Markov random fields and socionics

Socionics comes across as the more serious and academic eastern European cousin of MBTI (which is much better known by English speakers). Although many of the same criticism that apply to personality theories/tools such as MBTI are also applicable to socionics, I find these models fascinating. Here I’ll use socionics to set up a fun application/demonstration of Markov random fields.

Personality profiles

These personality theories are often discredited as unscientific or useless (with the possible exception of the big five

  • also see IBM’s showcase of an automatic personality profiler that takes English text as input and spits out a profile of the author), but online interest seems to keeps the debate about socionics as a science alive. Ex-Socionist’s blog has more on this topic.

Socionics

Socionics (okay my understanding of one of the sub-understandings of Model A) posits that there are 4 ways that people process information (originally proposed by Jung): (F)eeling, (S)ensing, (T)hinking, and i(N)tuition. Each of these functions are further divided into (i)ntroverted and (e)xtraverted functions - depending on whether they are directed inward of outward.

Elements

We now end up with 8 information elements:

elements = ['Te', 'Fe', 'Se', 'Ne', 'Ti', 'Fi', 'Si', 'Ni']

They can be tidily poured into a 3 dimensional ndarray:

elements = np.array(elements).reshape(2, 2, 2)

# Assign an index to each element
e2i = {element: i for i, element in enumerate(elements.flatten())}
e2ii = {elements[tuple(i)]: i for i in indices.T}

elements
> array([[['Te', 'Fe'],
         ['Se', 'Ne']],

         [['Ti', 'Fi'],
          ['Si', 'Ni']]], 
        dtype='|S2')

Types

The theory further postulates that each person uses all the elements, but that people differ in their preferences and the roles that they assign to each element. For example, a person’s preferred conscious element is called the leading function, while the secondary element is called the creative function because it assists the leading function to produce something new. All eight elements are given roles like these two.

The slots that the elements can take are highly constrained, however. We only need to specify two functions to completely specify a type. So personality types are often specified by their leading and creative functions. We are left with sixteen valid combinations of elements:

indices = np.indices((2, 2, 2)).reshape(3, 8)
types = ['{}{}'.format(elements[tuple(i)], elements[tuple(i - np.array([1, 1, j]))]) 
         for i in indices.T for j in [0, 1]]

# Assign an index to each type
t2i = {typ: index for index, typ in enumerate(types)}
i2t = {index: typ for index, typ in enumerate(types)}

types
> ['TeSi', 'TeNi', 'FeNi', 'FeSi',
   'SeTi', 'SeFi', 'NeFi', 'NeTi',
   'TiSe', 'TiNe', 'FiNe', 'FiSe',
   'SiTe', 'SiFe', 'NiFe', 'NiTe']

Relations

Socionics is also concerned with stereotypical interactions between types. Types have elements that they are strong with and therefore understand in others, but also elements that they are weak with and appreciate in others. There are, however, also weak and strong elements that they don’t care about or are irritated by if strongly expressed by others.

All these different combinations of type interactions are summed up in sixteen relation types:

relation_deltas = [
    ('duality', 'm', (-1, 0, -1), (-1, 0, -1)),
    ('activation', 'r', (-1, 0, -1), (-1, 0, -1)),
    ('semi-duality', 'm', (-1, 0, -1), (-1, 0, 0)),
    ('mirage', 'm', (-1, 0, 0), (-1, 0, -1)),
    ('mirror', 'r', (0, 0, 0,), (0, 0, 0)),
    ('identity', 'm', (0, 0, 0), (0, 0, 0)),
    ('cooperation', 'm', (0, 0, -1), (0, 0, 0)),
    ('congenerity', 'm', (0, 0, 0), (0, 0, -1)),
    ('quasi-identity', 'r', (-1, 0, 0), (-1, 0, 0)),
    ('extinguishment', 'm', (-1, 0, 0), (-1, 0, 0)),
    ('super-ego', 'm', (0, 0, -1), (0, 0, -1)),
    ('conflict', 'r', (0, 0, -1), (0, 0, -1)),
    ('requester', 'r', (-1, 0, -1), (-1, 0, 0)),
    ('request-recipient', 'r', (-1, 0, 0), (-1, 0, -1)),
    ('supervisor', 'r', (0, 0, 0), (0, 0, -1)),
    ('supervisee', 'r', (0, 0, -1), (0, 0, 0))
]
r2i = {rel: i for i, (rel, _, _, _) in enumerate(relation_deltas)}
i2r = {i: rel for i, (rel, _, _, _) in enumerate(relation_deltas)}

We are now ready to try and put some people into their boxes.

Markov social network

Let’s build a model that assumes each person is one of the sixteen types but we’re not sure exactly which one. The model is therefore a joint probability function over all types and relations for all the persons in our network. We’ll represent this model as a Markov random field using pyugm.

Types

For each person there is thus also a distribution over types. This is modelled by a factor for each person that encodes our uncertainty about their type:

# Helper to contruct a potential table over types for a person.
# Potentials for each types are set to one, except those 
#     types that are passed as arguments.
def prob_type(name, *prob_typ):
    type_data = np.ones(16)
    for typ, potential in prob_typ:
        type_data[t2i[typ]] = potential
    type_factor = DiscreteFactor([('{}_type'.format(name), 16)], type_data)
    return type_factor

factors = []
people = ['Lev', 'Dil', 'Uli', 'Yre',
          'Ani', 'Itu', 'Urb', 'Rya',
          'Owa', 'Ako']

factors.append(prob_type('Lev', ('SiFe', 2.0)))
factors.append(prob_type('Dil', ('FiNe', 2.0)))
factors.append(prob_type('Itu', ('TiNe', 2.0), ('NeTi', 2.5)))

Elements

Sometimes we rather want to specify the distribution over elements that someone likely has as leading or creative functions:


# Helper to contruct a factor over the 
#     first two elements for a person.
def prob_has(name, *prob_elements):
    data = np.ones(8)
    for element, potential in prob_elements:
        data[e2i[element]] = potential
    prim_factor = DiscreteFactor([('{}_primary_function'.format(name), 8)], data)
    sec_factor = DiscreteFactor([('{}_secondary_function'.format(name), 8)], data)
    return (prim_factor, sec_factor)

factors.extend(prob_has('Uli', ('Ti', 1.5), ('Se', 1.5),
                               ('Si', 1.3), ('Te', 1.4)))
factors.extend(prob_has('Yre', ('Ne', 1.5)))
factors.extend(prob_has('Ani', ('Fe', 1.5)))
factors.extend(prob_has('Urb', ('Ne', 2.5)))
factors.extend(prob_has('Rya', ('Si', 1.2)))
factors.extend(prob_has('Owa', ('Si', 1.1), ('Ti', 1.2),
                               ('Te', 1.2)))
factors.extend(prob_has('Ako', ('Ni', 1.5), ('Ti', 1.2),
                               ('Se', 1.1), ('Si', 1.1)))

At the moment each person has two factors that are completely unconnected - they don’t share any variables. But we know that the first two elements and a person’s type are deterministically tied. Let’s therefore add another factor for each person to capture the dependence between the function and type variables:

element_type = np.zeros((16, 8, 8))
for typ in types:
    el1 = typ[:2]
    el2 = typ[2:]
    element_type[t2i[typ], e2i[el1], e2i[el2]] = 1
for person in people:
    variables = [('{}_type'.format(person), 16),
                 ('{}_primary_function'.format(person), 8),
                 ('{}_secondary_function'.format(person), 8)]
    factors.append(DiscreteFactor(variables, element_type))

Relations

If we strongly believe that two people’s relationship is of a certain type, then that will constrain the types that each of them can be. Let’s therefore add a random variable relation between all the pairs of persons.

# Helper to construct the relation factors.
def prob_has(name, *prob_elements):
    data = np.ones(8)
    for element, potential in prob_elements:
        data[e2i[element]] = potential
    prim_factor = DiscreteFactor([('{}_primary_function'.format(name), 8)], data)
    sec_factor = DiscreteFactor([('{}_secondary_function'.format(name), 8)],data)
    return (prim_factor, sec_factor)

relations = {
    ('Lev', 'Itu'): [('duality', 1.5), ('activation', 1.5)],
    ('Uli', 'Dil'): [('duality', 1.5), ('activation', 1.5)],
    ('Rya', 'Urb'): [('duality', 1.5), ('activation', 1.5)],
    ('Dil', 'Rya'): [('quasi-identity', 1.5), ('super-ego', 1.5),
                     ('conflict', 1.5)],
    ('Uli', 'Urb'): [('duality', 1.2), ('activation', 1.2),
                     ('semi-duality', 1.2), ('mirage', 1.2),
                     ('congenerity', 1.2)],
    ('Itu', 'Owa'): [('semi-duality', 1.2), ('requester', 1.2)],
    ('Itu', 'Uli'): [('cooperation', 1.2), ('congenerity', 1.2)],
}
for i, name1 in enumerate(people):
    for name2 in people[i + 1:]:
        factors.append(prob_relation(name1, name2, *relations.get((name1, name2), [])))
 

We then capture the dependency between the types of the two persons and the relation between them with a further deterministic factor.

types_relation = np.zeros((16, 16, 16))
for i, typ1 in enumerate(types):
    el1 = typ1[:2]
    el2 = typ1[2:]
    for rel, direction, delta1, delta2 in relation_deltas:
        if direction == 'm':  
            # In some relations, the other type's leading function
            #    is defined by this type's leading function
            el2_1 = elements[tuple(e2ii[el1] + np.array(delta1))]
            el2_2 = elements[tuple(e2ii[el2] + np.array(delta2))]
        else:
            # In others, by this type's creative function
            el2_1 = elements[tuple(e2ii[el2] + np.array(delta1))]
            el2_2 = elements[tuple(e2ii[el1] + np.array(delta2))]
        typ2 = '{}{}'.format(el2_1, el2_2)
        types_relation[t2i[typ1], t2i[typ2], r2i[rel]] = 1.0

for i, name1 in enumerate(people):
    for name2 in people[i + 1:]:
        variables = [('{}_type'.format(name1), 16),
                     ('{}_type'.format(name2), 16),
                     ('{}_{}_relation'.format(name1, name2), 16)]
        factors.append(DiscreteFactor(variables, types_relation))

Inference

Now that the model is specified we can run inference and calculate the marginal probabilities of the different characters’ types.

model = Model(factors)
infer = LoopyBeliefUpdateInference(model)
infer.calibrate()

# Helper to find the highest probability types for a certain person.
def get_top_types(name, inference):
    marginals = inference.get_marginals('{}_type'.format(name))[0].data
    return sorted([(i2t[i], p) 
                   for i, p in enumerate(marginals)], 
                  key=lambda x: -x[1])[:10]
get_top_types('Owa', infer)
> [('SiTe', 0.07492),
   ('TeSi', 0.07291),
   ('NeTi', 0.06628),
   ('NiTe', 0.06628),
   ('SeTi', 0.06628),
   ('TiSe', 0.06628),
   ('TiNe', 0.06628),
   ('TeNi', 0.06628),
   ('SiFe', 0.06075),
   ('FeSi', 0.06075)] 

get_top_types('Itu', infer)
> [('NeTi', 0.13804),
   ('TiNe', 0.11043),
   ('SeFi', 0.05372),
   ('FiSe', 0.05372),
   ('SeTi', 0.05371)
    ... ]

The relations can be found similarly:

def get_top_rels(name1, name2, inference):
    marginals = inference.get_marginals('{}_{}_relation'.format(name1, name2))[0].data
    return sorted([(i2r[i], p) for i, p in enumerate(marginals)], key=lambda x: -x[1])[:10]
get_top_rels('Itu', 'Urb', infer)
> [('mirror', 0.06969),
   ('identity', 0.06966),
   ('congenerity', 0.06586),
   ('supervisee', 0.06586),
    ... ]

Other specifications

A few helper functions like prob_type, prob_has, and prob_relation had to be created to specify the network. In a way we’re building a relational modeling language.

It would be interesting to see how difficult/tedious other relational learning frameworks like Markov logic networks, Probabilistic soft logic, and Relational Markov networks are for the same problem.

Empirical profiles

Social networking sites (and dating sites) have tons of interaction data that can be used to validate models like these. Maybe the most likely latent-variable model with four dimensions corresponds roughly to one of the existing models. Or maybe they’re not even close. Or maybe there isn’t a (smallish) point of diminishing returns when modeling human interactions. It’s going to be interesting to see how these models evolve.

Written on September 16, 2015