As We May Dream – Think Probabilistic

A state of consciousness epitomized by a withdrawal of all bodily sensory functions, a state of deep-felt alertness; reminiscent of wielding the sharpest of swords and yet feeling as tranquil as an eternally unperturbed ocean. The wheel of time first spun backward, and then forward, until it was impossible to decipher which direction it spun anymore. I strayed beyond all concepts and precepts of time and space, thrust into a dimension of the pristine in what seemed like a jettison of immutable wholesomeness and profound compassion. There are many journeys in one’s life. Most of them are borne of the necessities, vicissitudes, and vagaries of being a life form on this planet. And then there the inimitable ones, when it becomes manifestly evident that the place one is being transported to is an incontrovertible clarion call. This was one such journey. I was being summoned, traveling with the uncertainty of not knowing where I was going but with the certainty that there was boundless energy in the destination. A mysterious experience of being both firmly grounded and escaping gravity at the same time.

The first rays of light soon appeared, and I found myself standing at the edge of a river surrounded by what looked like the preceding hours of dawn. An oil lamp illuminated a statute at the edge of the river. A statue of a frog laying eggs, its face writhing in excruciating pain, yet being protected by a fierce snake with a sizeable hood. What predator likes to preserve its natural prey, replacing the natural urge to maim with the higher force of compassion? It was in that instant that I knew where I stood. This was Sringeri, the first of the nondual monasteries established by Śankara. Somewhere deep inside of me, I felt as if the cottonwick of an oil lamp was kindled, as I noticed a pencil of light rays emerging from the windows of the main temple on a plateau, reachable by a flight of stairs. I climbed the stairs and began to walk towards the main temple, like a baby drawn towards pure maternal energy. Incense escaped from the imposing high windows, as I approached the main entrance and its multifaceted adorned doors, impregnated by Vēdic inscriptions and a pair of consummate swans.

I walked through an antechamber and into the main temple hall. No sooner had I stepped into the main fall than I felt the radiance of a kind that could only be made of ageless and irreducible energy. There were many people seated on the floor, surrounded by what seemed like many soft oil lamps. I surveyed the room to recognize many loved ones. What were they all doing here? And before a flicker of a thought, my presence in the main hall seemed to be noticed. There was mirth of collective laughter. Subbulakshmi, who was seated next to her singing companion Radha Vishwanathan and instrumental accompanists, spoke in a glowing and affectionate voice. “My child, you finally arrived! Do come here and take your seat, we were just about to begin”, she said, pointing to a mat on which there was a weaving of a peacock. I sat upon this peacock mat, facing Subbulakshmi and the entire gathering, letting go of the urge to make sense of it all and surrendering to sapient energy that characterized my newfound world.

One accompanist began playing the tanpura, the harmonic bourdon or drone symbolizing the jīva or life in carnātic music, as Subbulakshmi and Radha started to sing the dakshinamūrthy stotram followed by the kriti śri mahāganapati ravatu mām in gowla rāgam, set to ādi tālam rhythm. Listening to Subbulakshmi in person, being a witness to her sublime state of mind, opulent with devotion and spirit, her absolute control and total mastery of tone, pitch & timbre — spoke of a pristine language. A language of primordial cosmic vibrations representing the sound heard before the origin of time. Each swara adorned and embellished with aesthetic emotion, landing on the ears as if stamped with the fingerprints of the infinite, formless and timeless from an abode of eternal compassion.

I. Kalpanaswaram and Infinite Exchangeability

“My child, I think you have a question for me”, said Subbulakshmi after delivering that song. “Amma, the just as the self-luminous energy of the sun is a secret wrapped in a mystery, so is the power that sustains the kalpanaswaram (creative improvisation and structured permutations of solfege notes) in your singing. What is the secret of kalpanaswaram in your singing?”, I asked. “Of course of my child”, replied Subbulakshmi smilingly.

“The kalpanaswaram is like a ladder. In the first steps of the ladder, it is important to understand that kalpanaswaram is a type of Manodharma. Manodharma means creative improvisation, but improvisation within the bounds of a set of rules. As you know, each rāgam has its own grammar of swaras: the ascending sequence or ārohana and the descending scale or avarõhana. Each song, sung in a rāgam must obey the rāgam’s grammar. It must also obey the rules of rhythm. For example in ādi tālam, we have a system of rotating eight beats per cycle, and each beat can carry a swara. Many a time, each beat will carry more than one swara. In kalpanaswaram, the singer selects a position in the song for singing a sequence of swaras that obeys the grammar of the rāgam and the tālam. For example, if the artist starts singing on the 5th beat of a tālam cycle, she must end the improvised sequence at the exact 5th beat of another subsequent cycle. The difficulty of adhering to the rāgam and tālam cycle increases as one selects positions in the tālam cycle near the very beginning and the end. Since each beat can have no swara, one swara or more than one swara, the singer has to construct a sequence that is not only improvised on the spot, but also has to strictly obey the set of rules.

It is easy to obey the rules of the grammar if the improvisation is easy. True mastery is achieved by rigorous practice of increasing levels of difficulty. My child, but I have only explained to you what you already know. Now listen carefully.

Kalpanaswaram is first and foremost a disciplined practice of cultivating the spirit of the rāgam. It has as a prerequisite the inculcation of bhakti bhāva or a mindset of fervent and unshakeable devotion to the ascription of the dharma (rules). The singer has to let the rāgam seep into their very being and understand the gravitas of each swara in the rāgam. The singer must assign great importance to the promotion and sustenance of aesthetic emotion in their daily life, for the swara dēvathas (divine forces) in the rāgam lend easily to the mind that understands their sheer power. Moving the ladder of difficulty in kalpanaswaram must parallel the fortitude of sustaining bhakti bhāva even amidst the difficulties of life. And so climbs the singer, step by step, ascending the ladder of kalpanaswaram and by so doing, gains the depth of mastery and strength that accompany it. Invariance of the disciplined practice to the vicissitudes of life, forged with the right mindset and catapulted by devotion is the means to ascending the ladder of kalpanaswaram. This is one of the many secrets of the kalpanaswaram. Now I want you to describe what I said in a language more familiar to you.”

The transmission of this knowledge triggered a synesthetic analogy, and it felt as if Subbulakshmi was summoning a response from me which even I didn’t know existed inside of me.

“Amma, in Bayesian statistics, we have the principle of infinite exchangeability of random variables. A sequence of independent and identically distributed random variables is said to be inifinitely exchangeable if the joint probability of the sequence is unchanged by any permutation. So if we imagine each swara to be a random variable, independent of each other and identical in their importance, any joint distribution of random variables sequences is the same irrespective of any permutation! This is akin to the importance of each swara in a rāgam.

This can be explained by Bruno de Finetti\’s theorem! Given a sequence of swaras as random variables $x_{1}, x_{2},.., x_{n}$ , if these swaras are infinitely exchangeable, their joint distribution $p(x_{1}, x_{2},...,x_{n})$ is invariant to permutation! That means
that:

$p(x_{1}, x_{2},...,x_{n}) = p(x_{\pi_{1}},x_{\pi_{2}},...,x_{\pi_{n}})$

The swaras in this avatar are identically and independently distributed, and hence their joint distribution is infinitely exchangeable. The same isn’t true in reverse, if the swaras were infinitely exchangeable, they wouldn’t necessarily be identically and independently distributed. Each swara oscillates, with its unique import, carrying equal gravitas and meaning, but it is their collective that is their strength, their joint distribution is invariant to any permutation. According to De Finetti:

$p(x_{1}, x_{2},...,x_{n}) = \int \prod_{j=1}^{n} p(x_{j} | z) P(dz)$

where $P$ is some measure on the conditioned on $z$ . If the distribution $P$ on $z$ has a density, we can rewrite $P$ as $p(z)dz$ .

We can see immediately that $\prod_{j=1}^{n} p(x_{j} | z)$ is invariant to permutation of $n$ and become infinitely exchangeable! And this property of infinite exchangeability provides layers of pristine meaning for those random variables:

1. Just as the kalpanaswaram is based on the foundation of discipline and embodying the spirit of the rāgam that is otherwise hidden, the sequence of random variables, like the swaras are exchangeable because there exists a hidden parameter $z$ .

2 . Just us the singing of the aesthetic emotions laden in the kalpanaswaram must obey the spirit the grammar of the rāgam and not just any haphazard sequence of swaras, so too exists a likelihood $p(x|z)$ of the of the random variables on the parameter $z$ , meaning there is a distribution $P$ on $z$ and a prior.

3 . Just as the mastering the kalpanaswaram is incontrovertibly accompanied by the evolution of the capacity to tap into a hidden reservoir of strength to remain unperturbed by the difficulties of life, so too, is the concept of the latent variable $z$ . The swaras
themselves are observed as the kalpanaswaram is sung, they bring with them what is also not seen, but felt, just as there are observed and latent variables.

Amma, generative models, made of the seen and the unseen, in an effulgent alignment of harmony and profound equanimity!”

“Yes, my child”, beamed Subbulakshmi, extending the palm of her hands and touching my cheeks. “Bruno is just sitting over there!”

II. Anti-Realism and Meditative Priors

“Indeed I am”, said a well-groomed man with a sharp jawline. Was this really Bruno de Finetti? If so what was he doing Śringeri? “Excuse me sir, the only pictures I’ve seen of you are those in which you’re in a suit, I’m flummoxed as to imagine what brings you here to Śringeri”, I asked him bewilderingly.

“You have to learn to understand the paradoxes and embrace them my dear boy”, said de Finetti. “Maybe you thought and still think the probability of me being in a place such as this, so far away from Rome (which is a very new city compared to this ancient site) was infinitesimally small. I was listening to that description of my theorem of infinite exchangeability, and I thought there was a lot there. But going back to the expectation or otherwise, of me being here, you thought the probability of that was rather close to negligible, didn’t you lad?”, asked de Finetti with a strange grin on his face.

“Of course sir, I did. I mean, you’re here wearing a kurta pajama, and I can’t seem to the connection of anything from your world in a place such as this, without qualification or prejudice sir!”, I replied candidly. De Finetti let out a burst of playful laughter, soon to join in chuckling by others in the grand hall.

“My dear boy! You only assigned a close to zero probability to the event of me being here because you are focused on the realism of the conjecture that there exists a cardinal set of truth or true events somewhere. You see probability does not exist on its own, sans the construction of a prior expectation of an observer. Or in other words, it exists only in your mind. Your previous notions of the event inform your anticipation of the posterior, so to speak, so you have exteriorized your projections into what falsely appears as an objective reality. If for instance, you saw that much of my work was inspired by Einstein, Schrodinger, and other physicists, you would have plainly understood that given they were inspired by the teachings of nonduality, I would, by a transitive closure, also be inspired by the same precepts!

If we habitually project our perceptions onto the external world, then the futile pursuit of realism — this relentless focus on an ‘objective’ computation of probability is, by any measure of logic, quite silly. Understand, therefore that just as we do in everyday life, it is essential to understand the subjective nature of probability itself in statistics. The prior distribution is a subjective one, and represents, in a reductionist by a helpful way. The prior distribution is often informed by an assessment of the posterior, and this iterative cycle of refining priors by examining the posterior is a method that honors that precept of subjectivism. Understand, also the impetus this subjective view has on tools for assessment and interpretation. In other words, an estimated posterior that is a first-class object of a subjective, individual exercise cannot be merely analyzed using a breathless and ultimately futile search for an objective assessment function. The assessment and interpretative tools must also be tied to the subjective prior! So, given that you have now seen me here in Śringeri, inside this ancient temple, you have reasonably updated your prior. This might cause you, not to be shocked, if, for instance, you find Mr Schrodinger in a saucy bar in the city of New York!”, laughed de Finetti.

“Do you mean to say that it is important to capture the ‘lens’ of the observant, sir?”, I asked with excitement. “Maybe you mean, sir, to say that this lens does indeed get refined by the observer after her assessment of the posterior! The evaluation of the posterior is rather an exercise in confrontation and examination of the operational worldview that forms that prior, isn’t it, sir?. Maybe this exercise, this lensing of a posterior is what should inform the construction and evaluation of all latent variable models!

Just as it is the worldview of the observer that informs her perception of the world, so too does a prior belief inform the assessment of the posterior
Just as prevalent in the anti-realist teachings of nonduality that the notions of a ‘self’ view of the world appearing as distinct and objective, so too is the tendency to treat probability of an event as objective and distinct, as if it bore no connection to its subjective bearings
Just advaita prescribes a method of challenging that illusory worldview of an objective existence of an event formed in the image of the self as opposed to the Self, so too must the evaluation of the lens or posterior challenge the assumptions encoded in the construction of the prior derived from an individual’s worldview

A flash of a deep smile appeared on de Finetti’s face. “That is a good direction to go in, yes, my dear boy!” he said, exchanging a look of great satisfaction with others in the hall. “My dear boy, refining the lens in the light of new data is deeply advaitic and being able to do this is to perch oneself on the boundaries of closed and open-world reasoning? Such is the value of a meditative prior!”

III. Energy and Model Criticism & Elaboration

“Just a minute, let us not get swept away. Moderation is superior to total abstinence, Bruno!”, began a handsome middle-aged man, seated opposite to de Finetti. “Well I expected you to interject. Nikola, you never cease not to amaze me”, retorted de Finetti, grinning with satistifaction. “Nikola, who?”, I enquired, before I knew who it was. “You don’t happen to be Nikola Tesla, do you?!”.

“Of course my dear boy. There is an ambient cloud of magnificent and pristine energy in this ancient hall. I know you can feel it, just as I can. But if we take Bruno’s argument here, this high energy exists only inside my mind and in yours. Is that conjecture at all plausible, that the experience of this energy precludes its independent existence? Photosynthesis harnesses the tremendous self-luminous and radiant energy of the sun, but can it be said that the heat of the sun cannot exist independently of the plant harnessing its great power? It cannot exist. The sun is primordial and represents the supreme strength of an unimaginable and gargantuan magnitude and exists independent of any existence of a plant here on this planet.

What must be understood, therefore, is while one might accord provisional validity to the objective existence of the sun, the subjective and experiential act of harnessing its light by a plant completes the validation. Of course, the plant can be deluded into thinking that it draws its energy from the wind instead of the light of the sun, and it is always possible to misattribute a subjective experience to a source that does not exist. In such cases, the wind as a source of energy for photosynthesis exists only in a mistaken personal worldview!”

I was positively smitten. “This speaks to the explorations of assumptions, does it not?”, I asked excitedly. “In generative latent variable models, for instance, we make assumptions behind the generative processes that produce the data in front of us. Which set of unobservable and latent variables, aka, which model encompassing these assumptions might have generated the data in front of an observer? Of course, the test would be in model criticism and elaboration would it not? The observer would carefully select a series of discrepancy functions to compare the data in front of her against the data generated by her model. If the scores estimated by the discrepancy functions for both datasets align, then it observer’s assumptions are on a firmer footing, but if they aren’t aligned, then she has to abandon the model and go back to the drawing board and re-examine her assumptions again.

For example, let $y$ represent the observed data and $\theta$ a latent
variable that generatively produces $y$ . We then have the model:

$\theta \rightarrow y$

After we estimate and build a model we generated simulated data
$y^{rep}$ , we the posterior predictive distribution becomes:

$P(y^{rep}|y)= \int p(y^{rep}| \theta, y)p(\theta|y)d\theta$

If $g(x)$ is the observer\’s chosen discrepancy function, then we’re comparing the simulated data $y^{rep}$ to the actual observed data $y$ , or in other words, comparing $g(y^{rep}$ to $g(y)$ .

Just as the observer accords provisional validity to a set of assumptions, so does a statistician accord provisional validity to a set of assumptions about the data generating process itself.
Just as the observer subjects the provisionally valid assumptions made through a phenomenographic and experiential lens, so should a statistician subject her assumptions through the careful selection and construction of a discrepancy function informed by first-hand experience and reasoning about the context behind the dataset.
Just as the observer burnishes all candidate phenomenological theories through the furnace of phenomenographic tests, so should the statistician analyze the roots of all her assumptions, and in the event of failure, be willing to abandon the corresponding model and revamp her assumptions”.

“That is a good attitude and approach!” replied a beaming Tesla. “But how will one know how to choose the discrepancy function? An erroneous discrepancy function selected might lend the observer the insular comfort that her assumptions are indeed correct when they might not be”, he said. I looked around the room to find everyone smiling. A few more oil lamps inside the sanctum sanctorum light mysteriously self-kindled themselves.

IV. Quantum Mechanics and Phenomenography of Latent Structures

By this time, almost two hours had passed, and it was nearly five thirty in the morning, and the chanting of the morning ablutions had become. Young monks began to enter the hall. Some brought flowers and incense, and some seemed to be in a rapt conversation with Subbulakshmi. I looked around and recognized a great many faces. Faces of loved ones. The monks began to sit down on the floor with us.

“It is not enough to check one’s subjective lens through iterative model criticism”, said a voice. The accent was distinctly German. I turned to my right and recognized Erwin Schrödinger. “There are limits to the precision one can assign to whether one set of assumptions can are superior to others. There has to be an embrace of the inherent uncertainty”, he said, his face calm and tranquil.

“But what if the discrepancy functions show you that the underlying latent structure (assumptions) that one encoded is solid, isn’t that contrary to the principles of uncertainty? In that case, why should we be worried about inherent uncertainty”, I asked.

Schrödinger laughed. “But that is conditioned on the proposition that there exists only one underlying latent structure that generates a dataset”, replied Schrödinger, this voice assuming excitement. “Statistical knowledge that asserts that there is only one view of reality can be entirely wrong, young man! The permeating role of the observer cannot be emphasized enough. All observed eigenstates in quantum mechanics, for instance, are values measured at the time and of being observed by the observer. What does that tell you?”, he asked quizzingly.

“Are you saying then that there might be many possible underlying latent structures that can generate an observed dataset?”, I asked with a look of disbelief. “Of course, it is not good enough to ask what the underlying generative latent structure of a phenomenon is. We should rather ask the question ‘what do we want the underlying latent structure to be’, should we not, young man?”. Schrödinger’s teeth were as white as a perfect pearl and his smile as enchanting as a dancing peacock blissfully heralding the monsoon rain.

“So this means that it might be possible to have several latent structures that generate the same data, and the latent structure itself is determined by the observer”, I said. “Perhaps any model criticism exercise must be accompanied by an interpretive operation that attaches immense importance to how the phenomena being modeled is experienced by those closest to it. The context of phenomenography, this open-world reasoning beyond the purview of the observed variables at hand the set of latent structures in one’s mind, must be treated with utmost respect!

Just as the collapsed eigenstate is at the behest of the observer, so too is a Bayesian model tied to the phenomenographic perspective of the observer
Just as quantum mechanics embraces the limits of precise identification of a wave-function, so too must one embrace the limits of any model, however parsimonious it might be. Context and first-hand experience must be accorded the dignity and respect that it deserves
Just as the correspondence principles states that classical physics is a macroscopic approximation of quantum mechanics, the lesson for the Bayesian is to understand that parsimonious models are useful when they are built and used in the service of an inclusive and contextualizing mindset.”

Schrödinger flashed another grin, and Subbulakshmi rubbed the back of my head. By now the hall had filled with many more monks and looked rather full. I exchanged several looks of bewilderment and inexplicable happiness with all the familiar faces of loved ones.

V. Nondual Properties of the Dirichlet

The sanctum santorum was now lit brighter by the minute, and amidst the hustling at the doors and the incomprehensible chatter amongst those seated, I tried to close my eyes in an attempt to process the depth of the dialogue I was blessed to be witnessing. No sooner had I closed my eyes than I felt everyone in the hall standing up. I stood up, realizing that everyone’s gaze was fixated at the main door of the great hall. There was a sudden silence in the room, and the air inside seemed to expand with anticipation. The hall was suddenly so quiet that one could hear a distant chant from what sounded like a group of monks. Only that that chanting seemed to get louder by the second.

A train of chanting monks entered the great hall, nodding smilingly greeting faces and bearing the sanguine visage of a child entering the abode of its mother. Each monk flashed a namaskāram to Subbulakshmi as they formed a row on either side of the entrance of the sanctum santorum.

And then, I saw him. Entering the hall behind the monks was none other than Ādi Shankara, his face radiating with the effulgence of a thousand suns. He reverberated with a most profound frequency. What kind of energy was this, a barefoot and skinny monk with a simple robe, yet cloaked with what seemed like waves of pure and incontrovertible compassion. Shankara greeted Subbulakshmi and then smilingly looked straight into my face with a smile, and it felt as if a thousand oil lamps lit int he deepest recesses of my thought and being.

“So you are sitting on a peacock mat?”, Shankara asked me, and before I could reply, looked at Subbulakshmi and said: “Amma, śaravana bhava” with a broad smile. No words in the Sanskrit or English lexicon can meaningfully describe the depth of the communication that happened between Shankara and me at that instant. Shankara and the entire hall sat down on the floor.

“The spoken word cannot convey the idea of the Absolute, for when it tries to do that, it conjures the image of an object. A rope in a dark room can be mistaken for a snake. Adhyāsa or superimposition of an object of the snake on the rope leads to an erroneous inference that there exists a snake. But a flash of lightning punctures the darkness and brings with it the illuminating knowledge that the object mistaken for a snake is a harmless rope. In much the same way, one might say ‘I am happy’, or ‘I am sad’, or ‘I am’ anything, but this common notion of ‘I’ borne of superimposed empirical experience is incorrect. The ‘I’ or the ‘self’ is a superimposition upon the highest metaphysical reality, the Self. Every living being is endowed with an apriori experience of the Self, but it is common to see its true nature being obscured by superimpositions of the mind. There is no difference or duality between the Self and notions of ‘I’, ay more than there can be a difference between a spider and the web generated by it upon which it sits. You are the Self. You are the rising the and setting sun. Every sentient being is made of the same energy that forms the stars, and every star is made of the same energy present in every sentient being.

For example, consider your favorite distribution, the Dirichlet, the conjugate prior to the multinomial. The probability density function is given by:

$\frac{1}{B(\alpha)} \prod_{i=1}^{k}x_{i}^{\alpha_{i}-1}$

where $B(\alpha) = \frac{ \prod_{i=1}^{K} \Gamma (\alpha_{i})}{\Gamma \sum_{i=1}^{K}\alpha_{i}}$ and hyperparameter $\alpha = (\alpha_{1},...,\alpha_{K})$

The Dirichlet has a nondual aggregative property, almost fractal-like, in the sense that when one takes the sample space and breaks it into smaller chunks, the set of new chunks is also a Dirichlet distribution.

And so if $\{ Z_{1}, Z_{2},...,Z{n} \}$ is a chunked sample space over $\{ Z_{1}, Z_{2},...,Z{n} \}$ , then

$( \sum_{i_{A_{1}}} Q_{i}, \sum_{i_{A_{2}}} Q_{i},..., \sum_{i_{A_{n}}} Q_{i}) \sim Dir(\sum_{i_{A_{1}}} \alpha_{i}, \sum_{i_{A_{2}}} \alpha_{i},..., \sum_{i_{A_{n}}} \alpha_{i})$
“

Shankara stretched his hand and wiped my forehead with his robe, and drew vibhuti on my forehead. He then turned towards Subbulakshmi, “Amma, pālinchu kāmākshi madyamāvathi rāgam, followed by śaravana bhava, madyamāvathi rāgam”.

Subbulakshmi and Radha began to sing, soon joined by the entire the hall. I looked at the sanctum sanctorum and saw that it was now entirely lit. It was impossible to discern whether the sound of the veena came from the Subbulakshmi’s accompanists or from inside the Shārada garbagraha. I joined in the singing, surrounded by my loved ones and watched by Shankara. Each swara and string, reverberating with pristine rhythm and aesthetic melody, seemed to oscillate with numinous and divine energy, settling on every bit of thought and being until all the world dissolved in an ocean of profound bliss.

I. Kalpanaswaram and Infinite Exchangeability

II. Anti-Realism and Meditative Priors

III. Energy and Model Criticism & Elaboration

IV. Quantum Mechanics and Phenomenography of Latent Structures

V. Nondual Properties of the Dirichlet

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