Installation
Using CAMEL
Tutorials

EUCLIDSchoolS. Plaszczynski, Frejus July 2017. On this page... (hide) Foreword : take some time to answer the questions in bold in the following. 1. Session 1 : Likelihoodbased statistics1.1 prerequisites
1.2 CAMEL InstallationThe first timeDue to a high CPU load when running interactively, we will work on a remote cluster (DEC at CPPM thanks to A. Tilquin and his team for support) To avoid nodes saturation each users will be assigned a number (let say XX) and you should log as binomeXX@mardecXX.in2p3.fr (note that XX appears twice). If you are using X2go fill in (with XX being you id)
ssh Y4C binomeXX@mardecXX.in2p3.fr Check with the command users that you are the only binomeXX logged. Then check you are using the bash shell printenv SHELL should return /bin/bash, otherwise inform the instructor. Copy the initialization scripts to your home directory (ie. where you log in): cp /dec/users/plaszczy/euclidschool/euclid_setup.sh . source euclid_setup.sh ... and have a look at them with your preferred editor:we (strongly) suggest you use an editor without an X window as "nano, (x)emacs nw, vim ..." clone CAMEL euclidschool branch: git clone b euclidschool https://gitlab.in2p3.fr/cosmotools/CAMEL.git CAMEL/HEAD This will pull the CAMEL code and put it into CAMEL/HEAD. HEAD is a special name for CMT (that you will see later) Then execute "camel_setup" from the CAMEL/HEAD/cmt directory : cd CAMEL/HEAD/cmt ln s requirementsdec requirements source camel_setup.sh which class version is used? (for git experts: is it the latest one?) CMT is a convenient tool to avoid writing complicated Makefiles. details in : http://www.cmtsite.net The Makefile is replaced by the much more userfriendly make This builds the libraries. If everything goes fine, construct the executables and test programs with: make test make exec where are the executables? The main path for CAMEL (ie. ~/CAMEL/HEAD) is defined through the environment variable $CAMELROOT : it appeared when you sourced In the following all paths given are relative w.r.t. this location. All the executables are located in Optional: if you want to have a look at the structure of the code see Main.CamelUse (but you don't need it) At each sessionyou always need to do source euclid_setup.sh cd CAMEL/HEAD/cmt source camel_setup.sh 1.3 Model+parametersLet us first create a directory where we will put our work. There is already a cd $CAMELROOT mkdir work/output cd work/output We will learn how to write a parameter file ("parFile"). The very first questions one must ask are
Here we will use the popular (flat) LambdaCDM one. How many free parameters does it contain? In the following for the data we will use a (fake) matter power spectrum measurement (embedded in a likelihood) and work with 3 parameters only (for CPU reasons) ie. we assume the other parameters are known and fixed. In linear theory, the matter powerspectrum shape is most sensitive to the Hubble constant, and matter densities where one should distinguish between the standard "baryonic" one and the "cold dark matter" contribution.
In the following we will call your choice of parameters "par1,par2,par3" and the first values as "guess1,guess2,guess3" Edit a new file called for instance fix par1 cosmo guess1 fix par2 cosmo guess2 fix par3 cosmo guess3 class sBBN\ file bbn/sBBN.dat
The starting keyword "class" can be used to transmit any (valid) string parameter to CLASS (as in explanayory.ini).
Validation:
the ..../amd64_linux26/writeSpectraPk lcdm.par 2000 output.fits 0.5 (where "...." means the relative path from where you work to the amd64_linux26 directory) The third argument is lmax (here 2000) for Cl(CMB), the next is the name for the fits output file, and the last one (here 0.5) is the redshif (z) at which the matter spectrum is computed: You may have a look at the spectrum (but do not spend much time on this) which is in the 3d header of the fits file. You can do it remotely with: ipython pylab or retrieve the Here is how to read a FITS file in python: from astropy.io import fits h= fits.open("output.fits") pkdata=h[3].data k=pkdata['k'] pk=pkdata['pklin'] loglog(k,pk) 1.4 LikelihoodA likelihood is a real function of a set of parameters "pars" (a vector, here (par1,par2,par3)) that is supposed to return a high value when the model (or associated observable) matches the data and low values otherwise. If correctly written (a big job in real life) it contains all the information concerning some measurements(s). We will equally use the term "chi2" which by definition is chi2(pars)=2 ln[L(pars)] so that a "high likelihood" means a "low chi2". An essential piece of information is the value of pars for which L(pars) is maximal or chi2(pars) minimal. This is the Maximum Likelihood Estimator (MLE) and is most often the best one can do to infer which set of pars describes best the data. We will now work with a very basic likelihood, which assumes we did measure a set of "Pk" measuremnts at redshifts of z=0.5,1,1.5 over some k range and we know exactly the associated Gaussian errors (this is a very oversimplified example but it will really happen soon with the EUCLID/LSST missions, with a much higher precision than what we are playing with today). Here, the likelihood value (or chi2, a single number) of a set of cosmological parameters pars=(par1,par2,par3) is in fact computed in two steps:
To use these likelihoods (that are by default summed up) add the following lines to your parFile fakePk1=fakepk/pk_0.5.lik fakePk2=fakepk/pk_1.lik fakePk3=fakepk/pk_1.5.lik
class z_pk XX class P_k_max_1/Mpc YY To check that the parFile is correctly configured we then run the ..../amd64_linux26/writeChi2 lcdm.par 'What does happen? The (galactic) bias is a nuisance parameter, that comes from the fact that one does not measure directly the full matter power spectrum but the power spectrum inferred from some population of galaxies. It is not well known (and actually varies with the galaxy selection and redshift, here for pedagogical reasons we assume it is the same in all the likelihoods) There are generally much more nuisance parameters qualifying for instance possible systematics in the measurement: they impact directly the likelihood (not the cosmological model) and can have some external constraints depending on independent measurements/apriori knowledge. So finally what is called "parameters" is a set of (cosmos,nuisances) ones. The likelihood actually depends on them through :
'''Complete your parFile adding the nuisance parameter (in this case instead of "cosmo", use the keyword is "nui").
But how to find a good fist guess for this parameter?
You may use for that the ..../amd64_linux26/ScanParam lcdm.par bias 1 2 100 will vary the bias parameter linearly in the [1,2] interval with 100 steps. Look for the best chi2 and use the corresponding bias value Rerun 1.5 The Maximum Likelihood Estimator, a.k.a. "bestfit"Given some measurement(s), we are looking for the set of parameters (within a model) that is the most likely to produce those data (note the "likely" semantic, it is not the same than "probable"). Technically this is performed by looking for the parameters that minimize the chi2(pars) function (or maximizes the likelihood). "pars" is a vector that includes both cosmological and all the nuisance parameters. Here it has low dimensionality but it is frequent to go up to 10, 30 or 100 parameters. So we are dealing with the minimization of a multidimensional function. There are several algorithms to do that but there is no magical one (unless your likelihood is linear which is rarely the case). Furthermore we don't have access here to the analytical gradients of the function which complicates the work. We use in CAMEL the MINUIT package which has been widely used in High Energy Physics. However remember: no magic, you always have to experiment with such tools to ensure your really captured the minimum chi2 solution. This means trying different starting points and check that the solution is not blocked at some boundary for example. So far our parameters were fixed. Let us release one.
Modify one of your parameter adding a (rough) estimate of its error and some bounds within which it should be varied
with guesses for the "error" and "bounds" (use spaces or tabs to separate the fields) Also to see what happens during the minimization add to your parFile verbose=true Then run ("...." should be replaced by the proper relative path) ..../Minimize lcdm.par bf.txt hess.txt (the two output arguments are the bestfit/hessian outputs written in external files) What is shown is then the values that are being tried by MINUIT: the last 2 columns are of interest also: they gives respectively the chi2 value for this set of parameters and the CLASS time computation of the model (in s). Don't be worried at the end by this kind of message: WARNING: FunctionMinimum is invalid. which is hard to avoid because of the numerical noise that was discussed. The minimum can be actually valid (but maybe not the estimate of its error, see the Hessian part). Modify slightly the input value and rerun. Do you obtain the same minimum? Which one is better? It may be strange that the minimizer does not always find the same minimum depending on the starting values. In fact it is a challenging problem to go the minimum very precisely (we will see later that we want the chi2_min value below 1 in precision). First because we do not have the gradients to help the descent. But essentially because we are using a numerical code (CLASS) to compute the observables (here Pk). Those computations have soem accuracy ( cutoffs, binnings etc.) which leads to some numerical noise: the derivatives with respect to the cosmological parameters (ie. d_Pk/d_omega_b) do not vary very smoothly (as with an anlytical function) and any minimizer does hate that. We have somewhat tuned the minimization stategy with MINUIT to survive to these effects but still sometimes the minimizer finds a better route to the very precise minimum. In real life estimation we often shoot a set of randomly shaked starting values (within the bounds of the parFile): the receipe is then simple: always take the solution with the lowest chi2_min value. (there are some scripts in camel to automatize this process). 1.6 PrecisionRecall that CLASS is solving numerically some complicated equations. One can control the level of accuracy of many steps in the code. What is nice in CLASS is that all the precision parameters (cutoffs,steps etc.) can be specified in a single place. CLASS has some reasonable values but you may need to adapt them if your need some really high precision. Of course this will take more time so it is always a compromise between speed and precision. When we try to minimize a multidimensional function the algorithms assume that the function is smooth wrt the parameters. It is not the case here since within CLASS the "error" and "bounds" in finite precision of the computations brings up numerical noise because chi2(pars)=chi2(Pk(pars)) where the Pk operation is performed by CLASS. This complexify the minimizer's task of precisely finding the minimum. CAMEL provides some higherprecision parameters to CLASS, still within reasonable computation time. Add the following line to your parFile precisionFile=class_pre/hpjul2.pre (you may look at what is in this file, recall relative paths are wrt to the /lik directory) Rerun and see if there is some difference. Now release all the parameters (ie. modify the "fix" lines by "par" ones) and look for the global bestfit. You may first keep the nuisance fixed (to 1) then release it. You may try to figure out the MINUIT strategy by looking at the output. Does the number of calls before convergence scale as the dimensionality of the minimum? 1.7 The Hessian matrixYou notice as you run that MINUIT finally gives you some "Values" and "Errors" for the released parameters. While the Values are really those that produced the final Chi2 value, the "errors" are much more uncertain. They (should) represent the curvature of the function at the minimum (the so called "Hessian"). However
However we will see in the 2d session, that they can still be interesting, so once you are happy with your "bestfit" keep the "hess.txt" file in a safe place (or rename it). Let's read it in python ( For convenience here is a python function that will nicely plot the array: from pylab import * def plotcor(cor,par): covmat = np.tril(cor, k=1) covmat[covmat==0] = NaN imshow(covmat,vmin=1,vmax=1,interpolation='none') colorbar() yticks(range(len(cor)),par) xticks( range(len(covmat)),par) for (j,i),label in ndenumerate(covmat): if i<j : text(i,j,'{:2.3f}'.format(label),ha='center',va='center') show() why are there so many blank? If you want to know more about the original discussions on the meaning of "errors" in MINUIT see James(1980) 1.8 Are you frequentist or Bayesian?The MLE gives you a solution in a multidimensional space which is not very convenient to visualize. What can we say about individual parameters? If the Hessian is not reliable what should we do? Up to now, we were only dealing with classical statistics, but now you have to choose your religion and decide what a probability means! It's time for you to answer this very disturbing question: '''are you a Bayesian or a frequentist?
Certainly the most sane approach for a scientist is to be agnostic and compare both approaches (which, when the problem is well constrained give in practice similar results). 1.9 ProfilelikelihoodThis method is the frequentist answer to the interval problem. Rather than writing a lot of theory, let see in practice how it works. We choose one variable, here the bias and we will profile it:
NB: here "a bit" means that you are looking only to chi2 values that fluctuates within a few units (let's say ~5) wrt to the lowest chi2 value
Think about it: where is the minimum located wrt to the globalbest fit (ie. when all the parameters free)? Finally write your statement about the bias that was present in those data , using the "+" notation. (then ask God for what She/He put in the simulation) If you have time, try the same with H0. 2. Session 2 : MCMC2.1 New sessionremember to do: source euclid_setup.sh cd CAMEL/HEAD/cmt source camel_setup.sh
2.2 MCMC for realMonte Carlo Markov Chains are popular in cosmology. Although it may look as an evolved technique it is in fact mathematically a very simple one (the MetropolisHastings algorithm, father of all, takes about 10 lines to code, you should try). But anyone who practiced them in real life conditions (ie data) knows that a lot of cooking (on CPU, proposal matrix, convergence, priors) is generally forgotten/hidden. So let see in practice how it works.
The MCMC is therefore only a method to shoot samples from some (complicated) multidimensional function. It is not especially "Bayesian" (it can be used to compute integrals) and is quite inefficient. However with some practice you can manage to sample about everything. Here is how MetroplisHastings (MH) works in practice to sample a function F around its maximum :
Run this for "some" time, then read the file and do histograms. Isn't that simple? However:
Both these points are very intricate: if you have a "good" proposal you can run the chain for a shorter time. In fact theory says the best proposal is to input the "real" covariance of your parameters: you can determine it from the samples themselves but only if your samples have a good proposal! In MH, people try to make some "guess" for this covariance matrix, make some (very long) run, estimate empirically the covariances using these samples, input this matrix as a new proposal, and redo some run(s). In cosmology, it can take more than one week. In CAMEL we did implement some adaptive strategy to reconstruct a covariance matrix on the fly. It is not completely trivial and one should care not destroying the Markovian nature of the chain: samples do not depend only on the previous one but here also on the previous proposal (in practice the "adaptation" of the proposal is gradually decreased until it gets "frozen". We will learn how to use that. 2.3 Bayesian inferenceSo what does this has to do with cosmological parameter estimation? Here the sampled function F is simply the data likelihood. According to Bayes theorem
(there is a denominator, but it's however unimportant for parameter estimation)
Unfortunately most people forget about the priors and more or less implicitly put them to 1 arguing this is the most uninformative one (that's wrong see Jeffreys priors). So that finally in most cases, one uses elaborate statistical terms to justify laziness to pretend that the posterior distribution is equal to the likelihood (which is again wrong, think if something "likely" is the same than something "probable"). A serious Bayesian approach is to figure out how your result varies depending on your priors parametrization. An even simpler one is to compare the output of this method to the frequentist profilelikelihood one which does not have any buildin notion of prior. 2.4 Adaptive Metropolis (AM)So let us see how to run the AM algorithm in CAMEL the lazy way (flat priors actually taken from the bounds in your parFile).
algo=ada length=1000 bunchSize=100 It means we use the AM algorithm, to generate 1000 samples(vectors) that will be written into the file every 100 steps. Then we need some rough estimate of the proposal matrix: why not use the Hessian build in session1? Assuming your Hessian file is called "hessian.txt" then add proposal_cov=hessian.txt Finally AM has a number of parameters to tune the run. try: t0=100 ts=0 do_move=true scale=1 There are actually 2 things the algorithm is adapting from the data
"do_move" allows to jump randomly the first sample, so that each run will start with a different value. Note that the seed of the random number generator changes for each run. Then run ..../mcmc lcdm_mc.par chain1.txt This should last about 3mins. If you are ensure that everything is fine, you can check regularly the number of lines written in chain1.txt (which should increase with time by chuncks of 100) wc l chain1.txt 2.5 Checking for convergenceIn order to sample correctly the distribution, the samples distribution must be in a stationary state known as correct "mixing". Then its distribution "converges" to the right function (posterior/likelihood here). This requires running the algorithm sometimes for a long time (steps) before it forgets its origin and go into a region of the parameter space where the proposal matrix is well adapted. There is no way to be 100% sure a chain is mixing correctly, but you may often clearly see when it did not! Here are various ways: Trace plotsThe very first thing to do once you have your chain (file) is simply to plot each variable to see how it evolved (parameters+chi2 see 1st line header for the names) looking for the moment were the chain is in a somewhat "stable" regime (stationarity) In python read the (text) file, for instance with Have all variables converged? In your view, when do the distributions begin to be stationary (if anywhere)? With these plots (pedantically called "trace plots") you will identify 99% of the problems (not saying that the remaining 1% is OK) Acceptance rateHave a look at the chain1.txt values. What do you notice?
The acceptance rate is a mean value of times the jump was accepted (so value changed in the file).
A rule of thumb is that it should be close to 1/4 (meaning samples are repeated 3/4 of the times). It was established only for Gaussian distribution but in practice this rule also works well for other distributions.
CAMEL computes a this running mean (every 100 steps) and writes it in the file (you should have) named Finally if you wish to see how the scale factor was adapted the file 2.6 Multiple chainsOnce you are happy with your chain a finer insight can be gained running several chains and comparing their characteristics, so construct 4 independent chains ..../mcmc lcdm_mc.par chain2.txt ..../mcmc lcdm_mc.par chain3.txt ..../mcmc lcdm_mc.par chain4.txt They should be different each time since you start from a different starting point (do_move=true) and the seed of the random number generator is (implicitely) different in each run. We will now use some features of the Compare how the chains evolved for in each run : from pylab import * import camel #read the chains in a single list chains=camel.extract_chains('chain',4) #read the variables name from the 1st line of "chain1" with open('chain1.txt', 'r') as f: names= f.readline().split() #overplot the 4 chains for each variable ivar=0 [plot(c[ivar]) for c in chains] title(names[ivar]) #look at the other variables too The GelmanRubin testNow we can go on with a deeper convergence test. Its idea is, for each parameter, to compare the variance of each variables within each chain to the variance among the chains. This allows to build the "R" statistics and a good rule of thumb to see where the variable converged is to look for:
(or 0.01 if you are more demanding) Fortunately camel.py does already everything for you: #Compute GelmanRubin statistic for each parameter it,R = camel.GelmanRubin(chains,gap=100,length_min =0) figure() plot(it,R1.) legend(names[1:]) axhline( 0.03, color='k', ls='')
2.7 Building posteriorsgetting mean values and intervalsNow we are more confident that the chains sample correctly the likelihood around its maximum after some number of steps (lets say N0), we construct one single chain with our "best samples" by concatenating the last part of each chain: #define your N0 value N0= #construct a single chain from the last samples (using samples above N0 steps) chain=camel.mergeMC("chain",num=(1,2,3,4),burnin=N0) #names were defined previously  you can still use the following convenience names=camel.MCparnames("chain") #plot the histograms + basic statitics # the first variable is "chi2", we don't want it for ivar in range(1,5): subplot(2,2,ivar) parname=names[ivar] hist(chain[parname]) title(parname) print "%15s " % parname, "\t %f \t %f \t %f" % camel.getci( np.percentile(chain[parname], [16,50,84])) tight_layout() Now show God your result and ask Him for what She/He put in the simulation. Triangle plotOne can produce the famous "triangle plot" which not only shows the (smoothed) 1D histograms but 2D's too. camel.py contains some quick functions for doing that #names defined previously #chain is from mergeMC camel.triangle(chain,params=names[1:],smooth=5) Here we are short on statistics (in real life one runs for much longer on batch workers).
We can also use the Now starting again from "chain" as the mergedMC values and names as the column names: import camel from getdist import plots, MCSamples #see previously how chain/par are defined #build samples without chi2 mcmc = [chain[par] for par in names[1:]] #the follwing translates to some nice fonts for param names labels=[camel.parname.setdefault(n,n).replace('$','') for n in names[1:]] #getDist functions: samples=[MCSamples( samples=mcmc, names=names[1:],labels=labels)] g = plots.getPlotter() # creates the "plotting" object g.triangle_plot(samples,names[1:],contour_colors='blue',filled=True) 2.8 Closing the loop
construct the empirical covariance and compare it to the Hessian. 3. Want more?From the camel documentation construct a parFile including both the latest BOSSDR12 and JLA (SN1a) measurements and try to find the bestfit. Note that in this real life scenario you may need to
Running interactively with all parameters free is probably irrealistic but you may release a few ones (the full minmization should be run in batch mode) Now you are a real expert, time to have a look at this more complete tutorial 