is co-authored by Felipe Bandeira, Giselle Fretta, Thu Than, and Elbion Redenica. We additionally thank Prof. Carl Scheffler for his assist.
Introduction
Parameter estimation has been for many years some of the essential matters in statistics. Whereas frequentist approaches, reminiscent of Most Chance Estimations, was the gold commonplace, the advance of computation has opened house for Bayesian strategies. Estimating posterior distributions with Mcmc samplers grew to become more and more frequent, however dependable inferences rely upon a activity that’s removed from trivial: ensuring that the sampler — and the processes it executes below the hood — labored as anticipated. Preserving in thoughts what Lewis Caroll as soon as wrote: “If you happen to don’t know the place you’re going, any highway will take you there.”
This text is supposed to assist knowledge scientists consider an usually missed facet of Bayesian parameter estimation: the reliability of the sampling course of. All through the sections, we mix easy analogies with technical rigor to make sure our explanations are accessible to knowledge scientists with any degree of familiarity with Bayesian strategies. Though our implementations are in Python with PyMC, the ideas we cowl are helpful to anybody utilizing an MCMC algorithm, from Metropolis-Hastings to NUTS.
Key Ideas
No knowledge scientist or statistician would disagree with the significance of sturdy parameter estimation strategies. Whether or not the target is to make inferences or conduct simulations, having the capability to mannequin the information era course of is an important a part of the method. For a very long time, the estimations have been primarily carried out utilizing frequentist instruments, reminiscent of Most Chance Estimations (MLE) and even the well-known Least Squares optimization utilized in regressions. But, frequentist strategies have clear shortcomings, reminiscent of the truth that they’re targeted on level estimates and don’t incorporate prior data that might enhance estimates.
As a substitute for these instruments, Bayesian strategies have gained recognition over the previous a long time. They supply statisticians not solely with level estimates of the unknown parameter but additionally with confidence intervals for it, all of that are knowledgeable by the information and by the prior data researchers held. Initially, Bayesian parameter estimation was performed via an tailored model of Bayes’ theorem targeted on unknown parameters (represented as θ) and recognized knowledge factors (represented as x). We are able to outline P(θ|x), the posterior distribution of a parameter’s worth given the information, as:
[ P(theta|x) = fractheta) P(theta){P(x)} ]
On this components, P(x|θ) is the chance of the information given a parameter worth, P(θ) is the prior distribution over the parameter, and P(x) is the proof, which is computed by integrating all doable values of the prior:
[ P(x) = int_theta P(x, theta) dtheta ]
In some instances, because of the complexity of the calculations required, deriving the posterior distribution analytically was not doable. Nevertheless, with the advance of computation, working sampling algorithms (particularly MCMC ones) to estimate posterior distributions has turn out to be simpler, giving researchers a robust device for conditions the place analytical posteriors are usually not trivial to seek out. But, with such energy additionally comes a considerable amount of accountability to make sure that outcomes make sense. That is the place sampler diagnostics are available in, providing a set of beneficial instruments to gauge 1) whether or not an MCMC algorithm is working effectively and, consequently, 2) whether or not the estimated distribution we see is an correct illustration of the actual posterior distribution. However how can we all know so?
How samplers work
Earlier than diving into the technicalities of diagnostics, we will cowl how the method of sampling a posterior (particularly with an MCMC sampler) works. In easy phrases, we are able to consider a posterior distribution as a geographical space we haven’t been to however have to know the topography of. How can we draw an correct map of the area?
One in all our favourite analogies comes from Ben Gilbert. Suppose that the unknown area is definitely a home whose floorplan we want to map. For some cause, we can not immediately go to the home, however we are able to ship bees inside with GPS gadgets hooked up to them. If all the pieces works as anticipated, the bees will fly round the home, and utilizing their trajectories, we are able to estimate what the ground plan appears like. On this analogy, the ground plan is the posterior distribution, and the sampler is the group of bees flying round the home.
The explanation we’re writing this text is that, in some instances, the bees gained’t fly as anticipated. In the event that they get caught in a sure room for some cause (as a result of somebody dropped sugar on the ground, for instance), the information they return gained’t be consultant of the complete home; reasonably than visiting all rooms, the bees solely visited a couple of, and our image of what the home appears like will finally be incomplete. Equally, when a sampler doesn’t work accurately, our estimation of the posterior distribution can also be incomplete, and any inference we draw based mostly on it’s more likely to be fallacious.
Monte Carlo Markov Chain (MCMC)
In technical phrases, we name an MCMC course of any algorithm that undergoes transitions from one state to a different with sure properties. Markov Chain refers to the truth that the following state solely is determined by the present one (or that the bee’s subsequent location is barely influenced by its present place, and never by all the locations the place it has been earlier than). Monte Carlo signifies that the following state is chosen randomly. MCMC strategies like Metropolis-Hastings, Gibbs sampling, Hamiltonian Monte Carlo (HMC), and No-U-Flip Sampler (NUTS) all function by establishing Markov Chains (a sequence of steps) which can be near random and regularly discover the posterior distribution.
Now that you simply perceive how a sampler works, let’s dive right into a sensible situation to assist us discover sampling issues.
Case Examine
Think about that, in a faraway nation, a governor needs to know extra about public annual spending on healthcare by mayors of cities with lower than 1 million inhabitants. Quite than taking a look at sheer frequencies, he needs to know the underlying distribution explaining expenditure, and a pattern of spending knowledge is about to reach. The issue is that two of the economists concerned within the venture disagree about how the mannequin ought to look.
Mannequin 1
The primary economist believes that each one cities spend equally, with some variation round a sure imply. As such, he creates a easy mannequin. Though the specifics of how the economist selected his priors are irrelevant to us, we do have to remember that he’s attempting to approximate a Regular (unimodal) distribution.
[
x_i sim text{Normal}(mu, sigma^2) text{ i.i.d. for all } i
mu sim text{Normal}(10, 2)
sigma^2 sim text{Uniform}(0,5)
]
Mannequin 2
The second economist disagrees, arguing that spending is extra advanced than his colleague believes. He believes that, given ideological variations and price range constraints, there are two sorts of cities: those that do their greatest to spend little or no and those that aren’t afraid of spending so much. As such, he creates a barely extra advanced mannequin, utilizing a combination of normals to replicate his perception that the true distribution is bimodal.
[
x_i sim text{Normal-Mixture}([omega, 1-omega], [m_1, m_2], [s_1^2, s_2^2]) textual content{ i.i.d. for all } i
m_j sim textual content{Regular}(2.3, 0.5^2) textual content{ for } j = 1,2
s_j^2 sim textual content{Inverse-Gamma}(1,1) textual content{ for } j=1,2
omega sim textual content{Beta}(1,1)
]
After the information arrives, every economist runs an MCMC algorithm to estimate their desired posteriors, which will probably be a mirrored image of actuality (1) if their assumptions are true and (2) if the sampler labored accurately. The primary if, a dialogue about assumptions, shall be left to the economists. Nevertheless, how can they know whether or not the second if holds? In different phrases, how can they make sure that the sampler labored accurately and, as a consequence, their posterior estimations are unbiased?
Sampler Diagnostics
To judge a sampler’s efficiency, we are able to discover a small set of metrics that replicate totally different components of the estimation course of.
Quantitative Metrics
R-hat (Potential Scale Discount Issue)
In easy phrases, R-hat evaluates whether or not bees that began at totally different locations have all explored the identical rooms on the finish of the day. To estimate the posterior, an MCMC algorithm makes use of a number of chains (or bees) that begin at random places. R-hat is the metric we use to evaluate the convergence of the chains. It measures whether or not a number of MCMC chains have blended effectively (i.e., if they’ve sampled the identical topography) by evaluating the variance of samples inside every chain to the variance of the pattern means throughout chains. Intuitively, because of this
[
hat{R} = sqrt{frac{text{Variance Between Chains}}{text{Variance Within Chains}}}
]
If R-hat is near 1.0 (or under 1.01), it signifies that the variance inside every chain is similar to the variance between chains, suggesting that they’ve converged to the identical distribution. In different phrases, the chains are behaving equally and are additionally indistinguishable from each other. That is exactly what we see after sampling the posterior of the primary mannequin, proven within the final column of the desk under:
The r-hat from the second mannequin, nonetheless, tells a unique story. The very fact now we have such giant r-hat values signifies that, on the finish of the sampling course of, the totally different chains had not converged but. In observe, because of this the distribution they explored and returned was totally different, or that every bee created a map of a unique room of the home. This basically leaves us with out a clue of how the items join or what the whole flooring plan appears like.
Given our R-hat readouts have been giant, we all know one thing went fallacious with the sampling course of within the second mannequin. Nevertheless, even when the R-hat had turned out inside acceptable ranges, this doesn’t give us certainty that the sampling course of labored. R-hat is only a diagnostic device, not a assure. Generally, even when your R-hat readout is decrease than 1.01, the sampler won’t have correctly explored the total posterior. This occurs when a number of bees begin their exploration in the identical room and stay there. Likewise, if you happen to’re utilizing a small variety of chains, and in case your posterior occurs to be multimodal, there’s a chance that each one chains began in the identical mode and did not discover different peaks.
The R-hat readout displays convergence, not completion. With a purpose to have a extra complete concept, we have to verify different diagnostic metrics as effectively.
Efficient Pattern Measurement (ESS)
When explaining what MCMC was, we talked about that “Monte Carlo” refers to the truth that the following state is chosen randomly. This doesn’t essentially imply that the states are totally impartial. Although the bees select their subsequent step at random, these steps are nonetheless correlated to some extent. If a bee is exploring a front room at time t=0, it should most likely nonetheless be in the lounge at time t=1, despite the fact that it’s in a unique a part of the identical room. On account of this pure connection between samples, we are saying these two knowledge factors are autocorrelated.
On account of their nature, MCMC strategies inherently produce autocorrelated samples, which complicates statistical evaluation and requires cautious analysis. In statistical inference, we regularly assume impartial samples to make sure that the estimates of uncertainty are correct, therefore the necessity for uncorrelated samples. If two knowledge factors are too related to one another, the correlation reduces their efficient info content material. Mathematically, the components under represents the autocorrelation operate between two time factors (t1 and t2) in a random course of:
[
R_{XX}(t_1, t_2) = E[X_{t_1} overline{X_{t_2}}]
]
the place E is the anticipated worth operator and X-bar is the advanced conjugate. In MCMC sampling, that is essential as a result of excessive autocorrelation signifies that new samples don’t educate us something totally different from the previous ones, successfully lowering the pattern measurement now we have. Unsurprisingly, the metric that displays that is known as Efficient Pattern Measurement (ESS), and it helps us decide what number of really impartial samples now we have.
As hinted beforehand, the efficient pattern measurement accounts for autocorrelation by estimating what number of really impartial samples would offer the identical info because the autocorrelated samples now we have. Mathematically, for a parameter θ, the ESS is outlined as:
[
ESS = frac{n}{1 + 2 sum_{k=1}^{infty} rho(theta)_k}
]
the place n is the whole variety of samples and ρ(θ)ok is the autocorrelation at lag ok for parameter θ.
Sometimes, for ESS readouts, the upper, the higher. That is what we see within the readout for the primary mannequin. Two frequent ESS variations are Bulk-ESS, which assesses mixing within the central a part of the distribution, and Tail-ESS, which focuses on the effectivity of sampling the distribution’s tails. Each inform us if our mannequin precisely displays the central tendency and credible intervals.
In distinction, the readouts for the second mannequin are very unhealthy. Sometimes, we need to see readouts which can be at the very least 1/10 of the whole pattern measurement. On this case, given every chain sampled 2000 observations, we must always anticipate ESS readouts of at the very least 800 (from the whole measurement of 8000 samples throughout 4 chains of 2000 samples every), which isn’t what we observe.
Visible Diagnostics
Other than the numerical metrics, our understanding of sampler efficiency could be deepened via the usage of diagnostic plots. The primary ones are rank plots, hint plots, and pair plots.
Rank Plots
A rank plot helps us determine whether or not the totally different chains have explored all the posterior distribution. If we as soon as once more consider the bee analogy, rank plots inform us which bees explored which components of the home. Subsequently, to judge whether or not the posterior was explored equally by all chains, we observe the form of the rank plots produced by the sampler. Ideally, we wish the distribution of all chains to look roughly uniform, like within the rank plots generated after sampling the primary mannequin. Every colour under represents a sequence (or bee):
Underneath the hood, a rank plot is produced with a easy sequence of steps. First, we run the sampler and let it pattern from the posterior of every parameter. In our case, we’re sampling posteriors for parameters m and s of the primary mannequin. Then, parameter by parameter, we get all samples from all chains, put them collectively, and get them organized from smallest to largest. We then ask ourselves, for every pattern, what was the chain the place it got here from? This may permit us to create plots like those we see above.
In distinction, unhealthy rank plots are simple to identify. Not like the earlier instance, the distributions from the second mannequin, proven under, are usually not uniform. From the plots, what we interpret is that every chain, after starting at totally different random places, obtained caught in a area and didn’t discover everything of the posterior. Consequently, we can not make inferences from the outcomes, as they’re unreliable and never consultant of the true posterior distribution. This might be equal to having 4 bees that began at totally different rooms of the home and obtained caught someplace throughout their exploration, by no means overlaying everything of the property.
KDE and Hint Plots
Just like R-hat, hint plots assist us assess the convergence of MCMC samples by visualizing how the algorithm explores the parameter house over time. PyMC offers two kinds of hint plots to diagnose mixing points: Kernel Density Estimate (KDE) plots and iteration-based hint plots. Every of those serves a definite function in evaluating whether or not the sampler has correctly explored the goal distribution.
The KDE plot (often on the left) estimates the posterior density for every chain, the place every line represents a separate chain. This enables us to verify whether or not all chains have converged to the identical distribution. If the KDEs overlap, it means that the chains are sampling from the identical posterior and that mixing has occurred. However, the hint plot (often on the correct) visualizes how parameter values change over MCMC iterations (steps), with every line representing a unique chain. A well-mixed sampler will produce hint plots that look noisy and random, with no clear construction or separation between chains.
Utilizing the bee analogy, hint plots could be considered snapshots of the “options” of the home at totally different places. If the sampler is working accurately, the KDEs within the left plot ought to align intently, displaying that each one bees (chains) have explored the home equally. In the meantime, the correct plot ought to present extremely variable traces that mix collectively, confirming that the chains are actively transferring via the house reasonably than getting caught in particular areas.
Nevertheless, in case your sampler has poor mixing or convergence points, you will note one thing just like the determine under. On this case, the KDEs is not going to overlap, that means that totally different chains have sampled from totally different distributions reasonably than a shared posterior. The hint plot will even present structured patterns as an alternative of random noise, indicating that chains are caught in several areas of the parameter house and failing to totally discover it.
Through the use of hint plots alongside the opposite diagnostics, you’ll be able to determine sampling points and decide whether or not your MCMC algorithm is successfully exploring the posterior distribution.
Pair Plots
A 3rd sort of plot that’s usually helpful for diagnostic are pair plots. In fashions the place we need to estimate the posterior distribution of a number of parameters, pair plots permit us to look at how totally different parameters are correlated. To know how such plots are fashioned, suppose once more concerning the bee analogy. If you happen to think about that we’ll create a plot with the width and size of the home, every “step” that the bees take could be represented by an (x, y) mixture. Likewise, every parameter of the posterior is represented as a dimension, and we create scatter plots displaying the place the sampler walked utilizing parameter values as coordinates. Right here, we’re plotting every distinctive pair (x, y), ensuing within the scatter plot you see in the midst of the picture under. The one-dimensional plots you see on the sides are the marginal distributions over every parameter, giving us further info on the sampler’s conduct when exploring them.
Check out the pair plot from the primary mannequin.
Every axis represents one of many two parameters whose posteriors we’re estimating. For now, let’s give attention to the scatter plot within the center, which reveals the parameter mixtures sampled from the posterior. The very fact now we have a really even distribution signifies that, for any specific worth of m, there was a variety of values of s that have been equally more likely to be sampled. Moreover, we don’t see any correlation between the 2 parameters, which is often good! There are instances after we would anticipate some correlation, reminiscent of when our mannequin entails a regression line. Nevertheless, on this occasion, now we have no cause to consider two parameters ought to be extremely correlated, so the actual fact we don’t observe uncommon conduct is constructive information.
Now, check out the pair plots from the second mannequin.
Provided that this mannequin has 5 parameters to be estimated, we naturally have a better variety of plots since we’re analyzing them pair-wise. Nevertheless, they give the impression of being odd in comparison with the earlier instance. Specifically, reasonably than having a good distribution of factors, the samples right here both appear to be divided throughout two areas or appear considerably correlated. That is one other means of visualizing what the rank plots have proven: the sampler didn’t discover the total posterior distribution. Under we remoted the highest left plot, which incorporates the samples from m0 and m1. Not like the plot from mannequin 1, right here we see that the worth of 1 parameter significantly influences the worth of the opposite. If we sampled m1 round 2.5, for instance, m0 is more likely to be sampled from a really slim vary round 1.5.
Sure shapes could be noticed in problematic pair plots comparatively continuously. Diagonal patterns, for instance, point out a excessive correlation between parameters. Banana shapes are sometimes related to parametrization points, usually being current in fashions with tight priors or constrained parameters. Funnel shapes would possibly point out hierarchical fashions with unhealthy geometry. When now we have two separate islands, like within the plot above, this will point out that the posterior is bimodal AND that the chains haven’t blended effectively. Nevertheless, remember that these shapes would possibly point out issues, however not essentially achieve this. It’s as much as the information scientist to look at the mannequin and decide which behaviors are anticipated and which of them are usually not!
Some Fixing Methods
When your diagnostics point out sampling issues — whether or not regarding R-hat values, low ESS, uncommon rank plots, separated hint plots, or unusual parameter correlations in pair plots — a number of methods can assist you deal with the underlying points. Sampling issues usually stem from the goal posterior being too advanced for the sampler to discover effectively. Advanced goal distributions might need:
- A number of modes (peaks) that the sampler struggles to maneuver between
- Irregular shapes with slim “corridors” connecting totally different areas
- Areas of drastically totally different scales (just like the “neck” of a funnel)
- Heavy tails which can be tough to pattern precisely
Within the bee analogy, these complexities symbolize homes with uncommon flooring plans — disconnected rooms, extraordinarily slim hallways, or areas that change dramatically in measurement. Simply as bees would possibly get trapped in particular areas of such homes, MCMC chains can get caught in sure areas of the posterior.
To assist the sampler in its exploration, there are easy methods we are able to use.
Technique 1: Reparameterization
Reparameterization is especially efficient for hierarchical fashions and distributions with difficult geometries. It entails remodeling your mannequin’s parameters to make them simpler to pattern. Again to the bee analogy, think about the bees are exploring a home with a peculiar structure: a spacious front room that connects to the kitchen via a really, very slim hallway. One facet we hadn’t talked about earlier than is that the bees need to fly in the identical means via the complete home. That signifies that if we dictate the bees ought to use giant “steps,” they may discover the lounge very effectively however hit the partitions within the hallway head-on. Likewise, if their steps are small, they may discover the slim hallway effectively, however take without end to cowl the complete front room. The distinction in scales, which is pure to the home, makes the bees’ job harder.
A basic instance that represents this situation is Neal’s funnel, the place the dimensions of 1 parameter is determined by one other:
[
p(y, x) = text{Normal}(y|0, 3) times prod_{n=1}^{9} text{Normal}(x_n|0, e^{y/2})
]
We are able to see that the dimensions of x relies on the worth of y. To repair this drawback, we are able to separate x and y as impartial commonplace Normals after which remodel these variables into the specified funnel distribution. As an alternative of sampling immediately like this:
[
begin{align*}
y &sim text{Normal}(0, 3)
x &sim text{Normal}(0, e^{y/2})
end{align*}
]
You may reparameterize to pattern from commonplace Normals first:
[
y_{raw} sim text{Standard Normal}(0, 1)
x_{raw} sim text{Standard Normal}(0, 1)
y = 3y_{raw}
x = e^{y/2} x_{raw}
]
This method separates the hierarchical parameters and makes sampling extra environment friendly by eliminating the dependency between them.
Reparameterization is like redesigning the home such that as an alternative of forcing the bees to discover a single slim hallway, we create a brand new structure the place all passages have related widths. This helps the bees use a constant flying sample all through their exploration.
Technique 2: Dealing with Heavy-tailed Distributions
Heavy-tailed distributions like Cauchy and Pupil-T current challenges for samplers and the best step measurement. Their tails require bigger step sizes than their central areas (just like very lengthy hallways that require the bees to journey lengthy distances), which creates a problem:
- Small step sizes result in inefficient sampling within the tails
- Massive step sizes trigger too many rejections within the middle
Reparameterization options embrace:
- For Cauchy: Defining the variable as a change of a Uniform distribution utilizing the Cauchy inverse CDF
- For Pupil-T: Utilizing a Gamma-Combination illustration
Technique 3: Hyperparameter Tuning
Generally the answer lies in adjusting the sampler’s hyperparameters:
- Improve whole iterations: The only method — give the sampler extra time to discover.
- Improve goal acceptance fee (adapt_delta): Scale back divergent transitions (attempt 0.9 as an alternative of the default 0.8 for advanced fashions, for instance).
- Improve max_treedepth: Permit the sampler to take extra steps per iteration.
- Prolong warmup/adaptation section: Give the sampler extra time to adapt to the posterior geometry.
Do not forget that whereas these changes might enhance your diagnostic metrics, they usually deal with signs reasonably than underlying causes. The earlier methods (reparameterization and higher proposal distributions) usually supply extra elementary options.
Technique 4: Higher Proposal Distributions
This resolution is for operate becoming processes, reasonably than sampling estimations of the posterior. It mainly asks the query: “I’m presently right here on this panorama. The place ought to I leap to subsequent in order that I discover the total panorama, or how do I do know that the following leap is the leap I ought to make?” Thus, selecting a superb distribution means ensuring that the sampling course of explores the total parameter house as an alternative of only a particular area. proposal distribution ought to:
- Have substantial chance mass the place the goal distribution does.
- Permit the sampler to make jumps of the suitable measurement.
One frequent alternative of the proposal distribution is the Gaussian (Regular) distribution with imply μ and commonplace deviation σ — the dimensions of the distribution that we are able to tune to determine how far to leap from the present place to the following place. If we select the dimensions for the proposal distribution to be too small, it’d both take too lengthy to discover the complete posterior or it should get caught in a area and by no means discover the total distribution. But when the dimensions is simply too giant, you would possibly by no means get to discover some areas, leaping over them. It’s like enjoying ping-pong the place we solely attain the 2 edges however not the center.
Enhance Prior Specification
When all else fails, rethink your mannequin’s prior specs. Obscure or weakly informative priors (like uniformly distributed priors) can typically result in sampling difficulties. Extra informative priors, when justified by area data, can assist information the sampler towards extra affordable areas of the parameter house. Generally, regardless of your greatest efforts, a mannequin might stay difficult to pattern successfully. In such instances, take into account whether or not an easier mannequin would possibly obtain related inferential targets whereas being extra computationally tractable. The most effective mannequin is commonly not probably the most advanced one, however the one which balances complexity with reliability. The desk under reveals the abstract of fixing methods for various points.
| Diagnostic Sign | Potential Challenge | Advisable Repair |
| Excessive R-hat | Poor mixing between chains | Improve iterations, alter the step measurement |
| Low ESS | Excessive autocorrelation | Reparameterization, improve adapt_delta |
| Non-uniform rank plots | Chains caught in several areas | Higher proposal distribution, begin with a number of chains |
| Separated KDEs in hint plots | Chains exploring totally different distributions | Reparameterization |
| Funnel shapes in pair plots | Hierarchical mannequin points | Non-centered reparameterization |
| Disjoint clusters in pair plots | Multimodality with poor mixing | Adjusted distribution, simulated annealing |
Conclusion
Assessing the standard of MCMC sampling is essential for making certain dependable inference. On this article, we explored key diagnostic metrics reminiscent of R-hat, ESS, rank plots, hint plots, and pair plots, discussing how every helps decide whether or not the sampler is performing correctly.
If there’s one takeaway we wish you to bear in mind it’s that it is best to all the time run diagnostics earlier than drawing conclusions out of your samples. No single metric offers a definitive reply — every serves as a device that highlights potential points reasonably than proving convergence. When issues come up, methods reminiscent of reparameterization, hyperparameter tuning, and prior specification can assist enhance sampling effectivity.
By combining these diagnostics with considerate modeling choices, you’ll be able to guarantee a extra strong evaluation, lowering the chance of deceptive inferences resulting from poor sampling conduct.
References
B. Gilbert, Bob’s bees: the importance of using multiple bees (chains) to judge MCMC convergence (2018), Youtube
Chi-Feng, MCMC demo (n.d.), GitHub
D. Simpson, Maybe it’s time to let the old ways die; or We broke R-hat so now we have to fix it. (2019), Statistical Modeling, Causal Inference, and Social Science
M. Taboga, Markov Chain Monte Carlo (MCMC) methods (2021), Lectures on chance concept and mathematical Statistics. Kindle Direct Publishing.
T. Wiecki, MCMC Sampling for Dummies (2024), twecki.io
Stan Person’s Information, Reparametrization (n.d.), Stan Documentation
