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calc_mle_linear_model(A, y_obs, covar)
Computes the maximum likelihood estimator for θ for the linear model y = A θ where measurements errors of y_obs are normally distributed with covariance matrix covar about the true values of y.
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Of particular interest are models where $\Omega_m + \Omega_\Lambda = 1$, leading to a flat universe. Of course, we're not certain if the universe is flat or not. So it would be interesting to investigate whether the data provide evidence that the universe deviates from the model for a flat universe. Therefore, we'll consider both the three parameter model in model_ΛCDM and a simplified model model_ΛCDM_flat that requires only two parameters, $\left(H_0, \Omega_m\right)$.
It incorporated data from 833 supernovae (SNe), drawn from 19 datasets. Of these, 580 SNe passed usability cuts. We present new data from the HST Cluster Survey. All SNe were fit using a single lightcurve fitter (SALT2-1) and uniformly analyzed. All analyses and cuts were developed in a blind manner, i.e. with the cosmology hidden. The Union2.1 paper is available at Suzuki et al. (2012).
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Q1b: How does the value of χ² at the MLE compare to the expected distribution of χ² if the model were correct (shown below)? What does that suggest about the usefulness of this model?
The main difference from the similar exercise in Lab3 is that the design matrix will now contains two columns with the velocities and the squared velocities: $A = \left( \begin{matrix} v_1 & v_1^2\\
v_2 & v_2^2 \\
... & ... \\
v_{N_{\mathrm{obs}}} & v_{N_{\mathrm{obs}}}^2 \end{matrix} \right)$ As a result, our best-fit parameters will now be a vector containing two coefficients.
And the luminosity distance ($d_L$) is related to the comoving distance as
$$d_L(z) = (1+z) \, d_C(z).$$
You can learn more about these in a class on cosmology or general relativity. (I found some nice lecture notes from Prof. George Djorgovski.) For this exercise, we'll simply take them as an example of how physics can provide a set of non-linear equations that we'd like to incorporate into our model. You can see how we've implemented this model in the cells below.
The contribution of radiation is small for most of the history of the universe, so we'll set Ωᵣ=0. That leaves us with three parameters to fit: $\left(H_0, \Omega_m, \Omega_\Lambda\right)$. Now, we'll add the predictions of the ΛCDM model to our plot of the SN data for a representative set of values.
Q2f: How do the χ² of the best-fit ΛCDM models compare the expected value of χ², given our assumptions about the measurement process? How do the χ² of the best-fit ΛCDM models compare the χ² of the MLE quadratic and cubic models? How does this influence your thinking about the relative merits of using the physical model over the descriptive model?
We can quickly look at the posterior mean for each parameter (as well as several additioanl diagnostics that can be useful to assess convergence of the MCMC algorithm.
mimetext/htmlrootassigneelast_run_timestampAȆ
ٰpersist_js_state·has_pluto_hook_features§cell_id$48e246bc-4937-40d6-b686-73524f3ce057depends_on_disabled_cells§runtime published_object_keysdepends_on_skipped_cells§errored$f8249dfb-8564-4e7c-b51f-a59541f597e7queued¤logsrunning¦outputbodyCurious why we are not starting from the same initial state?
We're going to perform a few separate MCMC simulations, each starting from a different initial set of parameter values. This allows us to test whether the results from each simulation are consistent with each other. If the initial states are too similar, then it might be harder to recognize if they all just happened to get "stuck" in a local minimum.
mimetext/htmlrootassigneelast_run_timestampAȒ v2persist_js_state·has_pluto_hook_features§cell_id$f8249dfb-8564-4e7c-b51f-a59541f597e7depends_on_disabled_cells§runtime8published_object_keysdepends_on_skipped_cells§errored$2c6c5b06-9253-4819-bf73-ef67e9bbdda0queued¤logsrunning¦outputbodyQModeResult with maximized lp of -3434.46
[70.38264393630517, 0.29686916055770973]mimetext/plainrootassigneemle_ΛCDM_flat_try1last_run_timestampAș9Xpersist_js_state·has_pluto_hook_features§cell_id$2c6c5b06-9253-4819-bf73-ef67e9bbdda0depends_on_disabled_cells§runtime g[+published_object_keysdepends_on_skipped_cells§errored$a3429ed7-638c-42f3-ba89-98bf683369acqueued¤logsrunning¦outputbody
Q2h: Considering the similarities and differences in the posterior predictive distributions under the two models, is it practical to distinguish between the two models given our data set? Would collecting more data points be likely to improve our ability to distinguish between the two models? Why or why not? What might be more helpful if we wanted to determine whether the universe is flat or curved?
One option would be to include ever higher order terms, e.g., $v^3$, $v^4$. The table and plot below show the χ² for the maximum liklelihood estimator as a function of the order of the polynomial fit.
Computes the Fisher information matrix for θ for the linear model y = A θ where measurements errors of y_obs are normally distributed and have covariance covar. Note that y_obs is not passed to this function, because it does not affect results.
Q2i: Look at the run time for the two cells above that called sample. How did the runtime required to compute posterior samples compare for the two models? How might this influence you choice of which model to use?
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Q2b: Rerun the above two cells a dozen times. For each, do you always get identical outputs? Nearly identical outputs? What do you think is likely causing the differences? Why is one of the point estimators above less robust than the other?
Since both the flat ΛCDM and quadratic model have two unknown parameters, we'll start by fitting a flat ΛCDM model and comparing it to the quadratic model. We'll condition on the data and compute the MLE and MAP estimators using the model_ΛCDM_flat model.
For a model with parameter that affect the predictions in a non-linear way, the loss function is not a quadratic function of the model parameters. Therefore, we are not guaranteed that the loss function has a single global minimum. As a result, if we make different initial guesses and itteratively improve them ("walk downhill"), we aren't guaranteed to find the same global mode. When we called optimize above, it drew random initial guesses from the prior distriubution for the model parameters. To help it find the best-fit parameters, we can instruct the optimizer to start from our prefered initial guess.
mimetext/htmlrootassigneelast_run_timestampAȆpersist_js_state·has_pluto_hook_features§cell_id$48d57307-dfc9-4a95-a98f-52698ba5de0cdepends_on_disabled_cells§runtime published_object_keysdepends_on_skipped_cells§errored$e4fef95f-d237-46a3-94cd-b9646625352equeued¤logsrunning¦outputbody
mimetext/htmlrootassigneelast_run_timestampAȒ persist_js_state·has_pluto_hook_features§cell_id$e4fef95f-d237-46a3-94cd-b9646625352edepends_on_disabled_cells§runtime qpublished_object_keysdepends_on_skipped_cells§errored$47308e43-4311-4c03-9807-df006c51c00equeued¤logsrunning¦outputbodyQModeResult with maximized lp of -3439.70
[70.38154092144411, 0.29691650659799873]mimetext/plainrootassigneemap_ΛCDM_flat_try1last_run_timestampAȚҰpersist_js_state·has_pluto_hook_features§cell_id$47308e43-4311-4c03-9807-df006c51c00edepends_on_disabled_cells§runtimeΰ"qpublished_object_keysdepends_on_skipped_cells§errored$1c8238b3-a968-4b5a-bfaf-018249969d97queued¤logsrunning¦outputbodymimetext/plainrootassigneelast_run_timestampAȚZVpersist_js_state·has_pluto_hook_features§cell_id$1c8238b3-a968-4b5a-bfaf-018249969d97depends_on_disabled_cells§runtime]Cpublished_object_keysdepends_on_skipped_cells§errored$4544d226-aaa1-43c0-9b6f-ed73858b613aqueued¤logsrunning¦outputbodyPModeResult with maximized lp of -3439.70
[70.38154092144417, 0.2969165065979948]mimetext/plainrootassigneemap_ΛCDM_flatlast_run_timestampAȚ
Opersist_js_state·has_pluto_hook_features§cell_id$4544d226-aaa1-43c0-9b6f-ed73858b613adepends_on_disabled_cells§runtime f.published_object_keysdepends_on_skipped_cells§errored$c433e938-31f7-4417-b3c0-a2f49fea8640queued¤logsrunning¦outputbody
Check once you're ready to draw posterior samples from the ΛCDM models:
One possible generalization of Hubble's law would be to consider a quadratic function for the local expansion rate. $d = \frac{v}{H_0} + a_2 v^2.$
On one hand, in this model the predicted distance is a non-linear function of velocity. However, fitting for the model parameters $\left(H_0^{-1}, a_2\right)$ is still a linear problem, since those coefficients only appear linearly in the model if the observed velocities are held fixed. Thus, we can again compute the maximum likelihood estimator (MLE) with the quadratic model using linear algebra and the functions from Lab 3, as shown below.
So far, we've sought a descriptive model for our data. That is, a model that would fit the data well, but without a clear physical motivation or interpretation for the each model parameter. Next, we'll fit an explanatory model. In this approach, we first identify a physically motivated model and use it to make predictions for our data. Our hope is that by incorporating additional laws of physics, our model will have more predictive power.
The expansion of a homogeneous and isotropic universe is described by the Friedman equations and Einstein field equations. They can be combined into a differential equation for the cosmological scale factor. The equation can be integrated yield the following relationship between redshift ($z$), and comoving distance ($d_C$) for a flat universe, as a functinon of the density of radiation ($\Omega_r$), matter ($\Omega_m$) and dark energy ($\Omega_\Lambda$) in the unvierse today.
In the figure above, there are histograms for the marginal posterior distribution for each of the model parameters along the diagonal. In the bottom corner, there is a scatter plot, showing posterior samples from the posterior probability densitiy function. Here we see that both H₀ and Ωₘ are well measured and there is a minor correlation between values of the the two model parameters.
Next, we'll look at a similar pair plot (also known as a corner plot), for the three-parameter ΛCDM model.
In a previous lab, we measured the Hubble constant ($H_0$), the rate of expansion of the local universe using observations of Type 1a Supernovae. But we also noticed that the model wasn't a great description of all the avaliable data, particularly those at larger redshifts. In this lab we'll apply multiple models and compare how well each performs. To keep things self-contained, we'll download and read data from the Supernovae Cosmology Project and the same transformations, so that we can plot velocity versus distance. But since we've alredy seen how to do that, I've placed the cells that do that at the bottom of the notebook.
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This time the two point estimates are noticable different. To understand what happened, we'll compare the χ² of the MLE and MAP estimators for the ΛCDM model below.
Computes the uncertainty in the maximum likelihood estimate of θ for the linear model y = A θ where measurements errors of y_obs are normally distributed and have covariance covar. Note that y_obs is not passed to this function, because it does not affect results.
The two posterior predictive distribution very nearly overlaps, so you can't easily distinguish the two curves (for the mean of the posterior predictive distribution at each velocity). The shaded region shows a region plus/minus 5 times the standard deviation of the predictive distribution at each velocity. (Exagerated, so you can see it.) Again, the two shaded regions very nearly overlap. However, the predictive distributions are not identical, as shown in the figures below.
While the posterior mean values of H₀ are similar, the standard deviation of the marginal posterior distributions (reported under std) are quite different. Similarly, the posterior mean values of Ωₘ in the two models are similar. So why did the MAP estimators differ so much? It'll be useful to look at the correlation of different parameters with each other.
Q2g: Compare the posterior distributions under the two models. How does the width of the marginal distributions for H₀ and Ωₘ change depending on whether we use the flat or general ΛCDM model? What is causing the change? What does this imply for the uncertainty for the model parameters?
mimetext/htmlrootassigneelast_run_timestampAȆe
persist_js_state·has_pluto_hook_features§cell_id$c78f0ad9-03c0-4066-a390-193e7bfa84b1depends_on_disabled_cells§runtime /published_object_keysdepends_on_skipped_cells§errored$575cd616-c08c-4f84-9b68-76f6cda207b5queued¤logsrunning¦outputbodymimetext/plainrootassigneelast_run_timestampAȚ persist_js_state·has_pluto_hook_features§cell_id$575cd616-c08c-4f84-9b68-76f6cda207b5depends_on_disabled_cells§runtime 0published_object_keysdepends_on_skipped_cells§errored$185cc5dc-360e-44be-ba5b-4eca7455c8a8queued¤logsrunning¦outputbodymimetext/plainrootassigneelast_run_timestampAȒ Ұpersist_js_state·has_pluto_hook_features§cell_id$185cc5dc-360e-44be-ba5b-4eca7455c8a8depends_on_disabled_cells§runtime=published_object_keysdepends_on_skipped_cells§errored$e753bcf1-f68b-4a55-9132-99bc2ea1cf92queued¤logsrunning¦outputbodyh
mimeimage/svg+xmlrootassigneelast_run_timestampAȔ9persist_js_state·has_pluto_hook_features§cell_id$e753bcf1-f68b-4a55-9132-99bc2ea1cf92depends_on_disabled_cells§runtime?ipublished_object_keysdepends_on_skipped_cells§errored$a45b4f4c-efc8-435b-b37c-e265dd204a9cqueued¤logsrunning¦outputbodymimetext/plainrootassigneelast_run_timestampAȑpersist_js_state·has_pluto_hook_features§cell_id$a45b4f4c-efc8-435b-b37c-e265dd204a9cdepends_on_disabled_cells§runtimeBpublished_object_keysdepends_on_skipped_cells§errored$813ee621-02a2-4998-8ec6-4c08dbab8445queued¤logsrunning¦outputbodymimetext/plainrootassigneelast_run_timestampAȝt9\persist_js_state·has_pluto_hook_features§cell_id$813ee621-02a2-4998-8ec6-4c08dbab8445depends_on_disabled_cells§runtime [̵published_object_keysdepends_on_skipped_cells§errored$86287560-cf47-4313-bfb2-27d99cbdb5b2queued¤logsrunning¦outputbodyQModeResult with maximized lp of -3434.46
[70.38264393630523, 0.29686916055770585]mimetext/plainrootassigneemle_ΛCDM_flatlast_run_timestampAȚ 3Hpersist_js_state·has_pluto_hook_features§cell_id$86287560-cf47-4313-bfb2-27d99cbdb5b2depends_on_disabled_cells§runtime spublished_object_keysdepends_on_skipped_cells§errored$8462ca0a-2e42-4781-84e4-ae3dfc0cfa13queued¤logsrunning¦outputbodyeModeResult with maximized lp of -3439.60
[59.008614837322696, 0.4225550732642944, 0.9999999999999667]mimetext/plainrootassigneemap_ΛCDMlast_run_timestampAȝpersist_js_state·has_pluto_hook_features§cell_id$8462ca0a-2e42-4781-84e4-ae3dfc0cfa13depends_on_disabled_cells§runtime Epublished_object_keysdepends_on_skipped_cells§errored$6d38c597-00aa-4743-a93a-ab5344f17338queued¤logsrunning¦outputbodyV
Now that we've thought about the point estimates from each model, we're ready to compare the posterior distributions under the two models. We'll use an MCMC algorithm to draw samplmes from the posterior distribution for the model parameters for both the general ΛCDM and flat-ΛCDM models.
Q2e: How similar/different are the values of χ² at the MLE and MAP estimates for the two models? What does this imply about our ability to discriminate between the two models?
One great thing about the Turing probabilistic programming language is that it makes it easy to perform statistical inference even using models that include complex functions (e.g., the integral in the ΛCDM model). We can easily implement the model for our data by building on the previous functions as follows:
This is the end of the lab. Below, I included some extra calculations in case they're useful for people to refer back to when it comes time to do something for their proejct.
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Iinstantiatedòinstalled_versionsCSV0.10.15DownloadsstdlibMCMCChains6.0.7PDMats0.11.32Distributions0.25.117PlutoUI0.7.61Plots1.40.9Optim1.10.0PlutoTeachingTools0.3.1QuadGK2.11.2LinearAlgebrastdlibDataFrames1.7.0Turing0.35.5terminal_outputsCSV@
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1/sqrt(Ωr * (1+z)^4 + Ωm * (1+z)^3 + ΩΛ)
endmetadatashow_logsèdisabled®skip_as_script«code_folded$556d8f17-8a03-42c8-82c7-52c8125b604dcell_id$556d8f17-8a03-42c8-82c7-52c8125b604dcode[begin
"""
`calc_mle_linear_model(A, y_obs, covar)`
Computes the maximum likelihood estimator for θ for the linear model
`y = A θ`
where measurements errors of `y_obs` are normally distributed with covariance matrix `covar` about the true values of `y`.
"""
function calc_mle_linear_model end
function calc_mle_linear_model(A::AbstractMatrix, y_obs::AbstractVector, covar::PDMat)
@assert size(A,1) == length(y_obs) == size(covar,1) == size(covar,2)
@assert size(A,2) >= 1
(A' * (covar \ A)) \ (A' * (covar \ y_obs) )
end
function calc_mle_linear_model(A::AbstractMatrix, y_obs::AbstractVector, covar::AbstractMatrix)
calc_mle_linear_model(A,y_obs,PDMat(covar))
end
function calc_mle_linear_model(A::AbstractMatrix, y_obs::AbstractVector, covar::AbstractVector)
calc_mle_linear_model(A,y_obs,PDiagMat(covar))
end
Docs.doc(calc_mle_linear_model)
endmetadatashow_logsèdisabled®skip_as_script«code_folded$a5f3ceb9-0f84-4601-97c0-2d431427638fcell_id$a5f3ceb9-0f84-4601-97c0-2d431427638fcode@model function model_ΛCDM_flat(z, d_obs, σ_d_obs)
@assert length(z) == length(d_obs) == length(σ_d_obs)
H₀ ~ LogNormal(log(100), 1.0) # km/s/Mpc
Ωₘ ~ Uniform(1e-4,1) # Universe needs some matter
ΩΛ = 1 - Ωₘ # For flat universe, Ωₘ + ΩΛ = 1
d_true = luminosity_distance.(z; Ωm=Ωₘ, ΩΛ=ΩΛ, H0=H₀)
d_obs ~ MvNormal(d_true, σ_d_obs)
endmetadatashow_logsèdisabled®skip_as_script«code_folded$3207708b-3f90-4e8b-b0be-ff96341befe1cell_id$3207708b-3f90-4e8b-b0be-ff96341befe1codemd"""
Of particular interest are models where $\Omega_m + \Omega_\Lambda = 1$, leading to a flat universe.
Of course, we're not certain if the universe is flat or not.
So it would be interesting to investigate whether the data provide evidence that the universe deviates from the model for a flat universe.
Therefore, we'll consider both the three parameter model in `model_ΛCDM` and a simplified model `model_ΛCDM_flat` that requires only two parameters, $\left(H_0, \Omega_m\right)$.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$cd4fc36e-74f1-4b33-b92c-f0b7913a7203cell_id$cd4fc36e-74f1-4b33-b92c-f0b7913a7203codedetails(
md"""
This data is from the [Supernovae Cosmology Project](https://supernova.lbl.gov/)'s ["Union2.1" SN Ia compilation](https://supernova.lbl.gov/Union/). Click if you're curious for more details.""",
md"""It incorporated data from 833 supernovae (SNe), drawn from 19 datasets. Of these, 580 SNe passed usability cuts. We present new data from the HST Cluster Survey.
All SNe were fit using a single lightcurve fitter (SALT2-1) and uniformly analyzed.
All analyses and cuts were developed in a blind manner, i.e. with the cosmology hidden.
The Union2.1 paper is available at [Suzuki et al. (2012)](https://ui.adsabs.harvard.edu/abs/2012ApJ...746...85S/abstract).""")metadatashow_logsèdisabled®skip_as_script«code_folded$2482544e-ad05-495c-9787-0797939445becell_id$2482544e-ad05-495c-9787-0797939445becodeٙmd"""
Maximum v to plot: $(@bind max_v_plt PlutoUI.Slider(max(0,ceil(minimum(df_all.v)/200)*200):200:maximum(df_all.v), default=maximum(df_all.v)))
"""metadatashow_logsèdisabled®skip_as_script«code_folded$3f85dc0d-e8c4-4683-bf77-ee227136fa1acell_id$3f85dc0d-e8c4-4683-bf77-ee227136fa1acode\χ²_ΛCDM_mle = loss(mle_ΛCDM.values[2], mle_ΛCDM.values[3], mle_ΛCDM.values[1], df_all)metadatashow_logsèdisabled®skip_as_script«code_folded$c33d1afc-760d-48f3-a4a3-2288a2ccbf7fcell_id$c33d1afc-760d-48f3-a4a3-2288a2ccbf7fcodefunction loss(Ωm::Number, ΩΛ::Number, H0::Number, data::AbstractDataFrame)
if !((0<Ωm) && (0<ΩΛ)) return Inf end
d_pred = luminosity_distance.(data.z; Ωm=Ωm, ΩΛ=ΩΛ, H0=H0)
sum(abs2.((data.dₗ - d_pred) ./data.σ_dₗ ))
endmetadatashow_logsèdisabled®skip_as_script«code_folded$bb426c11-572a-4b9d-a223-6c3ddc9cda51cell_id$bb426c11-572a-4b9d-a223-6c3ddc9cda51codemd"""
**Q1b:** How does the value of χ² at the MLE compare to the expected distribution of χ² if the model were correct (shown below)? What does that suggest about the usefulness of this model?
"""metadatashow_logsèdisabled®skip_as_script«code_folded$da9eb594-e0fe-4e9c-be69-4150c9a44fb9cell_id$da9eb594-e0fe-4e9c-be69-4150c9a44fb9codeCbegin
df_all = select(df_in,[:id,:z]) # select only the columns we need
# Perform variabile transofrmations informed by astrophysics
local c = speed_of_light_kmps # convention is to use km/s
df_all.v = v_from_z.(df_in.z) # see equations above
df_all.dₗ = @. 10.0^((df_in.μ-25)/5) # see equations above
df_all.σ_dₗ = @. log(10) * df_all.dₗ * df_in.σ_μ/5 # propagation of uncertainty
# Now that columns have dimensions, add those as metadata
colmetadata!(df_all, :dₗ, "label", "velocity")
colmetadata!(df_all, :v, "units", "km/s")
colmetadata!(df_all, :dₗ, "label", "luminosity distance")
colmetadata!(df_all, :σ_dₗ, "label", "uncertainty in luminosity distance")
colmetadata!(df_all, :dₗ, "units", "Mpc")
colmetadata!(df_all, :σ_dₗ, "units", "Mpc")
df_all
endmetadatashow_logsèdisabled®skip_as_script«code_folded$d19e7f11-d1ff-4624-8603-38ffec0c7beacell_id$d19e7f11-d1ff-4624-8603-38ffec0c7beacodemd"""
## Download the Data
"""metadatashow_logsèdisabled®skip_as_script«code_folded$c6f03d5d-18e2-4491-90fe-fe0008f774c4cell_id$c6f03d5d-18e2-4491-90fe-fe0008f774c4codemd"""
The main difference from the similar exercise in Lab3 is that the design matrix will now contains two columns with the velocities and the squared velocities:
$A = \left( \begin{matrix}
v_1 & v_1^2\\
v_2 & v_2^2 \\
... & ... \\
v_{N_{\mathrm{obs}}} & v_{N_{\mathrm{obs}}}^2
\end{matrix}
\right)$
As a result, our best-fit parameters will now be a vector containing two coefficients.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$8d2155e4-4e99-4a16-90d1-7b856f285e9ecell_id$8d2155e4-4e99-4a16-90d1-7b856f285e9ecode.if ismissing(response_2h)
still_missing()
endmetadatashow_logsèdisabled®skip_as_script«code_folded$b3542d91-7c08-4b36-ae18-704fec04b5e8cell_id$b3542d91-7c08-4b36-ae18-704fec04b5e8codeٓif ready_to_sample_prior
Plots.histogram(vec(samples_prior_ΛCDM_flat[:H₀]), label=:none, xlims=(0,1000), xlabel="H_0", ylabel = "Prior PDF")
endmetadatashow_logsèdisabled®skip_as_script«code_folded$62a701c6-45a1-450b-af5a-fa63bf47b55bcell_id$62a701c6-45a1-450b-af5a-fa63bf47b55bcodeCif !isnothing(chain_ΛCDM_flat)
local n = 500
local subsample = (nluminosity_distance.(z_from_v.(v_plt); Ωm=subsample[:Ωₘ][i], ΩΛ=1.0 .- subsample[:Ωₘ][i], H0=subsample[:H₀][i]), 1:n)
posterior_pred_mean_ΛCDM_flat = mean(pred, dims=2)
posterior_pred_std_ΛCDM_flat = std(pred, mean=posterior_pred_mean_ΛCDM_flat, dims=2)
local target_qs = [0.05, 0.5, 0.95]
posterior_pred_quantiles_ΛCDM_flat = stack(i->quantile(view(pred,i,:),target_qs), 1:size(pred,1))'
end;metadatashow_logsèdisabled®skip_as_script«code_folded$4518f1f7-a2a3-4783-86ae-4413fc4291b5cell_id$4518f1f7-a2a3-4783-86ae-4413fc4291b5codegbegin
df_in = CSV.read(filename, DataFrame, delim="\t", skipto=6, header=[:id,:z,:μ,:σ_μ,:prob_low_mass_gal])
metadata!(df_in,"source", url, style =:note)
colmetadata!(df_in, :z, "label", "redshift", style =:note)
colmetadata!(df_in, :μ, "label", "distance modulus")
colmetadata!(df_in, :σ_μ, "label", "uncertainty in distance modulus")
df_in
end;metadatashow_logsèdisabled®skip_as_script«code_folded$f78e5ae6-362f-4001-9f54-bab48fe13736cell_id$f78e5ae6-362f-4001-9f54-bab48fe13736codeif ready_to_run_mcmc
chain_ΛCDM_flat = sample(model_ΛCDM_flat_given_data, NUTS(0.65), MCMCThreads(), num_mcmc_samples, num_chains; initial_params=init_states_ΛCDM_flat)
else
chain_ΛCDM_flat = nothing
end;metadatashow_logsèdisabled®skip_as_script«code_folded$b24bf937-b288-4cce-ad1c-3128c5735696cell_id$b24bf937-b288-4cce-ad1c-3128c5735696codebegin
local A_quartic = hcat(df_all.v, df_all.v.^2, df_all.v.^3, df_all.v.^4)
coeff_quartic = calc_mle_linear_model(A_quartic, df_all.dₗ, Σ_all)
H₀_mle_quartic = 1/coeff_quartic[1]
#σ_coeff_quartic = calc_sigma_mle_linear_model(A_quartic, Σ_all)
#σ_H₀_quartic = abs(H₀_mle_quartic * σ_coeff_quartic[1] / coeff_quartic[1])
d_pred_quartic = predict_linear_model(A_quartic, coeff_quartic)
χ²_quartic = sum(abs2.((df_all.dₗ.-d_pred_quartic)./df_all.σ_dₗ))
local A_plt_quartic = hcat(v_plt, v_plt.^2, v_plt.^3, v_plt.^4)
d_plt_quartic = predict_linear_model(A_plt_quartic, coeff_quartic)
end;metadatashow_logsèdisabled®skip_as_script«code_folded$e58eb467-21b7-4be4-8b9c-220f21aabc23cell_id$e58eb467-21b7-4be4-8b9c-220f21aabc23code-md"""
# A more flexible descriptive model
"""metadatashow_logsèdisabled®skip_as_script«code_folded$e5a0edbd-fa6b-4920-bf4c-a31c906b8c26cell_id$e5a0edbd-fa6b-4920-bf4c-a31c906b8c26code"md"""
And the luminosity distance ($d_L$) is related to the comoving distance as
$d_L(z) = (1+z) \, d_C(z).$
You can learn more about these in a class on cosmology or general relativity. (I found some nice [lecture notes](https://sites.astro.caltech.edu/~george/ay21/Ay21_Lec03.pdf) from Prof. George Djorgovski.)
For this exercise, we'll simply take them as an example of how physics can provide a set of non-linear equations that we'd like to incorporate into our model.
You can see how we've implemented this model in the cells below.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$ab81f6ff-11c0-49c1-94a1-e6da38dc1dc3cell_id$ab81f6ff-11c0-49c1-94a1-e6da38dc1dc3code?md"""
The contribution of radiation is small for most of the history of the universe, so we'll set Ωᵣ=0.
That leaves us with three parameters to fit: $\left(H_0, \Omega_m, \Omega_\Lambda\right)$.
Now, we'll add the predictions of the ΛCDM model to our plot of the SN data for a representative set of values.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$9e3c0b9a-bc23-4539-ba87-579df44e833fcell_id$9e3c0b9a-bc23-4539-ba87-579df44e833fcodezmd"""
**Q2f:** How do the χ² of the best-fit ΛCDM models compare the expected value of χ², given our assumptions about the measurement process?
How do the χ² of the best-fit ΛCDM models compare the χ² of the MLE quadratic and cubic models?
How does this influence your thinking about the relative merits of using the physical model over the descriptive model?
"""metadatashow_logsèdisabled®skip_as_script«code_folded$8d53dc09-1687-4107-977a-6dfb13933e20cell_id$8d53dc09-1687-4107-977a-6dfb13933e20codeٰmd"""
We can quickly look at the posterior mean for each parameter (as well as several additioanl diagnostics that can be useful to assess convergence of the MCMC algorithm."""metadatashow_logsèdisabled®skip_as_script«code_folded$0236d055-0d6d-440d-a44b-0d5cb0ab2273cell_id$0236d055-0d6d-440d-a44b-0d5cb0ab2273coderesponse_2b = missing;metadatashow_logsèdisabled®skip_as_script«code_folded$48e246bc-4937-40d6-b686-73524f3ce057cell_id$48e246bc-4937-40d6-b686-73524f3ce057code*# hideall
md"""
# Setup & Helper Code
"""metadatashow_logsèdisabled®skip_as_script«code_folded$f8249dfb-8564-4e7c-b51f-a59541f597e7cell_id$f8249dfb-8564-4e7c-b51f-a59541f597e7codedetails("Curious why we are not starting from the same initial state?",
md"""
We're going to perform a few separate MCMC simulations, each starting from a different initial set of parameter values.
This allows us to test whether the results from each simulation are consistent with each other.
If the initial states are too similar, then it might be harder to recognize if they all just happened to get "stuck" in a local minimum.
""")metadatashow_logsèdisabled®skip_as_script«code_folded$2c6c5b06-9253-4819-bf73-ef67e9bbdda0cell_id$2c6c5b06-9253-4819-bf73-ef67e9bbdda0codeEmle_ΛCDM_flat_try1 = maximum_likelihood(model_ΛCDM_flat_given_data)metadatashow_logsèdisabled®skip_as_script«code_folded$a3429ed7-638c-42f3-ba89-98bf683369accell_id$a3429ed7-638c-42f3-ba89-98bf683369accodemd"""
**Q2h:** Considering the similarities and differences in the posterior predictive distributions under the two models, is it practical to distinguish between the two models given our data set?
Would collecting more data points be likely to improve our ability to distinguish between the two models? Why or why not?
What might be more helpful if we wanted to determine whether the universe is flat or curved?
"""metadatashow_logsèdisabled®skip_as_script«code_folded$e0e610d7-a237-42ea-8706-8e09858fba03cell_id$e0e610d7-a237-42ea-8706-8e09858fba03code.if ismissing(response_2g)
still_missing()
endmetadatashow_logsèdisabled®skip_as_script«code_folded$17ced7bb-8431-4106-b812-ce6229bc2416cell_id$17ced7bb-8431-4106-b812-ce6229bc2416code-if installed_makie
let
df = DataFrame(chain_ΛCDM)
select!(df,[:H₀,:Ωₘ,:ΩΛ])
fig = pairplot(
df => (
PairPlots.Contour(linewidth=1.5),
PairPlots.Scatter(filtersigma=2, color=:black),
PairPlots.MarginHist()
)
)
Label(fig[0,:], "General ΛCMD Model")
fig
end
endmetadatashow_logsèdisabled®skip_as_script«code_folded$dbd2f20e-4c64-4fa5-a5fc-ec5197d2427dcell_id$dbd2f20e-4c64-4fa5-a5fc-ec5197d2427dcodebegin
df_mle_polynomial = DataFrame(order = 1:4)
df_mle_polynomial.χ² = [χ²_linear, χ²_quadratic,χ²_cubic, χ²_quartic]
local plt = Plots.plot(xlabel="Polynomial Order", ylabel="χ²", legend=:none, widen=:true)
Plots.plot!(plt,df_mle_polynomial.order, df_mle_polynomial.χ², lc=:blue, lw=2)
Plots.scatter!(plt,df_mle_polynomial.order, df_mle_polynomial.χ², mc=:blue)
Plots.ylims!(plt,0, maximum(df_mle_polynomial.χ²))
plt
endmetadatashow_logsèdisabled®skip_as_script«code_folded$23ee2544-2a2f-4302-8a62-2dd868cbb5d1cell_id$23ee2544-2a2f-4302-8a62-2dd868cbb5d1coderesponse_2d = missing;metadatashow_logsèdisabled®skip_as_script«code_folded$88ed369e-cf54-4df8-be50-6400d68f578acell_id$88ed369e-cf54-4df8-be50-6400d68f578acode@aside(tip(md"""Drawing posterior samples can be computationally intensive. By default the box is unchecked, so that each time you changed your initial guess, it didn't trigger a new set of MCMC simulations. Be sure to save your work before clicking the box that you're reading to run MCMC analyses."""), v_offset=-160)metadatashow_logsèdisabled®skip_as_script«code_folded$7dc10251-cdf5-4bb5-ae4a-3a68eb84ffabcell_id$7dc10251-cdf5-4bb5-ae4a-3a68eb84ffabcodemd"""
One option would be to include ever higher order terms, e.g., $v^3$, $v^4$.
The table and plot below show the χ² for the maximum liklelihood estimator as a function of the order of the polynomial fit.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$ab94159b-72b4-48c4-a3cc-51fdfb154813cell_id$ab94159b-72b4-48c4-a3cc-51fdfb154813codebegin
df_plt = DataFrame( :v=> v_plt)
df_plt.d_pred_linear = d_plt_linear
df_plt.d_pred_quadratic = d_plt_quadratic
df_plt.d_pred_cubic = d_plt_cubic
df_plt.d_pred_quartic = d_plt_quartic
#df_plt.d_pred_quintic = d_plt_quintic
df_plt
end;metadatashow_logsèdisabled®skip_as_script«code_folded$aaa80da1-f7de-42ed-9691-c367e6bd9a28cell_id$aaa80da1-f7de-42ed-9691-c367e6bd9a28code# hideall
TableOfContents()metadatashow_logsèdisabled®skip_as_script«code_folded$1a5fd7e8-3bb9-47f8-82c5-5bceee80fa62cell_id$1a5fd7e8-3bb9-47f8-82c5-5bceee80fa62codelet
if !isnothing(chain_ΛCDM_flat)
plt1 = Plots.scatter(chain_ΛCDM_flat[:H₀], chain_ΛCDM_flat[:Ωₘ], ms=1, markerstrokewidth=0, xlabel="H₀ (km/s/Mpc)", ylabel="Ωₘ", legend=:none)
plt2 = Plots.histogram(chain_ΛCDM_flat[:H₀], xlabel="H₀ (km/s/Mpc)", alpha=0.5, legend=:none)
plt3 = Plots.histogram(chain_ΛCDM_flat[:Ωₘ], xlabel="Ωₘ", alpha=0.5, legend=:none)
Plots.plot(plt2,plt1,plt3, layout=@layout([a _; b c]))
end
endmetadatashow_logsèdisabled®skip_as_script«code_folded$9db3ff24-6f54-4a37-b787-de4b14694d95cell_id$9db3ff24-6f54-4a37-b787-de4b14694d95code"""
`calc_fisher_matrix_linear_model(A, covar)`
Computes the Fisher information matrix for θ for the linear model
`y = A θ`
where measurements errors of `y_obs` are normally distributed and have covariance `covar`.
Note that `y_obs` is not passed to this function, because it does not affect results.
"""
function calc_fisher_matrix_linear_model(A::AbstractMatrix, covar::AbstractMatrix)
@assert size(A,1) == size(covar,1) == size(covar,2)
@assert size(A,2) >= 1
(A' * (covar \ A))
endmetadatashow_logsèdisabled®skip_as_script«code_folded$0c0312da-558a-4bc1-9e4a-b02ecd1c0572cell_id$0c0312da-558a-4bc1-9e4a-b02ecd1c0572code.if ismissing(response_1c)
still_missing()
endmetadatashow_logsèdisabled®skip_as_script«code_folded$4eb123a1-f976-4fad-9a96-e3a61bdbf264cell_id$4eb123a1-f976-4fad-9a96-e3a61bdbf264coderesponse_2a = missing;metadatashow_logsèdisabled®skip_as_script«code_folded$4655178c-a633-4b50-8f92-6ee028e21d0ecell_id$4655178c-a633-4b50-8f92-6ee028e21d0ecode,md"""
## Apply Variable Transformations
"""metadatashow_logsèdisabled®skip_as_script«code_folded$4acbfe11-b8a9-4154-8d15-994f5ce1594dcell_id$4acbfe11-b8a9-4154-8d15-994f5ce1594dcode/md"""
## Implementing the statistical model
"""metadatashow_logsèdisabled®skip_as_script«code_folded$8764ccfe-81cd-4bff-985a-c112fc5c902fcell_id$8764ccfe-81cd-4bff-985a-c112fc5c902fcodemd"""Use the sliders above to explore how changing each parameter affects the predictions of the model and the quality of the fit.
**Q2a:** Based on your qualitative exploration, can you identify any potential complications when fitting this model? If so, explain your concern."""metadatashow_logsèdisabled®skip_as_script«code_folded$089c50b6-5494-4f27-af3e-a8af972e2869cell_id$089c50b6-5494-4f27-af3e-a8af972e2869coderesponse_2f = missing;metadatashow_logsèdisabled®skip_as_script«code_folded$3ce14ed0-03e6-4a19-b338-04d2dff98d29cell_id$3ce14ed0-03e6-4a19-b338-04d2dff98d29code\χ²_ΛCDM_map = loss(map_ΛCDM.values[2], map_ΛCDM.values[3], map_ΛCDM.values[1], df_all)metadatashow_logsèdisabled®skip_as_script«code_folded$cc220b0f-84f6-48c4-8d8b-8eb90457588acell_id$cc220b0f-84f6-48c4-8d8b-8eb90457588acoderχ²_ΛCDM_flat_map = loss(map_ΛCDM_flat.values[2], 1-map_ΛCDM_flat.values[2], map_ΛCDM_flat.values[1], df_all)metadatashow_logsèdisabled®skip_as_script«code_folded$b224ddfb-97a2-45e6-99c3-8295a1a84eeacell_id$b224ddfb-97a2-45e6-99c3-8295a1a84eeacodemd"""
**Q2i:** Look at the run time for the two cells above that called `sample`. How did the runtime required to compute posterior samples compare for the two models? How might this influence you choice of which model to use?
"""metadatashow_logsèdisabled®skip_as_script«code_folded$0863dd21-7d83-4043-8ba0-508be2917ab1cell_id$0863dd21-7d83-4043-8ba0-508be2917ab1codefunction comoving_distance(z; Ωm::Number, ΩΛ::Number, Ωr::Number=0.0, H0::Number = 1)
@assert z>=0
(integral, uncertainty) = quadgk(zz->distance_integrand(zz;Ωm=Ωm,ΩΛ=ΩΛ,Ωr=Ωr),0.0, z)
return (speed_of_light_kmps / H0) * integral
endmetadatashow_logsèdisabled®skip_as_script«code_folded$49d3f522-1605-4b44-a97b-921e2d9d605acell_id$49d3f522-1605-4b44-a97b-921e2d9d605acodelif ready_to_sample_prior
samples_prior_ΛCDM_flat =sample(model_ΛCDM_flat_given_data, Prior(), 10_000)
endmetadatashow_logsèdisabled®skip_as_script«code_folded$0144843f-b649-4ea2-a546-e8c14a928670cell_id$0144843f-b649-4ea2-a546-e8c14a928670codemd"""**Q2b:** Rerun the above two cells a dozen times. For each, do you always get identical outputs? Nearly identical outputs? What do you think is likely causing the differences? Why is one of the point estimators above less robust than the other?"""metadatashow_logsèdisabled®skip_as_script«code_folded$f1d0e73d-7ba5-4b02-9c68-65dbd9b4cc33cell_id$f1d0e73d-7ba5-4b02-9c68-65dbd9b4cc33coderesponse_1c = missing;metadatashow_logsèdisabled®skip_as_script«code_folded$a745d04b-c658-4f35-8cd3-b3f6d12da438cell_id$a745d04b-c658-4f35-8cd3-b3f6d12da438codemd"""
Since both the flat ΛCDM and quadratic model have two unknown parameters, we'll start by fitting a flat ΛCDM model and comparing it to the quadratic model. We'll condition on the data and compute the MLE and MAP estimators using the `model_ΛCDM_flat` model.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$48d57307-dfc9-4a95-a98f-52698ba5de0ccell_id$48d57307-dfc9-4a95-a98f-52698ba5de0ccodetmd"""
For a model with parameter that affect the predictions in a non-linear way, the loss function is not a quadratic function of the model parameters.
Therefore, we are not guaranteed that the loss function has a single global minimum.
As a result, if we make different initial guesses and itteratively improve them ("walk downhill"), we aren't guaranteed to find the same global mode.
When we called `optimize` above, it drew random initial guesses from the prior distriubution for the model parameters.
To help it find the best-fit parameters, we can instruct the optimizer to start from our prefered initial guess.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$e4fef95f-d237-46a3-94cd-b9646625352ecell_id$e4fef95f-d237-46a3-94cd-b9646625352ecodeRaside(tip(md"""For this exercise, we'll assume that convergence isn't a problem here. If we were planning to use these results for a scientific publication, then we'd run the sampling algorithm for several times longer and still need to perform convergence tests to convince ourselves that the results had converged.
"""), v_offset=-70)metadatashow_logsèdisabled®skip_as_script«code_folded$47308e43-4311-4c03-9807-df006c51c00ecell_id$47308e43-4311-4c03-9807-df006c51c00ecodeGmap_ΛCDM_flat_try1 = maximum_a_posteriori(model_ΛCDM_flat_given_data)metadatashow_logsèdisabled®skip_as_script«code_folded$1c8238b3-a968-4b5a-bfaf-018249969d97cell_id$1c8238b3-a968-4b5a-bfaf-018249969d97code?if !isnothing(chain_ΛCDM_flat)
describe(chain_ΛCDM_flat)
endmetadatashow_logsèdisabled®skip_as_script«code_folded$4544d226-aaa1-43c0-9b6f-ed73858b613acell_id$4544d226-aaa1-43c0-9b6f-ed73858b613acodeimap_ΛCDM_flat = maximum_a_posteriori(model_ΛCDM_flat_given_data; initial_params=init_guess_ΛCMD_flat )metadatashow_logsèdisabled®skip_as_script«code_folded$c433e938-31f7-4417-b3c0-a2f49fea8640cell_id$c433e938-31f7-4417-b3c0-a2f49fea8640codeهmd"""
Check once you're ready to draw posterior samples from the ΛCDM models: $(@bind ready_to_run_mcmc CheckBox(; default=false))
"""metadatashow_logsèdisabled®skip_as_script«code_folded$b7fa0347-5237-485b-9e69-c8cb9324c5edcell_id$b7fa0347-5237-485b-9e69-c8cb9324c5edcode.if ismissing(response_2c)
still_missing()
endmetadatashow_logsèdisabled®skip_as_script«code_folded$0eb2408f-3be3-4e13-895f-50b5151a2125cell_id$0eb2408f-3be3-4e13-895f-50b5151a2125code.if ismissing(response_2e)
still_missing()
endmetadatashow_logsèdisabled®skip_as_script«code_folded$fa745b36-7465-443e-9f50-c786fe418c6acell_id$fa745b36-7465-443e-9f50-c786fe418c6acode$Σ_all = PDiagMat(df_all.σ_dₗ.^2)metadatashow_logsèdisabled®skip_as_script«code_folded$af56d326-e258-48ed-bfbb-0876c2e7411ecell_id$af56d326-e258-48ed-bfbb-0876c2e7411ecoderesponse_1d = missing;metadatashow_logsèdisabled®skip_as_script«code_folded$e190b329-1cd4-499a-80a9-ed549e5390aacell_id$e190b329-1cd4-499a-80a9-ed549e5390aacodeainit_states_ΛCDM_flat = map(i->vec(map_ΛCDM_flat.values) .+ [5, 0.1] .* randn(2), 1:num_chains)metadatashow_logsèdisabled®skip_as_script«code_folded$7be54420-b2ec-4ef4-adc8-848817305d07cell_id$7be54420-b2ec-4ef4-adc8-848817305d07code%md"""
# Ingest & Prepare the Data
"""metadatashow_logsèdisabled®skip_as_script«code_folded$786a5179-68cb-4157-b8d1-52c39cc365cccell_id$786a5179-68cb-4157-b8d1-52c39cc365cccode_md"""
One possible generalization of Hubble's law would be to consider a quadratic function for the local expansion rate.
$d = \frac{v}{H_0} + a_2 v^2.$
On one hand, in this model the predicted distance is a non-linear function of velocity. However, fitting for the model parameters $\left(H_0^{-1}, a_2\right)$ is still a linear problem, since those coefficients only appear linearly in the model if the observed velocities are held fixed.
Thus, we can again compute the maximum likelihood estimator (MLE) with the quadratic model using linear algebra and the functions from Lab 3, as shown below.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$b1932413-9d9d-4d19-833c-29513a73fa41cell_id$b1932413-9d9d-4d19-833c-29513a73fa41coderχ²_ΛCDM_flat_mle = loss(mle_ΛCDM_flat.values[2], 1-mle_ΛCDM_flat.values[2], mle_ΛCDM_flat.values[1], df_all)metadatashow_logsèdisabled®skip_as_script«code_folded$303488cb-b2d5-49f0-a66e-24c4bba21ec0cell_id$303488cb-b2d5-49f0-a66e-24c4bba21ec0codeif !isnothing(chain_ΛCDM)
local n = 500
local subsample = (nluminosity_distance.(z_from_v.(v_plt); Ωm=subsample[:Ωₘ][i], ΩΛ=subsample[:ΩΛ][i], H0=subsample[:H₀][i]), 1:n)
posterior_pred_mean_ΛCDM = mean(pred, dims=2)
posterior_pred_std_ΛCDM = std(pred, mean=posterior_pred_mean_ΛCDM, dims=2)
local target_qs = [0.05, 0.5, 0.95]
posterior_pred_quantiles_ΛCDM = stack(i->quantile(view(pred,i,:),target_qs), 1:size(pred,1))'
end;metadatashow_logsèdisabled®skip_as_script«code_folded$0e956c7f-bd7a-45e5-a080-cd26e1a6df1bcell_id$0e956c7f-bd7a-45e5-a080-cd26e1a6df1bcodeamd"""
So far, we've sought a **descriptive model** for our data.
That is, a model that would fit the data well, but without a clear physical motivation or interpretation for the each model parameter.
Next, we'll fit an **explanatory model**.
In this approach, we first identify a physically motivated model and use it to make predictions for our data.
Our hope is that by incorporating additional laws of physics, our model will have more predictive power.
The expansion of a homogeneous and isotropic universe is described by the [Friedman equations](https://en.wikipedia.org/wiki/Friedmann_equations) and [Einstein field equations](https://en.wikipedia.org/wiki/Einstein_field_equations). They can be combined into a differential equation for the [cosmological scale factor](https://en.wikipedia.org/wiki/Scale_factor_(cosmology)).
The equation can be integrated yield the following relationship between redshift ($z$), and comoving distance ($d_C$) for a flat universe, as a functinon of the density of radiation ($\Omega_r$), matter ($\Omega_m$) and dark energy ($\Omega_\Lambda$) in the unvierse today.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$57245b90-4fa2-4050-9cf1-4caaba190347cell_id$57245b90-4fa2-4050-9cf1-4caaba190347codeٔ"""
`v_from_z(z)`
Computes velocity (km/s) from redshift (z).
"""
function v_from_z(z::Number)
c = speed_of_light_mps/1000
c - 2c/(1+(1+z)^2)
endmetadatashow_logsèdisabled®skip_as_script«code_folded$2b1f1652-6b1b-4cab-b0d8-d41c931b8193cell_id$2b1f1652-6b1b-4cab-b0d8-d41c931b8193code^md"""
Maximum v to plot: $(@bind max_v_plt2 Slider(1e4:2.5e5; default=maximum(df_all.v)))
"""metadatashow_logsèdisabled®skip_as_script«code_folded$16de306a-aee2-4126-92ae-68ef33493072cell_id$16de306a-aee2-4126-92ae-68ef33493072codedf_mle_polynomialmetadatashow_logsèdisabled®skip_as_script«code_folded$56d8e6c4-1f19-4bb4-a3db-7bf00675efd9cell_id$56d8e6c4-1f19-4bb4-a3db-7bf00675efd9code2# hideall
begin
using Downloads
using CSV, DataFrames
using PDMats, LinearAlgebra
using QuadGK
#using StatsBase,
using Distributions
using Turing, MCMCChains # Random
using Optim
using Plots
using PlutoUI, PlutoTeachingTools
installed_makie = false
#=
installed_makie = true
using PairPlots # Makes nice plots, but requires different plotting backend
using WGLMakie
using Bonito, Markdown
Page() # for Franklin, you still need to configure
WGLMakie.activate!()
Makie.inline!(true) # Make sure to inline plots into Documenter output!
=#
endmetadatashow_logsèdisabled®skip_as_script«code_folded$1ff06db1-1d65-4478-b046-525b5ea99a16cell_id$1ff06db1-1d65-4478-b046-525b5ea99a16codecif ready_to_sample_prior
samples_prior_ΛCDM = sample(model_ΛCDM_given_data, Prior(), 10_000)
endmetadatashow_logsèdisabled®skip_as_script«code_folded$719cbf29-20a4-40de-8a92-3e87dae433facell_id$719cbf29-20a4-40de-8a92-3e87dae433facodeكmd"""
$d_C(z) = \frac{c}{H_0} \int_0^z \frac{dz'}{\sqrt{\left(\Omega_r (1+z')^4 + \Omega_m (1+z')^3 + \Omega_\Lambda \right)}}$
"""metadatashow_logsèdisabled®skip_as_script«code_folded$db3880d8-5f31-4133-aec3-a6a3caf79632cell_id$db3880d8-5f31-4133-aec3-a6a3caf79632coderesponse_2c = missing;metadatashow_logsèdisabled®skip_as_script«code_folded$935aa855-f947-41ca-afb0-27547d421743cell_id$935aa855-f947-41ca-afb0-27547d421743code٥md"""
**Q1d:** Does including a cubic term result in further reduction in χ²? What about a quartic term? How would you decide when to stop adding more terms?
"""metadatashow_logsèdisabled®skip_as_script«code_folded$18c5f554-ef2d-4c17-a3d5-5eca179c4b7dcell_id$18c5f554-ef2d-4c17-a3d5-5eca179c4b7dcodeلmd"""**Q2d:** Try changing the initial guess significantly. Are the results very sensitive or insensitive to the initial guess?"""metadatashow_logsèdisabled®skip_as_script«code_folded$ad90c07f-41c8-41f7-b056-8e5bbe519be7cell_id$ad90c07f-41c8-41f7-b056-8e5bbe519be7codemd"""
In the figure above, there are histograms for the marginal posterior distribution for each of the model parameters along the diagonal.
In the bottom corner, there is a scatter plot, showing posterior samples from the posterior probability densitiy function.
Here we see that both H₀ and Ωₘ are well measured and there is a minor correlation between values of the the two model parameters.
Next, we'll look at a similar **pair plot** (also known as a **corner plot**), for the three-parameter ΛCDM model.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$a8f89fe8-110b-452e-9ecb-53769aa02a2ecell_id$a8f89fe8-110b-452e-9ecb-53769aa02a2ecodemd"""
#### Overview
In a previous lab, we measured the Hubble constant ($H_0$), the rate of expansion of the local universe using observations of Type 1a Supernovae.
But we also noticed that the model wasn't a great description of all the avaliable data, particularly those at larger redshifts.
In this lab we'll apply multiple models and compare how well each performs.
To keep things self-contained, we'll download and read data from the Supernovae Cosmology Project and the same transformations, so that we can plot velocity versus distance.
But since we've alredy seen how to do that, I've placed the cells that do that at the bottom of the notebook.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$9bac20ab-05ec-49f9-847c-85f714dfc448cell_id$9bac20ab-05ec-49f9-847c-85f714dfc448codeپif ready_to_run_mcmc
chain_ΛCDM = sample(model_ΛCDM_given_data, NUTS(0.65), MCMCThreads(), num_mcmc_samples, num_chains; initial_params=init_states_ΛCDM)
else
chain_ΛCDM = nothing
end;metadatashow_logsèdisabled®skip_as_script«code_folded$079033dc-4ae8-4d7d-b0e2-fd7bb8f646dacell_id$079033dc-4ae8-4d7d-b0e2-fd7bb8f646dacodeif !isnothing(chain_ΛCDM) && !isnothing(chain_ΛCDM_flat)
let
plt2 = Plots.plot(v_plt, posterior_pred_mean_ΛCDM.-posterior_pred_mean_ΛCDM_flat, label=:none, ylabel="Mpc", title="Difference in Mean of Predictive Distributions")
plt3 = Plots.plot(xlabel="v (km/s)", ylabel="Mpc", title="Width of Predictive Distributions")
Plots.plot!(plt3,v_plt, posterior_pred_std_ΛCDM_flat, alpha=0.7, label="Flat ΛCDM")
Plots.plot!(plt3,v_plt, posterior_pred_std_ΛCDM, alpha=0.7, label=" ΛCDM")
Plots.plot(plt2,plt3,layout=(2,1))
end
endmetadatashow_logsèdisabled®skip_as_script«code_folded$f94cf623-82eb-4f81-af82-aaaceeb69a03cell_id$f94cf623-82eb-4f81-af82-aaaceeb69a03codeٵmd"""
This time the two point estimates are noticable different.
To understand what happened, we'll compare the χ² of the MLE and MAP estimators for the ΛCDM model below.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$a01c7aad-429d-447b-8943-a4e89dbac6f6cell_id$a01c7aad-429d-447b-8943-a4e89dbac6f6code.if ismissing(response_2d)
still_missing()
endmetadatashow_logsèdisabled®skip_as_script«code_folded$6a573a58-1aac-4d5b-ada4-9d8bb7c4295ccell_id$6a573a58-1aac-4d5b-ada4-9d8bb7c4295ccode8if installed_makie
let
df = DataFrame(chain_ΛCDM_flat)
select!(df,[:H₀,:Ωₘ])
fig = pairplot(
df => (
PairPlots.Contour(linewidth=1.5),
PairPlots.Scatter(filtersigma=2, color=:black),
# New:
PairPlots.MarginHist()
)
)
Label(fig[0,:], "Flat ΛCMD Model")
fig
end
endmetadatashow_logsèdisabled®skip_as_script«code_folded$430fd336-2f9e-4efb-bb58-3ff1533c5018cell_id$430fd336-2f9e-4efb-bb58-3ff1533c5018codebegin
local A_quadratic = hcat(df_all.v, df_all.v.^2)
coeff_quadratic = calc_mle_linear_model(A_quadratic, df_all.dₗ, Σ_all)
H₀_mle_quadratic = 1/coeff_quadratic[1]
σ_coeff_quadratic = calc_sigma_mle_linear_model(A_quadratic, Σ_all)
σ_H₀_quadratic = abs(H₀_mle_quadratic * σ_coeff_quadratic[1] / coeff_quadratic[1])
d_pred_quadratic = predict_linear_model(A_quadratic, coeff_quadratic)
χ²_quadratic = sum(abs2.((df_all.dₗ.-d_pred_quadratic)./df_all.σ_dₗ))
local A_plt_quadratic = hcat(v_plt, v_plt.^2)
d_plt_quadratic = predict_linear_model(A_plt_quadratic, coeff_quadratic)
end;metadatashow_logsèdisabled®skip_as_script«code_folded$b0b3ced7-f6da-4c61-9d0b-57912ac1ea9ecell_id$b0b3ced7-f6da-4c61-9d0b-57912ac1ea9ecodeXbegin
local A_cubic = hcat(df_all.v, df_all.v.^2, df_all.v.^3)
coeff_cubic = calc_mle_linear_model(A_cubic, df_all.dₗ, Σ_all)
H₀_mle_cubic = 1/coeff_cubic[1]
#σ_coeff_cubic = calc_sigma_mle_linear_model(A_cubic, Σ_all)
#σ_H₀_cubic = abs(H₀_mle_cubic * σ_coeff_cubic[1] / coeff_cubic[1])
d_pred_cubic = predict_linear_model(A_cubic, coeff_cubic)
χ²_cubic = sum(abs2.((df_all.dₗ.-d_pred_cubic)./df_all.σ_dₗ))
local A_plt_cubic = hcat(v_plt, v_plt.^2, v_plt.^3)
d_plt_cubic = predict_linear_model(A_plt_cubic, coeff_cubic)
end;metadatashow_logsèdisabled®skip_as_script«code_folded$a45eecde-490e-41f4-aa9b-91dd8309157ccell_id$a45eecde-490e-41f4-aa9b-91dd8309157ccodeKlet
if !isnothing(chain_ΛCDM)
plt_12 = Plots.scatter(chain_ΛCDM[:H₀], chain_ΛCDM[:Ωₘ], ms=1, markerstrokewidth=0, xlabel="H₀ (km/s/Mpc)", ylabel="Ωₘ", legend=:none)
plt_13 = Plots.scatter(chain_ΛCDM[:H₀], chain_ΛCDM[:ΩΛ], ms=1, markerstrokewidth=0, xlabel="H₀ (km/s/Mpc)", ylabel="ΩΛ", legend=:none)
plt_23 = Plots.scatter(chain_ΛCDM[:Ωₘ], chain_ΛCDM[:ΩΛ], ms=1, markerstrokewidth=0, xlabel="Ωₘ", ylabel="ΩΛ", legend=:none)
plt_h1 = Plots.histogram(chain_ΛCDM[:H₀], xlabel="H₀ (km/s/Mpc)", alpha=0.5, legend=:none)
plt_h2 = Plots.histogram(chain_ΛCDM[:Ωₘ], xlabel="Ωₘ", alpha=0.5, legend=:none)
plt_h3 = Plots.histogram(chain_ΛCDM[:ΩΛ], xlabel="ΩΛ", alpha=0.5, legend=:none)
Plots.plot(plt_h1,plt_12,plt_h2,plt_13, plt_23, plt_h3, layout=@layout([a _ _ ; b c _ ; d e f]) )
end
endmetadatashow_logsèdisabled®skip_as_script«code_folded$4525ed16-ca55-4299-a473-55550ac78a69cell_id$4525ed16-ca55-4299-a473-55550ac78a69coderesponse_1a = missing;metadatashow_logsèdisabled®skip_as_script«code_folded$b0a53804-568c-4911-ac4e-2d5944a43f66cell_id$b0a53804-568c-4911-ac4e-2d5944a43f66codeٛluminosity_distance(z::Number; Ωm::Number, ΩΛ::Number, Ωr::Number = 0.0, H0::Number = 1) = (1+z) * comoving_distance(z;Ωm=Ωm,ΩΛ=ΩΛ,Ωr=Ωr,H0=H0)metadatashow_logsèdisabled®skip_as_script«code_folded$810937ec-fdc2-4a26-a85f-17d3af0ada1ccell_id$810937ec-fdc2-4a26-a85f-17d3af0ada1ccodeUmodel_ΛCDM_flat_given_data = model_ΛCDM_flat(df_all.z, df_all.dₗ, df_all.σ_dₗ)metadatashow_logsèdisabled®skip_as_script«code_folded$8e5c2017-b360-4c34-9832-4f85564a206dcell_id$8e5c2017-b360-4c34-9832-4f85564a206dcode&md"""
## Fitting Polynomial Models
"""metadatashow_logsèdisabled®skip_as_script«code_folded$f792f5df-0980-4b31-b9dc-a302234dc582cell_id$f792f5df-0980-4b31-b9dc-a302234dc582code5if !isnothing(chain_ΛCDM)
describe(chain_ΛCDM)
endmetadatashow_logsèdisabled®skip_as_script«code_folded$86148d44-9381-458c-8091-f98b5fbbbfeecell_id$86148d44-9381-458c-8091-f98b5fbbbfeecodemd"""
H₀: $(@bind H0_plt Slider(40:1:120; default=70))
Ωₘ: $(@bind Ωm_plt Slider(0.01:0.01:1; default=0.5))
ΩΛ: $(@bind ΩΛ_plt Slider(0:0.02:1; default=0.012))
Use logscale: $(@bind plt_logscale CheckBox())
"""metadatashow_logsèdisabled®skip_as_script«code_folded$adf117fe-0f54-4f2a-b107-0635f56263f2cell_id$adf117fe-0f54-4f2a-b107-0635f56263f2coderesponse_2g = missing;metadatashow_logsèdisabled®skip_as_script«code_folded$bf698723-ef13-4de3-a8dd-a6ff7b0e1cf5cell_id$bf698723-ef13-4de3-a8dd-a6ff7b0e1cf5code&v_plt = range(0,stop=2.4e5, length=20)metadatashow_logsèdisabled®skip_as_script«code_folded$f99b0352-fb0f-467c-840b-4fb3e9a9126acell_id$f99b0352-fb0f-467c-840b-4fb3e9a9126acodeهmd"""
**Q1c:** Try to think of a model that might result in even better match between predictions an data. Explain your thinking.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$673adbdb-b3a7-49a7-94e3-192609cd2780cell_id$673adbdb-b3a7-49a7-94e3-192609cd2780code'md"""
## Fitting a flat ΛCDM model
"""metadatashow_logsèdisabled®skip_as_script«code_folded$c086c05f-ad73-44c7-bf8b-a304d3533690cell_id$c086c05f-ad73-44c7-bf8b-a304d3533690code&"""
`calc_sigma_mle_linear_model(A, covar)`
Computes the uncertainty in the maximum likelihood estimate of θ for the linear model
`y = A θ`
where measurements errors of `y_obs` are normally distributed and have covariance `covar`.
Note that `y_obs` is not passed to this function, because it does not affect results.
"""
function calc_sigma_mle_linear_model(A::AbstractMatrix, covar::AbstractMatrix)
@assert size(A,1) == size(covar,1) == size(covar,2)
@assert size(A,2) >= 1
sqrt.(diag(inv(PDMat(calc_fisher_matrix_linear_model(A,covar)))))
endmetadatashow_logsèdisabled®skip_as_script«code_folded$b2750a60-98c1-49fd-be2f-49273db8fe1fcell_id$b2750a60-98c1-49fd-be2f-49273db8fe1fcodemd"""
The two posterior predictive distribution very nearly overlaps, so you can't easily distinguish the two curves (for the mean of the posterior predictive distribution at each velocity). The shaded region shows a region plus/minus 5 times the standard deviation of the predictive distribution at each velocity. (Exagerated, so you can see it.) Again, the two shaded regions very nearly overlap. However, the predictive distributions are not identical, as shown in the figures below.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$2d97268e-560b-4398-982e-1cf5cd9ab7ddcell_id$2d97268e-560b-4398-982e-1cf5cd9ab7ddcode.if ismissing(response_2b)
still_missing()
endmetadatashow_logsèdisabled®skip_as_script«code_folded$45b97d0a-d607-4e32-8484-ac1420f6b3a4cell_id$45b97d0a-d607-4e32-8484-ac1420f6b3a4coderesponse_2e = missing;metadatashow_logsèdisabled®skip_as_script«code_folded$ae018553-7f25-4ee1-8e44-c8d80fe894c0cell_id$ae018553-7f25-4ee1-8e44-c8d80fe894c0codezmd"""
While the posterior mean values of H₀ are similar, the standard deviation of the marginal posterior distributions (reported under `std`) are quite different.
Similarly, the posterior mean values of Ωₘ in the two models are similar.
So why did the MAP estimators differ so much?
It'll be useful to look at the correlation of different parameters with each other.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$7ee34b79-a3f2-47f8-8f2a-1aa6c589405acell_id$7ee34b79-a3f2-47f8-8f2a-1aa6c589405acodeٚmd"""
Now, we'll try fitting a non-linear model to the general ΛCDM model (that allows for non-flat geometries). First, we'll condition on the data.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$c78f0ad9-03c0-4066-a390-193e7bfa84b1cell_id$c78f0ad9-03c0-4066-a390-193e7bfa84b1code6md"""**Q2g:** Compare the posterior distributions under the two models. How does the width of the marginal distributions for H₀ and Ωₘ change depending on whether we use the flat or general ΛCDM model? What is causing the change? What does this imply for the uncertainty for the model parameters? """metadatashow_logsèdisabled®skip_as_script«code_folded$575cd616-c08c-4f84-9b68-76f6cda207b5cell_id$575cd616-c08c-4f84-9b68-76f6cda207b5codeUif ready_to_sample_prior && installed_makie
let
df = DataFrame(samples_prior_ΛCDM_flat)
select!(df,[:H₀,:Ωₘ])
fig = pairplot(
df => (
PairPlots.Contour(linewidth=1.5),
PairPlots.Scatter(filtersigma=2, color=:black),
PairPlots.MarginHist()
)
)
Label(fig[0,:], "Prior from flat-ΛCMD Model")
fig
end
endmetadatashow_logsèdisabled®skip_as_script«code_folded$185cc5dc-360e-44be-ba5b-4eca7455c8a8cell_id$185cc5dc-360e-44be-ba5b-4eca7455c8a8coderesponse_2i = missing;metadatashow_logsèdisabled®skip_as_script«code_folded$e753bcf1-f68b-4a55-9132-99bc2ea1cf92cell_id$e753bcf1-f68b-4a55-9132-99bc2ea1cf92codeٰlet
n = size(df_all,1)
x = range(0,stop=ceil(Int,1.5*n))
Plots.plot(x,pdf.(Chisq(n), x), label=:none, xlabel="χ²", ylabel="PDF", title="Expected Distribution of χ²")
endmetadatashow_logsèdisabled®skip_as_script«code_folded$a45b4f4c-efc8-435b-b37c-e265dd204a9ccell_id$a45b4f4c-efc8-435b-b37c-e265dd204a9ccoderesponse_1b = missing;metadatashow_logsèdisabled®skip_as_script«code_folded$813ee621-02a2-4998-8ec6-4c08dbab8445cell_id$813ee621-02a2-4998-8ec6-4c08dbab8445codedif !isnothing(chain_ΛCDM) && !isnothing(chain_ΛCDM_flat)
let
plt1 = Plots.plot(xlabel="v (km/s)", ylabel="d (Mpc)", title="Posterior Predictive Distributions")
Plots.plot!(plt1,v_plt, posterior_pred_mean_ΛCDM_flat, ribbon=5.0 .* posterior_pred_std_ΛCDM_flat, alpha=0.7, label="Flat ΛCDM")
Plots.plot!(plt1,v_plt, posterior_pred_mean_ΛCDM, ribbon=5.0 .* posterior_pred_std_ΛCDM, alpha=0.7, label="ΛCDM")
#=
Plots.plot!(plt1,v_plt, view(posterior_pred_quantiles_ΛCDM,:,2), ribbon=(
view(posterior_pred_quantiles_ΛCDM,:,2).-view(posterior_pred_quantiles_ΛCDM,:,1),
view(posterior_pred_quantiles_ΛCDM,:,3).-view(posterior_pred_quantiles_ΛCDM,:,2)
),alpha=0.7, label="Flat ΛCDM")
Plots.plot!(plt1,v_plt, view(posterior_pred_quantiles_ΛCDM_flat,:,2), ribbon=(
view(posterior_pred_quantiles_ΛCDM_flat,:,2).-view(posterior_pred_quantiles_ΛCDM_flat,:,1),
view(posterior_pred_quantiles_ΛCDM_flat,:,3).-view(posterior_pred_quantiles_ΛCDM_flat,:,2)
),alpha=0.7, label="ΛCDM")
=#
Plots.xlims!(plt1,0,max_v_plt2)
Plots.ylims!(plt1,0,maximum(posterior_pred_mean_ΛCDM[v_plt.<=max_v_plt2]))
end
endmetadatashow_logsèdisabled®skip_as_script«code_folded$86287560-cf47-4313-bfb2-27d99cbdb5b2cell_id$86287560-cf47-4313-bfb2-27d99cbdb5b2codegmle_ΛCDM_flat = maximum_likelihood(model_ΛCDM_flat_given_data; initial_params=init_guess_ΛCMD_flat )metadatashow_logsèdisabled®skip_as_script«code_folded$8462ca0a-2e42-4781-84e4-ae3dfc0cfa13cell_id$8462ca0a-2e42-4781-84e4-ae3dfc0cfa13codeZmap_ΛCDM = maximum_a_posteriori(model_ΛCDM_given_data; initial_params=init_guess_ΛCDM )metadatashow_logsèdisabled®skip_as_script«code_folded$6d38c597-00aa-4743-a93a-ab5344f17338cell_id$6d38c597-00aa-4743-a93a-ab5344f17338code0md"""
Now that we've thought about the point estimates from each model, we're ready to compare the posterior distributions under the two models.
We'll use an MCMC algorithm to draw samplmes from the posterior distribution for the model parameters for both the general ΛCDM and flat-ΛCDM models.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$d429454d-1466-4765-a908-0fb8d0c06c9acell_id$d429454d-1466-4765-a908-0fb8d0c06c9acode.if ismissing(response_1d)
still_missing()
endmetadatashow_logsèdisabled®skip_as_script«code_folded$61af3c11-d027-4eb8-a272-6f69c9ed9a3fcell_id$61af3c11-d027-4eb8-a272-6f69c9ed9a3fcode#md"""
You can view the prediction of the maximum likelihood estimate by selecting `Quadratic` in the drop down box just below the figure above.
**Q1a:** Based on inspecting the plot above, does the quadratic model appear to result in an improved fit to the data? If so, what improved?
"""metadatashow_logsèdisabled®skip_as_script«code_folded$7589030f-7b85-4243-bb0b-b39b6f02c7b8cell_id$7589030f-7b85-4243-bb0b-b39b6f02c7b8code\md"""
# A physical model
#### (and hopefully an explanatory model)
## Friedman Equations
"""metadatashow_logsèdisabled®skip_as_script«code_folded$95bc4b33-d619-4a95-af19-5771d209336dcell_id$95bc4b33-d619-4a95-af19-5771d209336dcodes# hideall
begin
title = "Data Science for Astronomy"
week = 4
topic = "Model Building II: Model Assessment"
end;metadatashow_logsèdisabled®skip_as_script«code_folded$4bdd7b2d-9a67-4161-a0a0-93073d9edcf0cell_id$4bdd7b2d-9a67-4161-a0a0-93073d9edcf0codeٻmd"""
## Plot the data and physical model
We'll start with a plot of the data to remind us of the dataset and to help us think about what alternative models we might want to consider.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$cad9502b-3c05-424d-90d9-2e8e821bfa21cell_id$cad9502b-3c05-424d-90d9-2e8e821bfa21code.if ismissing(response_2i)
still_missing()
endmetadatashow_logsèdisabled®skip_as_script«code_folded$6bd2122a-9892-48d5-a55a-d2be917ea427cell_id$6bd2122a-9892-48d5-a55a-d2be917ea427codemd"""
Order of Model to Plot: $(@bind order_to_plt PlutoUI.Select([1=>"Linear", 2=>"Quadratic", 3=>"Cubic", 4=>"Quartic"]))
"""metadatashow_logsèdisabled®skip_as_script«code_folded$2532e9de-8fa7-44ce-8502-c2473928aa8ecell_id$2532e9de-8fa7-44ce-8502-c2473928aa8ecode# hideall
WidthOverDocs()metadatashow_logsèdisabled®skip_as_script«code_folded$ac337a4f-6f87-4c94-9e35-15bccdf389a3cell_id$ac337a4f-6f87-4c94-9e35-15bccdf389a3codeKmodel_ΛCDM_given_data = model_ΛCDM(df_all.z, df_all.dₗ, df_all.σ_dₗ)metadatashow_logsèdisabled®skip_as_script«code_folded$8db610d0-64d5-4692-a829-20a69a280e03cell_id$8db610d0-64d5-4692-a829-20a69a280e03codeٜ"""
`z_from_c(z)`
Computes redshift (z) from velocity (km/s).
"""
function z_from_v(v::Number)
c = speed_of_light_mps/1000
sqrt((1+v/c)/(1-v/c)) - 1
endmetadatashow_logsèdisabled®skip_as_script«code_folded$2aa67e22-a0fd-4870-b621-2fc7abcd39f9cell_id$2aa67e22-a0fd-4870-b621-2fc7abcd39f9codelet
plt1 = Plots.scatter(df_all.v, df_all.dₗ, yerr=df_all.σ_dₗ, xlabel="v (km/s)", ylabel = "Distance (Mpc)", label=:none, legend=:topleft ) #, ylims=(0,1.25e4))
Plots.xlims!(plt1, 0, max_v_plt)
Plots.ylims!(plt1, 0, maximum((df_all.dₗ.+df_all.σ_dₗ)[df_all.v.<=max_v_plt]))
plt2 = Plots.scatter(xlabel="v (km/s)", ylabel = "Residual (Mpc)", legend=:none)
#df_all.v, df_all.dₗ.-df_all.dₗ, xlims=(0,2.2e5))
Plots.xlims!(plt2, 0, max_v_plt)
#ylims!(plt2, -300,300)
if order_to_plt == 1
χ²str = string(round(df_mle_polynomial.χ²[1],digits=1))
Plots.plot!(plt1, df_plt.v, df_plt.d_pred_linear, lc=:red, label="Linear χ²=" * χ²str)
Plots.scatter!(plt2, df_all.v, df_all.dₗ.-d_pred_linear, yerr=df_all.σ_dₗ, label="Linear")
Plots.ylims!(plt2, extrema((df_all.dₗ.-d_pred_linear)[df_all.v.<=max_v_plt]))
elseif order_to_plt == 2
χ²str = string(round(df_mle_polynomial.χ²[2],digits=1))
Plots.plot!(plt1, df_plt.v, df_plt.d_pred_quadratic, lc=:orange, label="Quadratic χ²=" * χ²str)
Plots.scatter!(plt2, df_all.v, df_all.dₗ.-d_pred_quadratic, yerr=df_all.σ_dₗ, label="Quadratic")
Plots.ylims!(plt2, extrema((df_all.dₗ.-d_pred_quadratic)[df_all.v.<=max_v_plt]))
elseif order_to_plt == 3
χ²str = string(round(df_mle_polynomial.χ²[3],digits=1))
Plots.plot!(plt1, df_plt.v, df_plt.d_pred_cubic, lc=:green, label="Cubic χ²=" * χ²str)
Plots.scatter!(plt2, df_all.v, df_all.dₗ.-d_pred_cubic, yerr=df_all.σ_dₗ, label="Cubic")
Plots.ylims!(plt2, extrema((df_all.dₗ.-d_pred_cubic)[df_all.v.<=max_v_plt]))
elseif order_to_plt == 4
χ²str = string(round(df_mle_polynomial.χ²[4],digits=1))
Plots.plot!(plt1, df_plt.v, df_plt.d_pred_quartic, lc=:purple, label="Quartic χ²=" * χ²str)
Plots.scatter!(plt2, df_all.v, df_all.dₗ.-d_pred_quartic, yerr=df_all.σ_dₗ, label="Quartic")
Plots.ylims!(plt2, extrema((df_all.dₗ.-d_pred_quartic)[df_all.v.<=max_v_plt]))
end
Plots.plot(plt1,plt2, layout=(2,1) )
endmetadatashow_logsèdisabled®skip_as_script«code_folded$5c63db36-cac9-4a3b-8363-ee9a99150e0ccell_id$5c63db36-cac9-4a3b-8363-ee9a99150e0ccodeٓif ready_to_sample_prior
Plots.histogram(vec(samples_prior_ΛCDM_flat[:Ωₘ]), label=:none, xlims=(0,1), xlabel="Ωₘ", ylabel = "Prior PDF")
endmetadatashow_logsèdisabled®skip_as_script«code_folded$030350e7-062e-45e4-abf0-b2762fe0e389cell_id$030350e7-062e-45e4-abf0-b2762fe0e389code/md"""
## Posterior Predictive Distributions
"""metadatashow_logsèdisabled®skip_as_script«code_folded$410d54ae-ec72-47ca-9616-78aad7d69d8fcell_id$410d54ae-ec72-47ca-9616-78aad7d69d8fcode.if ismissing(response_1a)
still_missing()
endmetadatashow_logsèdisabled®skip_as_script«code_folded$f6d3fa9e-acc2-4361-99ea-d88ea00d2483cell_id$f6d3fa9e-acc2-4361-99ea-d88ea00d2483codeXmle_ΛCDM = maximum_likelihood(model_ΛCDM_given_data; initial_params=init_guess_ΛCDM )metadatashow_logsèdisabled®skip_as_script«code_folded$2b3729fc-62b2-478d-b35c-e69e6b0efc93cell_id$2b3729fc-62b2-478d-b35c-e69e6b0efc93codemetadatashow_logsèdisabled®skip_as_script«code_folded$d2a2344a-3f2c-45a8-b44f-2cf4287ea586cell_id$d2a2344a-3f2c-45a8-b44f-2cf4287ea586codemd"""
## Posterior sampling
"""metadatashow_logsèdisabled®skip_as_script«code_folded$b3e1bb2c-f181-4181-85f7-60a25032a9dccell_id$b3e1bb2c-f181-4181-85f7-60a25032a9dccodemd"""
**Q2e:** How similar/different are the values of χ² at the MLE and MAP estimates for the two models? What does this imply about our ability to discriminate between the two models?
"""metadatashow_logsèdisabled®skip_as_script«code_folded$baac5588-10b2-49a9-8821-d70de668a279cell_id$baac5588-10b2-49a9-8821-d70de668a279code.if ismissing(response_1b)
still_missing()
endmetadatashow_logsèdisabled®skip_as_script«code_folded$c9eee5e7-6ed9-4df4-978b-cb18f7173fdacell_id$c9eee5e7-6ed9-4df4-978b-cb18f7173fdacode7md"""
And we'll compute the MLE and MAP estimators.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$5534cef2-d172-4637-8d89-16f76155acafcell_id$5534cef2-d172-4637-8d89-16f76155acafcodeWmd"""
Now we'll compute the posterior predictive distributions under both models.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$f163ae68-a48c-4a34-9df7-97764c9e6997cell_id$f163ae68-a48c-4a34-9df7-97764c9e6997code;begin
local A_linear = hcat(df_all.v)
coeff_linear = calc_mle_linear_model(A_linear, df_all.dₗ, Σ_all)
H₀_mle_linear = 1/coeff_linear[1]
σ_coeff_linear = calc_sigma_mle_linear_model(A_linear, Σ_all)
σ_H₀_linear = abs(H₀_mle_linear * σ_coeff_linear[1] / coeff_linear[1])
d_pred_linear = predict_linear_model(A_linear, coeff_linear)
χ²_linear = sum(abs2.((df_all.dₗ.-d_pred_linear)./df_all.σ_dₗ))
local A_plt_linear = hcat(v_plt)
d_plt_linear = predict_linear_model(A_plt_linear, coeff_linear)
end;metadatashow_logsèdisabled®skip_as_script«code_folded$18ad9375-4589-45cb-abc8-943dd7204e7acell_id$18ad9375-4589-45cb-abc8-943dd7204e7acodeCmd"""
One great thing about the Turing probabilistic programming language is that it makes it easy to perform statistical inference even using models that include complex functions (e.g., the integral in the ΛCDM model). We can easily implement the model for our data by building on the previous functions as follows:
"""metadatashow_logsèdisabled®skip_as_script«code_folded$234fade9-1139-4a4e-b53f-2b21dad9b316cell_id$234fade9-1139-4a4e-b53f-2b21dad9b316code"init_guess_ΛCMD_flat = [100,0.5];metadatashow_logsèdisabled®skip_as_script«code_folded$1570d98d-c654-4075-b195-ca590e5e2664cell_id$1570d98d-c654-4075-b195-ca590e5e2664code*md"""
## Fitting non-flat ΛCDM models
"""metadatashow_logsèdisabled®skip_as_script«code_folded$2c1c7d8a-eda6-4900-9be7-e4b908fed81acell_id$2c1c7d8a-eda6-4900-9be7-e4b908fed81acode3begin
num_mcmc_samples = 1000
num_chains = 3
end;metadatashow_logsèdisabled®skip_as_script«code_folded$2efb7fb2-0402-4eec-b58a-a52fd530e3f5cell_id$2efb7fb2-0402-4eec-b58a-a52fd530e3f5codemd"""
## Read the Data file
"""metadatashow_logsèdisabled®skip_as_script«code_folded$bf7ca6ea-4c07-46e4-baff-b205e1663505cell_id$bf7ca6ea-4c07-46e4-baff-b205e1663505code9md"""
**Astro 416, Lab 4, Ex 1**
#### $title
# $topic
"""metadatashow_logsèdisabled®skip_as_script«code_folded$a1eb8ff3-ff5b-4e50-a75c-480cf2a7d861cell_id$a1eb8ff3-ff5b-4e50-a75c-480cf2a7d861code"""
`predict_linear_model(A, θ)`
Computes the predictions of a linear model with design matrix `A` and parameters `θ`.
"""
function predict_linear_model(A::AbstractMatrix, b::AbstractVector)
@assert size(A,2) == length(b)
A*b
endmetadatashow_logsèdisabled®skip_as_script«code_folded$3f2ef930-d367-11ef-3b5c-95b4989c2e21cell_id$3f2ef930-d367-11ef-3b5c-95b4989c2e21codeYbegin
mkpath("data")
filename = joinpath("data","SCPUnion2.1_mu_vs_z.txt")
if split(pwd(),'/')[end] == "test"
filename = "../" * filename
end
url = "http://supernova.lbl.gov/Union/figures/SCPUnion2.1_mu_vs_z.txt"
if !(filesize(filename)>0)
Downloads.download(url,filename)
@assert filesize(filename)>0
end
endmetadatashow_logsèdisabled®skip_as_script«code_folded$ac427d57-928e-41a1-8b51-2412fbd56799cell_id$ac427d57-928e-41a1-8b51-2412fbd56799coderesponse_2h = missing;metadatashow_logsèdisabled®skip_as_script«code_folded$e09e30f7-838f-4571-bb9c-6d0b425807a8cell_id$e09e30f7-838f-4571-bb9c-6d0b425807a8codeX# hideall
md"""
+++
title = $title
week_num = $week
topic = $topic
+++
""" |> Base.Text;metadatashow_logsèdisabled®skip_as_script«code_folded$95be03d5-9438-40ca-b8db-49b301091153cell_id$95be03d5-9438-40ca-b8db-49b301091153codehbegin
const speed_of_light_mps = 299_792_458
const speed_of_light_kmps = speed_of_light_mps/1000.
end;metadatashow_logsèdisabled®skip_as_script«code_folded$33045766-a7c5-42ff-bdf8-bde1fce9c56bcell_id$33045766-a7c5-42ff-bdf8-bde1fce9c56bcode.if ismissing(response_2a)
still_missing()
endmetadatashow_logsèdisabled®skip_as_script«code_folded$3e703f5f-cb93-4bf7-98d5-8c3bf328fe35cell_id$3e703f5f-cb93-4bf7-98d5-8c3bf328fe35code#init_guess_ΛCDM = [100, 0.5, 0.5];metadatashow_logsèdisabled®skip_as_script«code_folded$edaa83ba-14ee-472b-8eed-05b5997705e0cell_id$edaa83ba-14ee-472b-8eed-05b5997705e0codeًif ready_to_sample_prior
Plots.scatter(vec(samples_prior_ΛCDM_flat[:H₀]),vec(samples_prior_ΛCDM_flat[:Ωₘ]), ms=0, xlims=(0,200))
endmetadatashow_logsèdisabled®skip_as_script«code_folded$707a47f4-0b2d-447b-9c15-6dd82316f0c5cell_id$707a47f4-0b2d-447b-9c15-6dd82316f0c5code{md"""**Q2c:** Rerun the above two cells several times. Now do you get consistent results across runs? Why or why not?"""metadatashow_logsèdisabled®skip_as_script«code_folded$75750e56-bc74-44a5-914c-20ae445b0f58cell_id$75750e56-bc74-44a5-914c-20ae445b0f58code`md"""
We can also compare these to the χ² at MLE and MAP values of the `model_ΛCDM_flat`.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$fb2545e1-d624-4669-9e07-c94a598ba8edcell_id$fb2545e1-d624-4669-9e07-c94a598ba8edcode.if ismissing(response_2f)
still_missing()
endmetadatashow_logsèdisabled®skip_as_script«code_folded$b89069b9-4cbb-4521-b3c4-ad9ae5c15f41cell_id$b89069b9-4cbb-4521-b3c4-ad9ae5c15f41code&md"""
## Implement the ΛCDM model
"""metadatashow_logsèdisabled®skip_as_script«code_folded$ff0d5280-39fe-490a-ab06-4121c0555ca9cell_id$ff0d5280-39fe-490a-ab06-4121c0555ca9code\init_states_ΛCDM = map(i->vec(mle_ΛCDM.values) .+ [5, 0.1, 0.1] .* randn(3), 1:num_chains)metadatashow_logsèdisabled®skip_as_script«code_folded$e92981aa-8386-44f6-9e01-1bcf3633f6aecell_id$e92981aa-8386-44f6-9e01-1bcf3633f6aecode9md"""
This is the end of the lab.
Below, I included some extra calculations in case they're useful for people to refer back to when it comes time to do something for their proejct.
# Sampling from Priors
Check if you'd like to sample from the prior: $(@bind ready_to_sample_prior CheckBox(; default=false))
"""metadatashow_logsèdisabled®skip_as_script«code_folded$aab05339-dea2-4c7f-8b38-bc7d7971f1d9cell_id$aab05339-dea2-4c7f-8b38-bc7d7971f1d9codemd"""
## Functions used
"""metadatashow_logsèdisabled®skip_as_script«code_folded$46eb1088-0265-46a6-91ec-44dfa3bd862dcell_id$46eb1088-0265-46a6-91ec-44dfa3bd862dcodeLlet
plt = Plots.scatter(xlabel="v (km/s)", ylabel="Luminosity Distance (Mpc)")
if plt_logscale
Plots.scatter!(plt,log10.(df_all.v), log10.(df_all.dₗ), yerr=(log10.(df_all.dₗ)-log10.(df_all.dₗ-df_all.σ_dₗ),log10.(df_all.dₗ+df_all.σ_dₗ).-log10.(df_all.dₗ)), label=:none)
else
Plots.scatter!(plt,df_all.v, df_all.dₗ, yerr=df_all.σ_dₗ, label=:none)
end
z_plt = range(0.0,step=0.01, stop=2)
v_plt = v_from_z.(z_plt)
d_plt = luminosity_distance.(z_plt;Ωm=Ωm_plt,ΩΛ=ΩΛ_plt,H0=H0_plt)
loss_plt = loss(Ωm_plt,ΩΛ_plt,H0_plt,df_all)
loss_plt = round(loss_plt,digits=1)
H₀_plt_rounded = round(H0_plt,digits=1)
if plt_logscale
v_plt = log10.(v_plt)
d_plt = log10.(d_plt)
end
Plots.plot!(v_plt,d_plt, lw=3, label="Ωm = $Ωm_plt, ΩΛ= $ΩΛ_plt: χ² = $loss_plt \nH₀=$H₀_plt_rounded km/s/Mpc ")
endmetadatashow_logsèdisabled®skip_as_script«code_folded$76729087-6d1d-499e-a808-7980637c6d68cell_id$76729087-6d1d-499e-a808-7980637c6d68codeO@model function model_ΛCDM(z, d_obs, σ_d_obs)
@assert size(z) == size(d_obs) == size(σ_d_obs)
H₀ ~ LogNormal(log(100), 1.0) # km/s/Mpc
Ωₘ ~ Uniform(1e-4,1) # Universe needs some matter
ΩΛ ~ Uniform(0,1)
d_true = luminosity_distance.(z; Ωm=Ωₘ, ΩΛ=ΩΛ, H0=H₀)
d_obs ~ MvNormal(d_true, σ_d_obs)
endmetadatashow_logsèdisabled®skip_as_script«code_folded«notebook_id$b8549f88-e39b-11ef-2396-db6cd9761657in_temp_dir¨metadata