Something has been bugging me about E.T. Jaynes' treatment of continuous parameters.

In his book Probability Theory: The Logic of Science, uses notation that I am unfamiliar with when getting probabilities for continuous parameters from their densities. For example, in chapter 4, page 420, equation 4-49, Jaynes "discretizes" the continuous parameter $f \in [0,1]$ by taking the interval $(f, f+\mathrm df)$ and putting a probability on it via $g$, the density of $f$:

.. so we will go back to the original probability from (4-3):

$$P(A|DX)=\frac{P(D|AX)P(A|X)}{P(D|X)}$$

Letting $A$ now stand for the proposition "The fraction of bad ones is in the range $(f, f+\mathrm df)$", there is a prior pdf

$$P(A|X)=g(f|X)~\mathrm df, $$

which gives the probability that the fraction of bad ones is in the range $\mathrm df;$

Why isn't the above written as $P(A|X)=\displaystyle\int_f^{f+\mathrm dx} g(t|X)~\mathrm dt$ (with dummy variable $t$)? $g(f|X)~\mathrm df$ for a finite $\mathrm df$ is the left-rectangle approximation (a la Riemann sums), not $P(A|X).$ If you were to partition the $[0,1]$ into many separate proposition and assigned probability as above, it may not sum to $1.$

Or does $\mathrm df$ denote some sort of infinitesimal notation I'm not familiar with? I've never seem that notation used 'alone'-- afaik it's only meaningful when used in context of differentiation/integration, like $\frac{\mathrm d}{\mathrm dx}$ or $\int \textrm{foo}()~\mathrm dx$. If it is some infinitesimal notation, this seems to run contrary to Jaynes' philosophy of sticking to finite number of propositions until the end, where the limit may be taken explicitly.

As another example, in chapter 15, page 1514, equation (15-38), he writes the bivariate normal probability with correlation $\rho$ as:

$$p(\mathrm dx~\mathrm dy|I)=\frac{\sqrt{1-\rho^2}}{2\pi} \exp\left[-\frac{1}{2}(x^2 + y^2 -2 \rho x y)\right]~\mathrm dx\mathrm dy$$

which is a little different again. Presumably $p(\mathrm dx~\mathrm dy|I)$ means "Probability that the true value of $(x,y)$ falls in $(x+\mathrm dx,y+\mathrm dy)$", but here again he writes the Riemann-sum-like approximation instead of the integral of the density on $[(x,x+\mathrm dx) \times (y,y+~\mathrm dy)].$

What am I missing?

Something has been bugging me about E.T. Jaynes' treatment of continuous parameters.

In his book Probability Theory: The Logic of Science, uses notation that I am unfamiliar with when getting probabilities for continuous parameters from their densities. For example, in chapter 4, page 420, equation 4-49, Jaynes "discretizes" the continuous parameter $f \in [0,1]$ by taking the interval $(f, f+df)$ and putting a probability on it via $g$, the density of $f$:

.. so we will go back to the original probability from (4-3):

$$P(A|DX)=\frac{P(D|AX)P(A|X)}{P(D|X)}$$

Letting $A$ now stand for the proposition "The fraction of bad ones is in the range $(f, f+df)$", there is a prior pdf

$$P(A|X)=g(f|X)~df, $$

which gives the probability that the fraction of bad ones is in the range $df;$

Why isn't the above written as $P(A|X)=\displaystyle\int_f^{f+dx} g(t|X)~dt$ (with dummy variable $t$)? $g(f|X)~df$ for a finite $df$ is the left-rectangle approximation (a la Riemann sums), not $P(A|X).$ If you were to partition the $[0,1]$ into many separate proposition and assigned probability as above, it may not sum to $1.$

Or does $df$ denote some sort of infinitesimal notation I'm not familiar with? I've never seem that notation used 'alone'-- as far as I know it's only meaningful when used in context of differentiation/integration, like $\frac{d}{dx}$ or $\int \textrm{foo}()~dx$. If it is some infinitesimal notation, this seems to run contrary to Jaynes' philosophy of sticking to finite number of propositions until the end, where the limit may be taken explicitly.

As another example, in chapter 15, page 1514, equation (15-38), he writes the bivariate normal probability with correlation $\rho$ as:

$$p(dx~dy|I)=\frac{\sqrt{1-\rho^2}}{2\pi}\,\exp\left[-\frac{1}{2}(x^2 + y^2 -2 \rho x y)\right]\,dx\,dy$$

which is a little different again. Presumably $p(dx~dy|I)$ means "Probability that the true value of $(x,y)$ falls in $(x+dx,y+dy)$", but here again he writes the Riemann-sum-like approximation instead of the integral of the density on $[(x,x+dx) \times (y,y+dy)].$

What am I missing?

Something has been bugging me about E.T. Jaynes' treatment of continuous parameters.

In his book Probability Theory: The Logic of Science, uses notation that I am unfamiliar with when getting probabilities for continuous parameters from their densities. For example, in chapter 4, page 420, equation 4-49, Jaynes "discretizes" the continuous parameter $f \in [0,1]$ by taking the interval $(f, f+df)$ and putting a probability on it via $g$, the density of $f$:

.. so we will go back to the original probability from (4-3):

$P(A|DX)=\frac{P(D|AX)P(A|X)}{P(D|X)}$

Letting A now stand for the proposition "The fraction of bad ones is in the range $(f, f+df)$", there is a prior pdf

$P(A|X)=g(f|X)df$,

which gives the probability that the fraction of bad ones is in the range $df$,

Why isn't the above written as $P(A|X)=\int_f^{f+dx} g(t|X)dt$ (with dummy variable $t$)? $g(f|X)df$ for a finite $df$ is the left-rectangle approximation (a la Riemann sums), not $P(A|X)$. If you were to partition the $[0,1]$ into many separate proposition and assigned probability as above, it may not sum to 1.

Or does $df$ denote some sort of infinitesimal notation I'm not familiar with? I've never seem that notation used 'alone'-- afaik it's only meaningful when used in context of differentiation/integration, like $\frac{d}{dx}$ or $\int foo()dx$. If it is some infinitesmal notation, this seems to run contrary to Jaynes' philosophy of sticking to finite number of propositions until the end, where the limit may be taken explicitely.

As another example, in chapter 15, page 1514, equation (15-38), he writes the bivariate normal probability with correlation $\rho$ as:

$p(dx\ dy|I)=\frac{\sqrt{1-\rho^2}}{2\pi}exp[-\frac{1}{2}(x^2 + y^2 -2 \rho x y)]dxdy$

Which is a little different again. Presumably $p(dx\ dy|I)$ means "Probability that the true value of $(x,y)$ falls in $(x+dx,y+dy)$", but here again he writes the Riemann-sum-like approximation instead of the integral of the density on $[(x,x+dx) \times (y,y+dy)]$.

What am I missing?

Something has been bugging me about E.T. Jaynes' treatment of continuous parameters.

In his book Probability Theory: The Logic of Science, uses notation that I am unfamiliar with when getting probabilities for continuous parameters from their densities. For example, in chapter 4, page 420, equation 4-49, Jaynes "discretizes" the continuous parameter $f \in [0,1]$ by taking the interval $(f, f+\mathrm df)$ and putting a probability on it via $g$, the density of $f$:

.. so we will go back to the original probability from (4-3):

$$P(A|DX)=\frac{P(D|AX)P(A|X)}{P(D|X)}$$

Letting $A$ now stand for the proposition "The fraction of bad ones is in the range $(f, f+\mathrm df)$", there is a prior pdf

$$P(A|X)=g(f|X)~\mathrm df, $$

which gives the probability that the fraction of bad ones is in the range $\mathrm df;$

Why isn't the above written as $P(A|X)=\displaystyle\int_f^{f+\mathrm dx} g(t|X)~\mathrm dt$ (with dummy variable $t$)? $g(f|X)~\mathrm df$ for a finite $\mathrm df$ is the left-rectangle approximation (a la Riemann sums), not $P(A|X).$ If you were to partition the $[0,1]$ into many separate proposition and assigned probability as above, it may not sum to $1.$

Or does $\mathrm df$ denote some sort of infinitesimal notation I'm not familiar with? I've never seem that notation used 'alone'-- afaik it's only meaningful when used in context of differentiation/integration, like $\frac{\mathrm d}{\mathrm dx}$ or $\int \textrm{foo}()~\mathrm dx$. If it is some infinitesimal notation, this seems to run contrary to Jaynes' philosophy of sticking to finite number of propositions until the end, where the limit may be taken explicitly.

As another example, in chapter 15, page 1514, equation (15-38), he writes the bivariate normal probability with correlation $\rho$ as:

$$p(\mathrm dx~\mathrm dy|I)=\frac{\sqrt{1-\rho^2}}{2\pi} \exp\left[-\frac{1}{2}(x^2 + y^2 -2 \rho x y)\right]~\mathrm dx\mathrm dy$$

which is a little different again. Presumably $p(\mathrm dx~\mathrm dy|I)$ means "Probability that the true value of $(x,y)$ falls in $(x+\mathrm dx,y+\mathrm dy)$", but here again he writes the Riemann-sum-like approximation instead of the integral of the density on $[(x,x+\mathrm dx) \times (y,y+~\mathrm dy)].$

What am I missing?

Something has been bugging me about E.T. Jaynes' treatment of continuous parameters.

In his book Probability Theory: The Logic of Science, uses notation that I am unfamiliar with when getting probabilities for continuous parameters from their densities. For example, in chapter 4, page 420, equation 4-49, Jaynes "discretizes" the continuous parameter $f \in [0,1]$ by taking the interval $(f, f+df)$ and putting a probability on it via $g$, the density of $f$:

.. so we will go back to the original probability from (4-3):

 

$P(A|DX)=\frac{P(D|AX)P(A|X)}{P(D|X)}$

 

Letting A now stand for the proposition "The fraction of bad ones is in the range $(f, f+df)$", there is a prior pdf

 

$P(A|X)=g(f|X)df$,

 

which gives the probability that the fraction of bad ones is in the range $df$,

Why isn't the above written as $P(A|X)=\int_f^{f+dx} g(t|X)dt$ (with dummy variable $t$)? $g(f|X)df$ for a finite $df$ is the left-rectangle approximation (a la Riemann sums), not $P(A|X)$. If you were to partition the $[0,1]$ into many separate proposition and assigned probability as above, it may not sum to 1.

Or does $df$ denote some sort of infinitesimal notation I'm not familiar with? I've never seem that notation used 'alone'-- afaik it's only meaningful when used in context of differentiation/integration, like $\frac{d}{dx}$ or $\int foo()dx$. If it is some infinitesmal notation, this seems to run contrary to Jaynes' philosophy of sticking to finite number of propositions until the end, where the limit may be taken explicitely.

As another example, in chapter 15, page 1514, equation (15-38), he writes the bivariate normal probability with correlation $\rho$ as:

$p(dx\ dy|I)=\frac{\sqrt{1-\rho^2}}{2\pi}exp[-\frac{1}{2}(x^2 + y^2 -2 \rho x y)]dxdy$

Which is a little different again. Presumably $p(dx\ dy|I)$ means "Probability that the true value of $(x,y)$ falls in $(x+dx,y+dy)$", but here again he writes the Riemann-sum-like approximation instead of the integral of the density on $[(x,x+dx) \times (y,y+dy)]$.

What am I missing?

Something has been bugging me about E.T. Jaynes' treatment of continuous parameters.

In his book Probability Theory: The Logic of Science, uses notation that I am unfamiliar with when getting probabilities for continuous parameters from their densities. For example, in chapter 4, page 420, equation 4-49, Jaynes "discretizes" the continuous parameter $f \in [0,1]$ by taking the interval $(f, f+df)$ and putting a probability on it via $g$, the density of $f$:

.. so we will go back to the original probability from (4-3):

$P(A|DX)=\frac{P(D|AX)P(A|X)}{P(D|X)}$

Letting A now stand for the proposition "The fraction of bad ones is in the range $(f, f+df)$", there is a prior pdf

$P(A|X)=g(f|X)df$,

which gives the probability that the fraction of bad ones is in the range $df$,

Why isn't the above written as $P(A|X)=\int_f^{f+dx} g(t|X)dt$ (with dummy variable $t$)? $g(f|X)df$ for a finite $df$ is the left-rectangle approximation (a la Riemann sums), not $P(A|X)$. If you were to partition the $[0,1]$ into many separate proposition and assigned probability as above, it may not sum to 1.

Or does $df$ denote some sort of infinitesimal notation I'm not familiar with? I've never seem that notation used 'alone'-- afaik it's only meaningful when used in context of differentiation/integration, like $\frac{d}{dx}$ or $\int foo()dx$. If it is some infinitesmal notation, this seems to run contrary to Jaynes' philosophy of sticking to finite number of propositions until the end, where the limit may be taken explicitely.

As another example, in chapter 15, page 1514, equation (15-38), he writes the bivariate normal probability with correlation $\rho$ as:

$p(dx\ dy|I)=\frac{\sqrt{1-\rho^2}}{2\pi}exp[-\frac{1}{2}(x^2 + y^2 -2 \rho x y)]dxdy$

Which is a little different again. Presumably $p(dx\ dy|I)$ means "Probability that the true value of $(x,y)$ falls in $(x+dx,y+dy)$", but here again he writes the Riemann-sum-like approximation instead of the integral of the density on $[(x,x+dx) \times (y,y+dy)]$.

What am I missing?