $C(x \cap d|x \cap c)$ with arbitrary numbers. The resulting expression is convex in the posterior choice probabilities $P(x|d)$, so we get a nice and clean objective function with a unique minimum. We can transform the above equality into a variational principle by replacing the individual choice costs $C(x \cap d|x \cap c)$ with arbitrary numbers. The resulting expression is convex in the posterior choice probabilities $P(x|d)$, so we get a nice and clean objective function with a unique minimum.
  
-We can even go a step further: by multiplying the expression by $-1$, we can treat the costs as utilities, obtaining+We can even go a step further: noticing that the variational problem is translationally invariant in the costs, and multiplying the expression by $-1$, we can treat the resulting "negative costs plus a constant" as utilities, obtaining
 \[ \[
   \sum_x P(x|d) U(x) - \frac{1}{\beta} \sum_x P(x|d) \log \frac{ P(x|d) }{ P(x|c) }.   \sum_x P(x|d) U(x) - \frac{1}{\beta} \sum_x P(x|d) \log \frac{ P(x|d) }{ P(x|c) }.
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