is co-authored by Felipe Bandeira, Giselle Fretta, Thu Than, and Elbion Redenica. We additionally thank Prof. Carl Scheffler for his help.
Introduction
Parameter estimation has been for many years one of the vital subjects in statistics. Whereas frequentist approaches, equivalent to Most Chance Estimations, was the gold normal, the advance of computation has opened area for Bayesian strategies. Estimating posterior distributions with Mcmc samplers turned 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. Retaining in thoughts what Lewis Caroll as soon as wrote: “For those who don’t know the place you’re going, any highway will take you there.”
This text is supposed to assist knowledge scientists consider an typically neglected side 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 stage 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 strong parameter estimation strategies. Whether or not the target is to make inferences or conduct simulations, having the capability to mannequin the info technology course of is a vital a part of the method. For a very long time, the estimations had been primarily carried out utilizing frequentist instruments, equivalent to Most Chance Estimations (MLE) and even the well-known Least Squares optimization utilized in regressions. But, frequentist strategies have clear shortcomings, equivalent to the truth that they’re centered on level estimates and don’t incorporate prior data that might enhance estimates.
As an alternative choice to these instruments, Bayesian strategies have gained reputation 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 info and by the prior data researchers held. Initially, Bayesian parameter estimation was executed by means of an tailored model of Bayes’ theorem centered on unknown parameters (represented as θ) and recognized knowledge factors (represented as x). We will outline P(θ|x), the posterior distribution of a parameter’s worth given the info, as:
[ P(theta|x) = fractheta) P(theta){P(x)} ]
On this formulation, P(x|θ) is the probability of the info 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 attainable values of the prior:
[ P(x) = int_theta P(x, theta) dtheta ]
In some instances, as a result of complexity of the calculations required, deriving the posterior distribution analytically was not attainable. Nevertheless, with the advance of computation, working sampling algorithms (particularly MCMC ones) to estimate posterior distributions has change into simpler, giving researchers a robust software for conditions the place analytical posteriors usually are not trivial to search out. But, with such energy additionally comes a considerable amount of duty to make sure that outcomes make sense. That is the place sampler diagnostics are available, providing a set of helpful instruments to gauge 1) whether or not an MCMC algorithm is working nicely and, consequently, 2) whether or not the estimated distribution we see is an correct illustration of the true 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 will 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?
Certainly one of our favourite analogies comes from Ben Gilbert. Suppose that the unknown area is definitely a home whose floorplan we want to map. For some purpose, we can not straight go to the home, however we will ship bees inside with GPS gadgets hooked up to them. If every part works as anticipated, the bees will fly round the home, and utilizing their trajectories, we will estimate what the ground plan appears to be like like. On this analogy, the ground plan is the posterior distribution, and the sampler is the group of bees flying round the home.
The rationale we’re writing this text is that, in some instances, the bees received’t fly as anticipated. In the event that they get caught in a sure room for some purpose (as a result of somebody dropped sugar on the ground, for instance), the info they return received’t be consultant of your entire home; somewhat than visiting all rooms, the bees solely visited just a few, and our image of what the home appears to be like like will in the end 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 subsequent state solely depends upon the present one (or that the bee’s subsequent location is barely influenced by its present place, and never by the entire locations the place it has been earlier than). Monte Carlo implies that the subsequent state is chosen randomly. MCMC strategies like Metropolis-Hastings, Gibbs sampling, Hamiltonian Monte Carlo (HMC), and No-U-Flip Sampler (NUTS) all function by developing Markov Chains (a sequence of steps) which can be near random and progressively 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 grasp extra about public annual spending on healthcare by mayors of cities with lower than 1 million inhabitants. Quite than sheer frequencies, he needs to grasp 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 every 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 understand 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 finances constraints, there are two sorts of cities: those that do their finest to spend little or no and those that aren’t afraid of spending rather a lot. 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 info arrives, every economist runs an MCMC algorithm to estimate their desired posteriors, which might 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 ensure that the sampler labored accurately and, as a consequence, their posterior estimations are unbiased?
Sampler Diagnostics
To judge a sampler’s efficiency, we will discover a small set of metrics that replicate completely 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 completely 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 nicely (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 implies 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, nevertheless, tells a distinct story. The actual fact we’ve such massive r-hat values signifies that, on the finish of the sampling course of, the completely different chains had not converged but. In apply, because of this the distribution they explored and returned was completely different, or that every bee created a map of a distinct room of the home. This basically leaves us with out a clue of how the items join or what the entire ground plan appears to be like like.
Given our R-hat readouts had been massive, 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 software, not a assure. Generally, even when your R-hat readout is decrease than 1.01, the sampler may not have correctly explored the complete 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 likelihood that every one chains began in the identical mode and didn’t discover different peaks.
The R-hat readout displays convergence, not completion. In an effort to have a extra complete concept, we have to verify different diagnostic metrics as nicely.
Efficient Pattern Measurement (ESS)
When explaining what MCMC was, we talked about that “Monte Carlo” refers to the truth that the subsequent state is chosen randomly. This doesn’t essentially imply that the states are totally impartial. Regardless that 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 can most likely nonetheless be in the lounge at time t=1, though it’s in a distinct 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 frequently 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 comparable to one another, the correlation reduces their efficient data content material. Mathematically, the formulation under represents the autocorrelation perform 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 implies that new samples don’t educate us something completely different from the previous ones, successfully decreasing the pattern measurement we’ve. Unsurprisingly, the metric that displays that is known as Efficient Pattern Measurement (ESS), and it helps us decide what number of actually impartial samples we’ve.
As hinted beforehand, the efficient pattern measurement accounts for autocorrelation by estimating what number of actually impartial samples would offer the identical data because the autocorrelated samples we’ve. 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 full variety of samples and ρ(θ)okay is the autocorrelation at lag okay 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 dangerous. Sometimes, we need to see readouts which can be at the very least 1/10 of the full pattern measurement. On this case, given every chain sampled 2000 observations, we should always count on ESS readouts of at the very least 800 (from the full 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 will be deepened by means of using diagnostic plots. The principle ones are rank plots, hint plots, and pair plots.
Rank Plots
A rank plot helps us determine whether or not the completely different chains have explored the entire 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 guage 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 would like the distribution of all chains to look roughly uniform, like within the rank plots generated after sampling the primary mannequin. Every coloration under represents a series (or bee):
Below 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, dangerous rank plots are simple to identify. In contrast to the earlier instance, the distributions from the second mannequin, proven under, usually are not uniform. From the plots, what we interpret is that every chain, after starting at completely different random places, acquired caught in a area and didn’t discover the whole lot 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 may be equal to having 4 bees that began at completely different rooms of the home and acquired caught someplace throughout their exploration, by no means overlaying the whole lot of the property.
KDE and Hint Plots
Much like R-hat, hint plots assist us assess the convergence of MCMC samples by visualizing how the algorithm explores the parameter area over time. PyMC gives 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 goal in evaluating whether or not the sampler has correctly explored the goal distribution.
The KDE plot (normally 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. Alternatively, the hint plot (normally on the precise) visualizes how parameter values change over MCMC iterations (steps), with every line representing a distinct 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 will be considered snapshots of the “options” of the home at completely different places. If the sampler is working accurately, the KDEs within the left plot ought to align intently, exhibiting that every one bees (chains) have explored the home equally. In the meantime, the precise plot ought to present extremely variable traces that mix collectively, confirming that the chains are actively transferring by means of the area somewhat 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 won’t overlap, which means that completely different chains have sampled from completely different distributions somewhat than a shared posterior. The hint plot may also present structured patterns as an alternative of random noise, indicating that chains are caught in several areas of the parameter area and failing to completely discover it.
Through the use of hint plots alongside the opposite diagnostics, you may 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 typically 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 completely different parameters are correlated. To grasp how such plots are shaped, suppose once more concerning the bee analogy. For those who think about that we’ll create a plot with the width and size of the home, every “step” that the bees take will be represented by an (x, y) mixture. Likewise, every parameter of the posterior is represented as a dimension, and we create scatter plots exhibiting 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 course of the picture under. The one-dimensional plots you see on the perimeters are the marginal distributions over every parameter, giving us further data on the sampler’s habits 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 exhibits the parameter mixtures sampled from the posterior. The actual fact we’ve a really even distribution implies that, for any specific worth of m, there was a variety of values of s that had been equally more likely to be sampled. Moreover, we don’t see any correlation between the 2 parameters, which is normally good! There are instances once we would count on some correlation, equivalent to when our mannequin includes a regression line. Nevertheless, on this occasion, we’ve no purpose to imagine two parameters must be extremely correlated, so the actual fact we don’t observe uncommon habits is constructive information.
Now, check out the pair plots from the second mannequin.
On condition that this mannequin has 5 parameters to be estimated, we naturally have a larger variety of plots since we’re analyzing them pair-wise. Nevertheless, they appear odd in comparison with the earlier instance. Specifically, somewhat than having an excellent distribution of factors, the samples right here both appear to be divided throughout two areas or appear considerably correlated. That is one other method of visualizing what the rank plots have proven: the sampler didn’t discover the complete posterior distribution. Beneath we remoted the highest left plot, which comprises the samples from m0 and m1. In contrast to the plot from mannequin 1, right here we see that the worth of 1 parameter tremendously 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 will be noticed in problematic pair plots comparatively often. Diagonal patterns, for instance, point out a excessive correlation between parameters. Banana shapes are sometimes linked to parametrization points, typically being current in fashions with tight priors or constrained parameters. Funnel shapes would possibly point out hierarchical fashions with dangerous geometry. When we’ve two separate islands, like within the plot above, this will point out that the posterior is bimodal AND that the chains haven’t blended nicely. Nevertheless, understand that these shapes would possibly point out issues, however not essentially accomplish that. It’s as much as the info scientist to look at the mannequin and decide which behaviors are anticipated and which of them usually are 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 will help you tackle 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 completely different areas
- Areas of drastically completely different scales (just like the “neck” of a funnel)
- Heavy tails which can be tough to pattern precisely
Within the bee analogy, these complexities signify homes with uncommon ground 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 will use.
Technique 1: Reparameterization
Reparameterization is especially efficient for hierarchical fashions and distributions with difficult geometries. It includes 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 by means of a really, very slim hallway. One side we hadn’t talked about earlier than is that the bees need to fly in the identical method by means of your entire home. That implies that if we dictate the bees ought to use massive “steps,” they’ll discover the lounge very nicely however hit the partitions within the hallway head-on. Likewise, if their steps are small, they’ll discover the slim hallway nicely, however take endlessly to cowl your entire 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 depends upon one other:
[
p(y, x) = text{Normal}(y|0, 3) times prod_{n=1}^{9} text{Normal}(x_n|0, e^{y/2})
]
We will see that the dimensions of x relies on the worth of y. To repair this downside, we will separate x and y as impartial normal Normals after which remodel these variables into the specified funnel distribution. As an alternative of sampling straight like this:
[
begin{align*}
y &sim text{Normal}(0, 3)
x &sim text{Normal}(0, e^{y/2})
end{align*}
]
You’ll be able to reparameterize to pattern from normal 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 system 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 comparable 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 perfect 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
- Giant step sizes trigger too many rejections within the heart
Reparameterization options embrace:
- For Cauchy: Defining the variable as a metamorphosis 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 complete iterations: The only method — give the sampler extra time to discover.
- Improve goal acceptance price (adapt_delta): Scale back divergent transitions (strive 0.9 as an alternative of the default 0.8 for advanced fashions, for instance).
- Improve max_treedepth: Enable the sampler to take extra steps per iteration.
- Prolong warmup/adaptation section: Give the sampler extra time to adapt to the posterior geometry.
Keep in mind that whereas these changes might enhance your diagnostic metrics, they typically deal with signs somewhat than underlying causes. The earlier methods (reparameterization and higher proposal distributions) usually supply extra basic options.
Technique 4: Higher Proposal Distributions
This resolution is for perform becoming processes, somewhat than sampling estimations of the posterior. It principally asks the query: “I’m presently right here on this panorama. The place ought to I soar to subsequent in order that I discover the complete panorama, or how do I do know that the subsequent soar is the soar I ought to make?” Thus, selecting a very good distribution means ensuring that the sampling course of explores the complete parameter area as an alternative of only a particular area. An excellent proposal distribution ought to:
- Have substantial likelihood mass the place the goal distribution does.
- Enable 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 normal deviation σ — the dimensions of the distribution that we will tune to determine how far to leap from the present place to the subsequent place. If we select the dimensions for the proposal distribution to be too small, it’d both take too lengthy to discover your entire posterior or it can get caught in a area and by no means discover the complete distribution. But when the dimensions is just too massive, you would possibly by no means get to discover some areas, leaping over them. It’s like taking part in 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 generally result in sampling difficulties. Extra informative priors, when justified by area data, will help information the sampler towards extra affordable areas of the parameter area. Generally, regardless of your finest efforts, a mannequin might stay difficult to pattern successfully. In such instances, take into account whether or not a less complicated mannequin would possibly obtain comparable inferential objectives whereas being extra computationally tractable. One of the best mannequin is commonly not essentially the most advanced one, however the one which balances complexity with reliability. The desk under exhibits the abstract of fixing methods for various points.
| Diagnostic Sign | Potential Difficulty | Advisable Repair |
| Excessive R-hat | Poor mixing between chains | Improve iterations, modify 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 completely 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 guaranteeing dependable inference. On this article, we explored key diagnostic metrics equivalent to 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 would like you to remember it’s that it’s best to at all times run diagnostics earlier than drawing conclusions out of your samples. No single metric gives a definitive reply — every serves as a software that highlights potential points somewhat than proving convergence. When issues come up, methods equivalent to reparameterization, hyperparameter tuning, and prior specification will help enhance sampling effectivity.
By combining these diagnostics with considerate modeling choices, you may guarantee a extra strong evaluation, decreasing the danger of deceptive inferences on account of poor sampling habits.
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 likelihood idea and mathematical Statistics. Kindle Direct Publishing.
T. Wiecki, MCMC Sampling for Dummies (2024), twecki.io
Stan Consumer’s Information, Reparametrization (n.d.), Stan Documentation
