Density functions
$\require{mathtools}
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% the following cause issues with real LaTeX tho :/ maybe consider naming it \fhi instead?
\let\fi\phi % because it looks like an f
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% .. operators
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% .. functions
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% .. analysis
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% .. sets
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% .... two-part
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% .. dotted versions (is it equal in the limit?)
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% .. named tensors
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% ... information theory
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% .. other divergences
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% .. classical
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% .. probabilistic
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% .. circuits
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% .. resources
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% .. keywords
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Relative densities
Let $P$ and $Q$ be two probability distributions such that $\supp(Q) \sse \supp(P)$. Then the relative density $\f Q P$ is very natural quantity if we think of $Q$ as the result of updating a prior $P$ based on some event $\BE$. Indeed, by the update rule we have $\f{Q(x)}{P(x)} \propto \Pr\bco{\BE}{\BX=x}$, so $\f Q P$ depends only on the relative likelihood of $\BE$ for different values of $x$. More precisely, by the update rule,
\[
\f{Q(x)}{P(x)} = \f{\Pr\bco{\BE}{\BX = x}}{\Pr[\BE]}.
\]
We will often denote density functions as $\phi(x) \ce \f{Q\p x}{P \p x}$, and by default we will imagine $x$ as drawn from $P$.
Reweighting expectations
The density function turns expectations over $P$ into expectations over $Q$:
\[
\E_{\Bx \sim P}\b{\phi(\Bx)f(\Bx)} = \sum_x P(x) \phi(x) f(x) = \sum_x Q(x)f(x) = \E_{\Bx \sim Q}\b{f(\Bx)}.
\]
In particular, the mean of any density function is $1$:
\[
\E_{\Bx \sim P}\b{\phi(\Bx)} = \E_{\Bx \sim Q}\b 1 = 1,
\]
and conversely, any $\phi$ that has mean $1$ under $P$ corresponds to $\f Q P$ for some posterior $Q$.
As a random variable
Sometimes it is convenient to talk about “the density $\f{Q(x)}{P(x)}$, with $x$ distributed according to $P$”. For that purpose, let’s define the shortcut
\[
\ff{Q}{P} \ce {}_{\Bx \sim P}\b{\f{Q(\Bx)}{P(\Bx)}}
\]
(where $_{\Bx \sim P}\b{f(\Bx)}$ denotes “the distribution of $f(x)$ for $x$ drawn from $P$”, as defined in Random expressions).
Intuitively, the double bar indicates that this expression is “over” $P$ in two different ways:
- $Q(x)$ is divided by $P(x)$
- and the probability is taken over $P$.
In particular, this allows to understand the notation $D^f\pff{Q}{P}$ for ratio divergences (and special cases such as the information divergence $\D\pff{Q}{P}$) as simply the dispersion operator $D^f$ applied to the distribution $\ff{Q}{P}$.
Absolute densities
#to-write
- in boolean cube, relative to uniform (or some other reference distribution, e.g. $p$-biased)
- in reals, relative to imaginary prior which puts density 1 everywhere
#to-write
- everything O’Donnell says about them
odonnell--boolean-analysis.pdf - the backlinks
- its Fourier coefficients are the corresponding expectations
- natural for e.g. the Kelley–Meka bounds on arithmetic progressions
- generally survey results that are more natural in either densities or Entropy deficit?
- density of a sum is convolution of densities
- see
lee--entropy-opt-1.pdfp.7 highlights