Interpretation and understanding of the relation for the photometric redshift in a given bin

In the context of photometric probe of surveys (like LSST), I need to understand the relation I have to use for photometric bins.

Considering $$p_{ph}(z_p|z)$$ the probability to measure a photometric redshift equal to $$z_p$$ knowing the real redshift is $$z$$ and given $$n(z)$$ the density distribution of objects, I have the following formula which gives the density $$n_i$$ of galaxies into $$i$$-th bin and that I would like to understand :

$$n_{i}(z)=\frac{\int_{z_{i}^{-}}^{z_{i}^{+}} \mathrm{d} z_{\mathrm{p}}\,n(z)\,p_{\mathrm{ph}}(z_{\mathrm{p}} | z)}{\int_{z_{min}}^{z_{max}} \int_{z_{i}^{-}}^{z_{i}^{+}}\,\mathrm{d} z \,\mathrm{d} z_{\mathrm{p}} \,n(z) \,p_{\mathrm{ph}}(z_{\mathrm{p}} | z)} \quad(1)$$

with $$\left(z_{i}^{-}, z_{i}^{+}\right)$$ which are the values of both sides (+ and -) of redshift $$i$$ bin.

Here my attempt to simplify : with Bayes theorem, one has :

$$p_{\mathrm{ph}}(z_{\mathrm{p}} | z) = p_{\mathrm{ph}}(z|z_{\mathrm{p}})\,\dfrac{p_{\mathrm{ph}}(z_{\mathrm{p}})}{p_{\mathrm{ph}}(z)}\quad(2)$$

and Moreover, the bottom term of $$(1)$$ gives :

$$\int_{z_{min}}^{z_{max}} \int_{z_{i}^{-}}^{z_{i}^{+}}\,\mathrm{d} z \,\mathrm{d} z_{\mathrm{p}} \,n(z) \,p_{\mathrm{ph}}(z_{\mathrm{p}} | z)$$

$$=\int_{z_{i}^{-}}^{z_{i}^{+}}\,n(z)\,p_{\mathrm{ph}}(z_{\mathrm{p}})\,\text{d}z_{\mathrm{p}}\quad(3)$$

So, by combining $$(2)$$ and $$(3)$$, we get :

$$n_{i}(z)=\dfrac{n(z)\,p_{\mathrm{ph}}(z_{\mathrm{p}})}{\int_{z_{i}^{-}}^{z_{i}^{+}}\,n(z)\,p_{\mathrm{ph}}(z_{\mathrm{p}})\,\text{d}z_{\mathrm{p}}}\quad(4)$$

BUT from this $$(4)$$ I don't know how conclude about the interpretation of this objects density into the redshift of bin $$i$$.

Surely there is an error in my attempt to simplify the expression $$(1$$) but I don't where.

If someone could help me to grasp better the subitilities of the relation $$(1)$$, from a mathematical or physics point of views, this would be nice to explain it.

UPDATE 1:

In particular, I don't agree with Equations (3) and (4). Indeed, using Bayes theorem in the bottom term of Equation (1) leads to

$$\int_{z_{min}}^{z_{max}} \int_{z_i^-}^{z_i^+} dz dz_p n(z) p_{ph}(z|z_p)\frac{p_{ph}(z_p)}{p_{ph}(z)}$$

But $$p_{ph}(z | z_p) dz$$ is not equal to $$p_{ph}(z)$$, so you can't simplify the fraction out, like you do in Equation (3)

In addition, $$dz_p n(z) p_{ph} (z_p | z)$$ has no reason to be constant, so you can't easily remove the integration sign, like you do in Equation (4).

For equation $$(3)$$, I wanted to simplify by :

$$\int_{z_{min}}^{z_{max}} \int_{z_{i}^{-}}^{z_{i}^{+}}\,\mathrm{d} z \,\mathrm{d} z_{\mathrm{p}} \,n(z) \,p_{\mathrm{ph}}(z_{\mathrm{p}} | z)= \int_{z_{i}^{-}}^{z_{i}^{+}}\,n(z)\,p_{\mathrm{ph}}(z_{\mathrm{p}})\,\text{d}z_{\mathrm{p}}\quad(5)$$

since we should have : $$p_{ph}(z) = \int_{z_{min}}^{z_{max}} p_{ph}(z | z_p)\,dz_p\quad(6)$$, shouldn't we ?

In comment above, you talk about the relation : $$dz_p\,p_{ph} (z_p | z)$$, not the equation $$(6)$$ above, do you agree ?

From an intuitive point of view, I agree that ratio between density $$n_i(z)$$ and total density $$n(z)$$ is equal to the ratio between the probability of getting objects of redshift $$z_p$$ knowing the true redshift $$z$$ by integrating over $$z_p$$ and the total density integrated over $$z_p$$ and $$z$$.

But as soon as we have to formulate it with mathematics and conditional probabilities, it's more difficult.

Maybe I shoud consider for example the following relation which is the equivalent of Bayes theorem but with density functions (called conditional density) :

$$g\left(x | y_{0}\right)=\frac{f\left(x, y_{0}\right)}{\int f\left(t, y_{0}\right) \mathrm{d} t}\quad(7)$$

But I don't know how to connect this equation $$(7)$$ with equation $$(1)$$.

Sorry if I misunderstood : your comment will be precious, that's why I want to clarify this step.

UPDATE 2 :

1)

The numerator of Equation (1) $$\int_{z_i^-}^{z_i^+} dz_p n(z) p_{ph}(z_p | z)$$ is simply the number of of samples in the $$i^{th}$$ photometric bin. Indeed, at a given redshift $$[z, z+dz]$$, there are $$n(z).dz$$ samples. Each of these samples has a probability $$p_{ph}(z_p|z)$$ of ending up in the $$i^{th}$$ bin. So by integrating over $$[z_i^-, z_i^+]$$, you get the number of samples with true redshift $$z$$ and a photometric redshift in that bin.

The problem is that, in the numerator of equation $$(1)$$ :

$$\int_{z_i^-}^{z_i^+} dz_p n(z) p_{ph}\,(z_p | z)$$

we integrate over $$z_p$$ and not $$z$$, so we can't consider the number of samples $$n(z)\,\text{d}z$$ and after say that we compute the probability to be in $$i-th$$ bin to have $$n_i(z)$$ samples/density value. Indeed, $$\text{dz}$$ doesn't appear in the numerator of equation $$(1)$$.

Do you agree with this issue for me ?

2) I provide you a figure that coul help someone to grasp the signification of $$eq(1)$$ :

You can see each color corresponding to the i-th redshift considered and computed from the $$eq(1)$$. I hope this will help.

I would like to make the connection between what I have put into my UPDATE 1, i.e the conditional density or maybe I should rather express the condditional probability like this :

$$g\left(x | y_{0}\right)=\dfrac{f\left(x, y_{0}\right)}{\int f\left(t, y_{0}\right) \mathrm{d} t}\quad(8)$$

This relation is qualified of "Bayes theorem" for continuous case :

But If I inegrate this expression $$(8)$$, I get :

$$\int\,g\left(x | y_{0}\right)\,\text{d}y_0=\int\,\dfrac{f\left(x, y_{0}\right)\,\text{d}y_0}{\int f\left(t, y_{0}\right) \mathrm{d} t}\quad\neq\quad P(X|y_0)\quad(9)$$

How to make appear the term $$P(X|y_0)$$ from equation $$(8)$$ ?

UPDATE 3: Really no one to explain problematics cited above in my UPDATE 2 and my attempt to simplify the equation $$(1)$$ at the beginning of my post (with eventual connection with conditional probability in equations $$(7$$ or $$(9)$$ ?

Any help/remark/suggestion is welcome

• I don't understand how you get (3) – usernumber Apr 1 '20 at 10:06
• Nor how you get the numerator in (4) – usernumber Apr 1 '20 at 10:11
• @usernumber I got them with Bayes theorem – youpilat13 Apr 1 '20 at 10:34

I don't really understand all the equations you wrote out and I'm not sure you can perform the simplification the way you suggest. In particular, I don't agree with Equations (3) and (4). Indeed, using Bayes theorem in the bottom term of Equation (1) leads to

$$\int_{z_{min}}^{z_{max}} \int_{z_i^-}^{z_i^+} dz dz_p n(z) p_{ph} (z | z_p)\frac{p_{ph}(z_p)}{p_{ph}(z)}$$

But $$p_{ph}(z | z_p) dz$$ is not equal to $$p_{ph}(z)$$, so you can't simplify the fraction out, like you do in Equation (3)

In addition, $$dz_p n(z) p_{ph} (z_p | z)$$ has no reason to be constant, so you can't easily remove the integration sign, like you do in Equation (4).

But I can try to explain the Equation (1) you are puzzled about.

The numerator of Equation (1) $$\int_{z_i^-}^{z_i^+} dz_p n(z) p_{ph} (z_p | z)$$ is simply the number of of samples in the $$i^{th}$$ photometric bin. Indeed, at a given redshift $$[z, z+dz]$$, there are $$n(z).dz$$ samples. Each of these samples has a probability $$p_{ph}(z_p | z)$$ of ending up in the $$i^{th}$$ bin. So by integrating over $$[z_i^-, z_i^+]$$, you get the number of samples with true redshift $$z$$ and a photometric redshift in that bin.

The denominator $$\int_{z_{min}}^{z_{max}} \int_{z_i^-}^{z_i^+} dz dz_p n(z) p_{ph} (z_p | z)$$ provides a normalization term, by integrating the numerator over all redshifts.

• Thanks for your quick answer. I put an UPDATE 1 to try grasping subtilities of this $n_i(z)$ definition. – youpilat13 Apr 1 '20 at 12:19
• could you please take a look at my UPDATE2 since I have not graspped all the subtilities of the definition of $n_i(z)$ ? – youpilat13 Apr 3 '20 at 14:53
• @usenumber . I have provided the results of the analytic formula (1) on a figure in section 2). Could you still think that I can't make the connection between what I have put into my UPDATE 1, i.e the conditional density or maybe I should rather express the condditional probability like this. Best regards – youpilat13 Mar 14 at 22:19